Lecture Notes on Smooth Manifolds

A Primer in Differential Geometry (for Undergraduates)

Luca Vitagliano

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First printing, September 2018

Contents

Introduction ...... 7 Notation and Conventions8

I Smooth Manifolds and Smooth Maps

1 Charts, Atlases and Smooth Manifolds ...... 13 1.1 Charts and Atlases 13 1.2 Topological Spaces 21 1.3 Smooth Manifolds 26

2 Smooth Maps and Submanifolds ...... 31 2.1 Smooth Functions 31 2.2 Smooth Maps 35 2.3 Submanifolds 42

3 Tangent Vectors ...... 55 3.1 Tangent Spaces 55 3.2 Tangent Maps 60 3.3 The Tangent Bundle 68

4 Full Rank Smooth Maps ...... 73 4.1 Full Rank Maps 73 4.2 Rank Theorem 76 4.3 Embeddings 81

II Vector Fields and Differential Forms

5 Vector Fields ...... 89 5.1 Vector Fields 89 5.2 Vector Fields and Fields of Vectors 92 5.3 Vector Fields and Smooth Maps 96

6 Flows and Symmetries ...... 103 6.1 Integral Curves of a Vector Field 103 6.2 Flow of a Vector Field 110 6.3 Symmetries and Inﬁnitesimal Symmetries 119

7 Covectors and Differential 1-Forms ...... 127 7.1 Covectors and the Cotangent Bundle 127 7.2 Differential 1-Forms and Fields of Covectors 131 7.3 Differential 1-Forms and Smooth Maps 138

8 Differential Forms and Cartan Calculus ...... 143 8.1 Algebraic Preliminaries: Alternating Forms 143 8.2 Higher Degree Differential Forms 151 8.3 Cartan Calculus 158

Index ...... 173 Introduction

Differential Geometry is one of the big branches of modern Geometry. It studies geometric problems with tools from Calculus, Linear Algebra and Commutative Algebra. Standard topics in Differential n Geometry are smooth curves and surfaces in the standard Euclidean space R . However, nowadays those notions are encompassed by the abstract definition of a smooth manifold. Smooth manifolds are the main objects in Differential Geometry, and these notes provide a short introduction to them and to calculus on them. Historically the definition of a smooth manifold has two main motivations: Defining a coordinate invariant calculus. In undergraduate Analysis courses we learn calcu- n lus on R and its open subsets. The standard Euclidean space is equipped with canonical coordinates t1,...,tn, and we traditionally build calculus in terms of those coordinates. However, it is often useful to change coordinates. For instance, polar coordinates in R3 are better suited to study the motion of a particle moving under the action of a central potential with center located at the origin. We are thus led to the question of how to translate the main constructions in calculus (derivatives, integrals, etc.) from standard coordinates to polar coordinates, or, more generally, any coordinate system. There is an even more fundamental question: does calculus have any coordinate invariant meaning, i.e. is there anything in calculus which is independent of the choice of coordinates? Notice that this question is particularly relevant in view of the fact that coordinates are actually a human artifact: there are no coordinates in nature! For instance, in the physical 3-dimensional space (or, better to say, 4-dimentional space-time) where we live, there are no distinguished coordinates, and using Cartesian coordinates, as we often do, is an arbitrary human choice, with some advantages in some cases, but still a human choice. In some sense, Differential Geometry is what remains of calculus after removing all the dependences on the arbitrary choice of a coordinate system. Defining a calculus on more general spaces. As we already stressed, in undergraduate n Analysis courses, we learn calculus on R and its open subsets. However there are more general spaces than that where it is useful to perform calculus. For instance, when describing the motion of a particle on the surface of the 2-dimensional sphere, it would be useful to be able to compute derivatives of curves and functions on the sphere, independently on how the sphere embeds into R3. Even more generally, there are physical systems whose configuration space (space of positions) cannot be embedded in the standard Euclidean space in any physically meaningful way. The typical 8 example is the rigid body. Let B be a rigid body moving in our 3-dimensional physical Euclidean space E. We choose Cartesian coordinates on E and think of it as R3. In particular, it is equipped with a standard orthonormal frame Rcan. Now let G be the center of mass of B, and let RB be an orthonormal reference frame attached to B and defining the same orientation as Rcan. Then, the positions of B in E are parameterized by the position of G, given by its coordinates, together with the position of RB with respect to Rcan, given by a special orthogonal matrix. Summarizing, the configuration space of B is (isomorphic to) R3 × SO(3), and a natural question is: is it possible to define a calculus on such space (so to study the motion of B)? Actually R3 × SO(3) is an example of a smooth manifold, and calculus on smooth manifolds is a part of Differential Geometry. So Differential Geometry was born to provide a coordinate invariant calculus on one side, and to define a calculus on more general spaces on the other side. Later, mostly for its deep conceptual contents, it became an important field of research with connections to several other branches of Mathematics including: Algebraic Topology, Algebraic Geometry, Mathematical Physics, Partial Differential Equations, etc.

Notation and Conventions If A,B are sets, C ⊆ A, D ⊆ B are subsets, and F : A → B is a map such that F(C) ⊆ D, we will usually denote again by F : C → D the restriction of F to C in the domain, and to D in the codomain. Despite these notation might seem confusing, it is very useful in practice. Notice that indicating explicitly the domain and the codomain in F : A → B, and F : C → D, should actually remove the ambiguity. The identity map of a set A is denoted by idA : A → A, or simply id : A → A. The inclusion of a subset A ⊆ B in a set B is usually denoted by iA : A ,→ B. It is the restriction to A, of the identity map idB : B → B. S A cover of a set A is a family F = {B} of parts such that A = B∈F B. We also say that F covers A. A countable set is a set which is either finite or in bijection with integers. n We denote by R the field of real numbers. Let n be a non-negative integer and let R be the standard n-dimensional Euclidean space. We will usually denote by t1,...,tn the standard n n coordinates on R interpreting them as real valued functions on R . As such, they can be seen as n n 1 n the components of the identity map idRn : R → R . The standard coordinates t ,...,t restrict to n n any open subset U ⊆ R , and we denote with the same symbols t1,...,t the restrictions. When n = 1, we usually denote by t (or x) the only coordinate. When n = 2 (resp. 3), we usually write x,y (resp. x,y,z) instead of t1,t2 (resp. t1,t2,t3). Occasionally, we will deviate from these conventions. n m Let n,m be non-negative integers, and let U ⊆ R and V ⊆ R be open subsets. By definition, a real function f : U → R is C∞ or smooth (at some point P ∈ U) if it possesses partial derivatives of arbitrarily high order (in P). Given smooth functions f ,g : U → R, the sum

f + g : U → R, P 7→ ( f + g)(P) := f (P) + g(P) and the product

f g : U → R, P 7→ ( f g)(P) := f (P)g(P) are smooth functions as well. A map F = (F1,...,Fm) : U → V is C∞ or smooth (at some point i P ∈ U) if all its components F , i = 1,...,m, are smooth (in P). The identity idU : U → U is a smooth map, and the composition of smooth maps is smooth. A smooth map F : U → V is a diffeomorphism if it is additionally invertible, and its inverse F−1 : V → U is smooth as well. In this case F−1 is also a diffeomorphism. The identity is a diffeomorphism and the composition of diffeomorphisms is a diffeomorphism. As both a diffeomorphism and its inverse are continuous, 9 a diffeomorphism F : U → V transforms open subsets into open subsets, i.e. U0 ⊆ U is an open subset if and only if so is F(U0) ⊆ V. n n Let U ⊆ R be an open subset, with standard coordinates t1,...,t , let P ∈ U be a point, and let f : U → R be a smooth function. The partial derivatives of f at P will be denoted

∂ 1 n ∂ ∂ f | 1 n f (t ,...,t ) or simply | f or (P). ∂ti (t ,...,t )=P ∂ti P ∂ti The partial derivative functions will be denoted

∂ f ∂ f (t1,...,tn) or simply . ∂ti ∂ti

d If n = 1, the only derivative will be denoted dt . i i=1,...,n Let M(n,m;R) be the space of n × m real matrices. For a matrix A = (a j) j=1,...,m ∈ M(n,m;R), the upper index i will usually range over the rows, and the lower index j will range over the columns. j j=1,...,m For instance, if B = (bk)k=1,...,q ∈ M(m,q;R) is another matrix, then the product A · B ∈ M(n,q;R) is given by

i=1,...,n m ! i j A · B = ∑ a jbk . j=1 k=1,...,q

Smooth Manifolds and I Smooth Maps

1 Charts, Atlases and Smooth Manifolds 13 1.1 Charts and Atlases 1.2 Topological Spaces 1.3 Smooth Manifolds

2 Smooth Maps and Submanifolds ..... 31 2.1 Smooth Functions 2.2 Smooth Maps 2.3 Submanifolds

3 Tangent Vectors ...... 55 3.1 Tangent Spaces 3.2 Tangent Maps 3.3 The Tangent Bundle

4 Full Rank Smooth Maps ...... 73 4.1 Full Rank Maps 4.2 Rank Theorem 4.3 Embeddings

1. Charts, Atlases and Smooth Manifolds

1.1 Charts and Atlases n Roughly, a smooth manifold is a space which looks like some standard Euclidean space R near by every point. In order to make this definition precise, we have to learn how to put coordinates on a space in such a way that the transition from one coordinate frame to the other is a smooth enough process. We begin defining charts on a set. So, let M be a set, and let n be a non-negative integer. Definition 1.1.1 — Chart. An n-dimensional chart on M is a map ϕ : U → Ub, where X U ⊆ M is a subset, n X Ub ⊆ R is a non-empty open subset, X ϕ is a bijection. A chart will be often denoted by (U,ϕ), and with Ub we always indicate the range of ϕ. Then U is called a coordinate domain, and ϕ the coordinate map. The coordinate map is a vector valued map with components usually denoted x1,...,xn (or y1,...,yn, etc.) and called coordinates on U determined by the chart (U,ϕ). Accordingly, we will often write (U,ϕ = (x1,...,,xn)). A chart on M around a point p ∈ M is a chart (U,ϕ) such that U 3 p. A chart (U,ϕ) around a point p ∈ M is centered at p if Ub = ϕ(U) 3 0 and ϕ(p) = 0. We now make some examples of charts on sets. n n n Example 1.1 — The Standard Chart on R . The identity map id : R → R is an n-dimensional n n n chart on R whose associated coordinates are the standard coordinates t1,...,t . The chart (R ,id) n n is also called the standard chart on R . The standard chart on R is centered at 0.

Example 1.2 Example 1.1 can be slightly generalized considering any n-dimensional real vector space V and any frame R = (e1,...,en) of V. Recall that the coordinate map associated to R is the vector space isomorphism

n 1 n ϕR : V → R , v 7→ ϕR(v) := (x (v),...,x (v))

mapping a vector v to its components (x1(v),...,xn(v)) in the frame R. In their turn, (x1(v),...,xn(v)) n i are implicitly given by v = ∑i=1 x (v)ei. It is clear that (V,ϕR) is an n-dimensional chart on V 14 Chapter 1. Charts, Atlases and Smooth Manifolds

M U p

P !"

ℝ$

Figure 1.1: A chart.

centered at 0, for any frame R.

Example 1.3 — The Stereographic Charts on the Sphere. Consider the (n + 1)-dimensional n+ n+ standard Euclidean space R 1 with standard coordinates t1,...,t 1. The n-dimensional sphere is n n+ the subset S ⊆ R 1 deﬁned by n n+1 S = P ∈ R : kPk = 1 , where k−k is the standard Euclidean norm: for P = (P1,...,Pn+1), q kPk := (P1)2 + ··· + (Pn+1)2. We now deﬁne two charts on Sn: the stereographic charts. To do this, we preliminarily denote by

P+ := (0,...,0,1) and P− = (0,...,0,−1) the intersections of the sphere with the tn+1-axis, and call them the north pole and the south pole respectively. We also denote n n U+ := S r {P+} and U− := S r {P−}. n+1 For any point P ∈ U+ the line through P and P+ intersects the hyperplane t = 0 in a point with 1 n coordinates denoted (X+(P),...,X+(P),0). The map n 1 n ϕ+ : U+ → R , P 7→ ϕ+(P) := (X+(P),...,X+(P)) is called the stereographic projection (from the north). n It is easy to see that (U+,ϕ+) is an n-dimensional chart on S (see Exercise 1.1), called the stereographic chart (from the north). Replacing P+ with P− and U+ with U−, we can deﬁne another n chart (U−,ϕ−) on S : n 1 n ϕ− : U− → R , P 7→ ϕ−(P) := (X−(P),...,X−(P)).

The coordinate map ϕ− is called the stereographic projection from the south, and the chart (U−,ϕ−) is the stereographic chart from the south. Notice that the stereographic projection from the north (U+,ϕ+) is centered at the south pole P−, and, similarly, the stereographic projection from the south (U−,ϕ−) is centered at the north pole P+. 1.1 Charts and Atlases 15

ℝ'!(

P+

)' P

0 j!(#)

Figure 1.2: Stereographic projection from the north.

Exercise 1.1 Consider the stereographic charts

1 n 1 n (U+,ϕ+ = (X+,...,X+)) and (U−,ϕ− = (X−,...,X−))

on the sphere, from Example 1.3. Show that they are indeed charts, and that the stereographic projections are explicitly given by

Pi Pi Xi (P) = and Xi (P) = , for P = (P1,...,Pn+1) ∈ Sn. + 1 − Pn+1 − 1 + Pn+1

Example 1.4 — More Charts on the Sphere. In this example, we define 2(n + 1) more charts n n n on S . We begin considering the standard n-dimensional open disk, i.e. the open subset D ⊆ R defined by n n D := {P ∈ R : kPk < 1}. n Now, for all i = 1,...,n + 1, let Ui,± ⊆ S be the subset defined by 1 n+1 n i Ui,± := (P ,...,P ) ∈ S : ±P > 0 . Consider the orthogonal projection (onto the hyperplane ti = 0)

n+1 n 1 n 1 n 1 i n+1 πi : R → R , (P ,...,P ) 7→ πi(P ,...,P ) := (P ,...,Pb ,...,P ), n where a hat “−b” denotes omission. The projection πi maps Ui,± to D , hence we can consider the restriction: n πi : Ui,± → D .

It is clear that the (Ui,±,πi) are all n-dimensional charts on the sphere. Notice that (Ui,±,πi) is a chart centered at (0,..., ±1 ,...,0). |{z} i-th place

16 Chapter 1. Charts, Atlases and Smooth Manifolds

tn+1 ℝ!"#

P+

0 p!"#(%)

Figure 1.3: Orthogonal projection on the hyperplane tn+1 = 0.

n+1 Example 1.5 — Affine Charts on the Projective Space. On R r {0} with standard coordinates t0,...,tn, consider the equivalence relation ∼ given by P ∼ Q ⇔ there exists a non-zero real number λ such that P = λQ. The n-dimensional (real) projective space is the quotient n n+1 RP := R r {0}/ ∼ n+ n of R 1 r {0} under the equivalence relation ∼. Equivalently, RP is the space of lines through n+ n n+ the origin in R 1. The equivalence class [P] of the point P = (P0,...,P ) ∈ R 1 r {0} will be denoted by [P0 : ··· : Pn], n n and the real numbers P0,...,P are called the homogeneous coordinates of [P] ∈ RP . In this n n example we define n-dimensional charts on RP . First of all, for every i = 0,...,n, let Ui ⊆ RP be the subset defined by 0 n i Ui := [P : ··· : P ] : P 6= 0 . The map ! P0 cPi Pn ϕ : U → n, [P] = [P0 : ··· : Pn] 7→ ϕ ([P]) := ,..., ,..., , i i R i Pi Pi Pi is well-defined (here, as in Example 1.4, “−b” denotes omission). Additionally, it is invertible, with inverse given by −1 n 1 n −1 1 n ϕ : R → Ui, Q = (Q ,...,Q ) 7→ ϕ (Q) := [Q : ··· : 1 : ··· : Q ]. i i |{z} i-th place

Hence, the (Ui,ϕi) are all n-dimensional charts on the n-dimensional projective space. They are n called the afﬁne charts on RP . The i-th afﬁne chart (Ui,ϕi) is centered at [0 : ··· : 1 : ··· : 0]. |{z} i-th place

1.1 Charts and Atlases 17

Given an n-dimensional chart (U,ϕ) on a set M, we can use the coordinate map ϕ to identify the coordinate domain U with Ub := ϕ(U). In turn, we can use this identification to transfer notions in calculus (smoothness, derivatives, etc.) from Ub to U. For instance, we could declare that a real − function f : U → R is smooth at a point p ∈ U if the composition f ◦ ϕ 1 : Ub → R is smooth at ϕ(p). If we want to do the same for the whole M, we need to have, at least, a family of charts S A = {(U,ϕ)} covering M in the sense that M = (U,ϕ)∈A U. Suppose we have such a family A , let f : M → R be a (real valued) function on M, and let p ∈ M. Then, it is natural to declare that f − is smooth at p if the composition f ◦ ϕ 1 : U → R is smooth, for some chart (U,ϕ) ∈ A around p. However, the latter condition may depend on (U,ϕ), unless A satisfies suitable additional conditions. The above discussion motivates the following two definitions. Definition 1.1.2 — Compatible Charts. Two n-dimensional charts (U,ϕ = (x1,...,xn)),(V,ψ = (y1,...,yn)) on a set M are compatible if either X U ∩V = ∅, or X U ∩V 6= ∅ and, additionally, the following two conditions are satisfied: (1) ϕ(U ∩V) ⊆ Ub and ψ(U ∩V) ⊆ Vb are (non-empty) open subsets, and (2) ψ ◦ϕ−1 : ϕ(U ∩V) → ψ(U ∩V) is a diffeomorphism (hence ϕ ◦ψ−1 : ψ(U ∩V) → ϕ(U ∩V) is a diffeomorphism as well). The map ψ ◦ϕ−1 : ϕ(U ∩V) → ψ(U ∩V) is called the transition map between the charts (U,ϕ) and (V,ψ) (or between the coordinates (x1,...,xn) and (y1,...,yn)).

p V U

j y

"# P y ∘ j $% ()

ℝ' ℝ'

Figure 1.4: Compatible charts.

Notice that the compatibility between charts is a reﬂexive and symmetric relation, but it’s not transitive in general (can you provide a counter-example?). Deﬁnition 1.1.3 — Atlas. An n-dimensional smooth atlas (or, simply, an atlas) on a set M is a family A = {(U,ϕ)} of n-dimensional charts such that S X A covers M, i.e. M = (U,ϕ)∈A U, and X any two charts in A are compatible. Before providing examples of atlases we want to stress that there are various ways how to 18 Chapter 1. Charts, Atlases and Smooth Manifolds produce a compatible chart from a given chart.

Example 1.6 — Subcharts. Let M be a set and let (U,ϕ) be a chart on M. Choose any non- empty open subset Vb ⊆ Ub in Ub = ϕ(U) and put V := ϕ−1(Vb). In particular, we can restrict ϕ to obtain a new map ϕ : V → Vb. It is clear that (V,ϕ) is again a chart, called a subchart of (U,ϕ). Additionally, the transition map between (U,ϕ) and (V,ϕ) is simply the identity id : Vb → Vb. Hence any subchart of (U,ϕ) is compatible with (U,ϕ).

Example 1.7 Let M be a set and let (U,ϕ) be an n-dimensional chart on M. Choose any n diffeomorphism Ψ : Ub → Wb onto an(other) open subset Wb ⊆ R . The composition Ψ ◦ ϕ : U → Wb is bijective, hence (U,Ψ ◦ ϕ) is a chart. Additionally, the transition map between (U,ϕ) and

(U,Ψ ◦ ϕ) is precisely Ψ : Ub → Wb . Hence (U,ϕ) and (U,Ψ ◦ ϕ) are compatible.

Exercise 1.2 Let M be a set, let p ∈ M be a point, and let (U,ϕ) be a chart on M around p.

Show that there exists a chart centered at p and compatible with (U,ϕ).

We now provide various examples of atlases. n n n Example 1.8 — The Standard Atlas on R . The standard chart (R ,id) on R forms an atlas n n {(R ,id)} called the standard atlas of R .

Example 1.9 Let V be an n-dimensional real vector space and let R be a frame of V. The chart (V,ϕR) from Example 1.2 forms an atlas {(V,ϕR)}. n n+1 Example 1.10 — Two Atlases on the Sphere. Consider the n-dimensional sphere S ⊆ R . n The stereographic projections (U+,ϕ+)(U−,ϕ−) (Example 1.3) cover S and are compatible (see Exercise 1.3 below). Hence they form an atlas on the sphere, called the stereographic atlas. n Similarly, the charts (Ui,±,πi) from Example 1.4 cover S and are pairwise compatible. Hence they form another atlas {(Ui,±,πi)}i=1,...,n+1 on the sphere. Finally, notice that the charts (Ui,±,πi) are also compatible with the stereographic projections (Exercise 1.3).

n n+ Exercise 1.3 Consider the n-dimensional sphere S ⊆ R 1. Show that (1) the stereographic projections (Example 1.3) form an n-dimensional atlas, (2) the charts (Ui,±,πi) from Example 1.4 form an n-dimensional atlas, (3) the charts (Ui,±,πi) are compatible with the stereographic projections.

Example 1.11 — The Affine Atlas of the Projective Space. Consider the n-dimensional n n projective space RP . The affine charts (Ui,ϕi) (Example 1.5) cover RP and are compatible (see Exercise 1.4 below). Hence they form an atlas, called the affine atlas of the projective space.

n Exercise 1.4 Consider the n-dimensional projective space RP . Show that the afﬁne charts (Example 1.5) form an n-dimensional atlas on it.

We will show in the next chapter that the notion of smoothness of a function f : M → R (at a point p ∈ M) is independent of the choice of a chart (around p) in a ﬁxed atlas. However, it may still depend on the atlas, unless we restrict ourselves to a class of compatible atlases. Deﬁnition 1.1.4 — Compatible Atlases. Two atlases A ,B on a set M are compatible if the charts of A are compatible with the charts of B, or, equivalently, if A ∪ B is an atlas as well.

Example 1.12 Let V be an n-dimensional real vector space and let R,S be frames of V. The charts (V,ϕR) and (V,ϕS ) are compatible. Indeed the transition map is

−1 n n ϕS ◦ ϕR : R → R 1.1 Charts and Atlases 19 which is a vector space isomorphism, hence a diffeomorphism. It follows that the atlases

{(V,ϕR)} and {(V,ϕS )} are compatible.

Example 1.13 According to Exercise 1.3, the stereographic atlas and the atlas {(Ui,±,πi)}i=1,...,n+1 consisting of the orthogonal projections are compatible atlases on the n-dimensional sphere.

Proposition 1.1.1 The compatibility between atlases on a set is an equivalence relation.

Proof. The compatibility between atlases is clearly reﬂexive and symmetric. It remains to prove that it is transitive. So, let A ,B,C be atlases on a set M and suppose that A is compatible with B, and B is compatible with C . We want to show that A is compatible with C . To do this, consider a chart (U,ϕ) in A and a chart (W, χ) in C and show that they are compatible. If U ∩W = ∅ there is nothing to prove. So let U ∩W 6= ∅ and look at the transition map: χ ◦ ϕ−1 : ϕ(U ∩W) → χ(U ∩W). (1.1)

We remark, preliminarily, that (1.1) is invertible. Now, ϕ(U ∩W) ⊆ Ub = ϕ(U) is an open subset. Indeed, let P ∈ ϕ(U ∩W), and let p = ϕ−1(P). As B is an atlas, there is a chart (V,ψ) in B around p. Additionally, (V,ψ) and (U,ϕ) are compatible, meaning that (1) ψ(U ∩V) ⊆ Vb and ϕ(U ∩V) ⊆ Ub are open subsets, and (2) the transition map ϕ ◦ ψ−1 : ψ(U ∩W) → ϕ(U ∩V). is a diffeomorphism. But (V,ψ) and (W, χ) are also compatible. In particular, ψ(V ∩W) ⊆ Vb is an open subset as well. As the intersection of two open subsets is an open subset, it follows that

ψ(U ∩V) ∩ ψ(V ∩W) = ψ(U ∩V ∩W) ⊆ Vb is an open subset. Hence −1 ϕ(U ∩V ∩W) = (ϕ ◦ ψ )(ψ(U ∩V ∩W)) ⊆ Ub is an open subset. More precisely, it is an open neighborhood of P in ϕ(U ∩W). It follows from the arbitrariness of P that ϕ(U ∩W) ⊆ Ub is an open subset (see Figure 1.5). Exchanging the roles of (U,ϕ) and (W, χ), we immediately see that χ(U ∩W) ⊆ Wb is also an open subset. Next, we show that the transition map (1.1) is smooth. It is enough to show that it is smooth around every point, i.e., for every P ∈ ϕ(U ∩W), there exists an open neighborhood U0 of P in ϕ(U ∩W) such that the restriction −1 (χ ◦ ϕ )|U0 : U0 → χ(U ∩W) is smooth. This is indeed the case, e.g. we can choose U0 = ϕ(U ∩V ∩W), where (V,ψ) is as in the ﬁrst part of the proof. Then −1 −1 −1 (χ ◦ ϕ )|U0 = (χ ◦ ψ ) ◦ (ψ ◦ ϕ )|U0 , which is the smooth composition of two smooth maps. Finally, exchanging again the roles of (U,ϕ) and (W, χ), reveals that ϕ ◦ χ−1 is also smooth around every point, hence it is smooth. This concludes the proof. Given an atlas A on a set M, it is often useful to consider all charts on M that are compatible with all charts in A . With this in mind, we give the following 20 Chapter 1. Charts, Atlases and Smooth Manifolds

V M

p W U

j c y

P '( c ∘ y'( !" y ∘ j %" #$

* ℝ* ℝ

ℝ*

Figure 1.5: Compatibility of atlases is transitive.

Deﬁnition 1.1.5 — Smooth Structure. An atlas A on a set M is maximal if it is not properly contained in any other (compatible) atlas, or, equivalently, if it already contains all charts compatible with its charts. A maximal n-dimensional atlas on a set M is also called an n- dimensional smooth structure on M.

Yet in other words, an atlas A is maximal if, whenever a chart (U,ϕ) is compatible with all the charts in A , we have (U,ϕ) ∈ A . For instance, a maximal atlas contains all subcharts of its charts, and all charts obtained composing a chart with a diffeomorphism.

Proposition 1.1.2 Let M be a set. Then (1) every atlas A on M is contained in a unique maximal atlas Amax, necessarily compatible with A , (2) two atlases are contained in the same maximal atlas if and only if they are compatible.

Proof. Left as Exercise 1.5.

Exercise 1.5 Prove Proposition 1.1.2. (Hint: for the ﬁrst part deﬁne

Amax = {charts (U,ϕ) on M : (U,ϕ) is compatible with every chart in A }.

For the second part use Proposition 1.1.1.)

For instance, the charts on a real vector space determined by frames are all in the same maximal atlas. Similarly, the stereographic atlas on the sphere and the atlas of orthogonal projections are contained in the same maximal atlas.

n Exercise 1.6 — The standard smooth structure on R . Consider the standard one-chart atlas n n n {id : R → R } on R . According to Proposition 1.1.2, it is contained in a unique maximal n atlas Acan, called the standard smooth structure on R . Show that Acan consists of charts (U,ϕ), n where U ⊆ R is any open subset, and ϕ : U → Ub is any diffeomorphism between open subsets n of R . 1.2 Topological Spaces 21

An atlas A on a set M determines a distinguished family of subsets of M according to the following Deﬁnition 1.1.6 A subset U ⊆ M is A -open (or, simply, open if this does not lead to confusion) if, for every point p ∈ U , there exists a chart (U,ϕ) ∈ A around p, such that ϕ(U ∩U) ⊆ Ub is an open subset.

Proposition 1.1.3 A subset U ⊆ M is A -open if and only if, for every chart (V,ψ) ∈ A , ψ(U ∩V) ⊆ Vb is an open subset.

Proof. The “if” part of the statement is obvious. Let’s prove the “only if” part. So let U ⊆ M be an A -open subset, and let (V,ψ) be a chart in A . We want to show that ψ(U ∩V) ⊆ Vb is an open subset. If U ∩V = ∅, then ψ(U ∩V) = ∅ ⊆ Vb is an open subset. On the other hand, − let U ∩V 6= ∅, pick a point P ∈ ψ(U ∩V), and let p = ψ 1(P). As U is open, there is a chart (U,ϕ) ∈ A around p such that ϕ(U ∩U) is open. As (V,ψ) and (U,ϕ) are in the same atlas, they are compatible, hence the transition map

ψ ◦ ϕ−1 : ϕ(U ∩V) → ψ(U ∩V)

n is a diffeomorphism between open subsets of R . It follows that

ϕ(U ∩U ∩V) = ϕ(U ∩U) ∩ ϕ(U ∩V) ⊆ Ub

is an open subset, and

ψ(U ∩U ∩V) = (ψ ◦ ϕ−1)(ϕ(U ∩U ∩V)) ⊆ ψ(U ∩V)

is an open subset as well. From the arbitrariness of P, ψ(U ∩V) ⊆ Vb is an open subset.

In the following, we denote by τA the family of A -open subsets in M. The main properties of τA are summarized in the following Proposition 1.1.4 — Topology Determined by a Smooth Structure. Let A be an atlas on a set M, and let τA be the family of A -open subsets of M. First of all, τA does only depend on the smooth structure containing A . In other words, any two compatible atlases A ,B determine the same family of open subsets: τA = τB. Additionally, (1) ∅,M are in τA , S (2) for every subfamily F ⊆ τA , the union U ∈F U is in τA , (3) for every ﬁnitely many U1,...,Uk ∈ τA , the intersection U1 ∩ ··· ∩ Uk is in τA .

Exercise 1.7 Prove Proposition 1.1.4. (Hint: use Proposition 1.1.3. For Properties (1), (2), (3), n use also the analogous properties enjoyed by the open subsets of R ) Properties (1), (2), (3) in Proposition 1.1.4 can be axiomatized to get what is called a topology on a set. A set equipped with a topology is called a topological space, and the branch of Mathematics studying topological spaces is Topology. In the next section we brieﬂy recall those fundamentals of Topology that will be useful in the rest of these notes.

1.2 Topological Spaces Let X be a set. 22 Chapter 1. Charts, Atlases and Smooth Manifolds

Deﬁnition 1.2.1 — Topological Space. A topology on X is a family τ = {U } of subsets of X such that (1) ∅,X ∈ τ, S (2) for every subfamily F ⊆ τ, the union U ∈F U is in τ, (3) for every ﬁnitely many U1,...,Uk ∈ τ, the intersection U1 ∩ ··· ∩ Uk is in τ. A set X equipped with a topology τ is a topological space. The elements of τ are called open subsets. A subset C ⊆ X is closed if its complement X rC is open.

Example 1.14 — Atlas Topology. Let (M,A ) be a set equipped with an atlas. The family τA of A -open subsets in M is a topology, called the atlas topology, which only depends on the smooth structure Amax containing A : τA = τAmax . Accordingly, M is a topological space, whose open subsets are A -open subsets. In what follows we will call them just open subsets if there is no risk of confusion. n Example 1.15 The open subsets in R form a topology denoted τcan and called the standard n n n topology of R . On the other hand the standard atlas A = {id : R → R } induces its own atlas n topology τA . It is easy to see that, actually, τA = τcan. Indeed a subset U ⊆ R is A -open if n n id(U ∩ R ) is open, i.e. U ∈ τcan. In the following, we will always interpret R as a topological space with the standard topology.

Example 1.16 On any set X there are at least two canonical topologies: X the smallest possible one, consisting of ∅ and X only, called the concrete topology, and denoted τcon, and X the largest possible one, consisting of all subsets in X, called the discrete topology, and denoted τdis. A topological space (X,τ) is concrete if τ = τcon, and it is discrete if τ = τdis. Notice that a topological space (X,τ) is discrete if and only if every singleton {p} ⊆ X, with p ∈ X, is an open subset. It immediately follows from this remark that a 0-dimensional atlas induces the discrete topology.

R Let (X,τ) be a topological space, and let σ be the family of closed subsets of X. It is easy to see that (1) ∅,X ∈ σ, T (2) for every subfamily G ⊆ σ, the intersection C∈G C is in σ, (3) for every ﬁnitely many C1,...,Ck ∈ σ, the union C1 ∪ ··· ∪Ck is in σ. Moreover a family σ of subsets of a set X enjoys Properties (1), (2), (3), if and only if it is the family of closed subset with respect to some (necessarily unique) topology on X.

Every subset Y ⊆ X in a topological space X is a topological space in a canonical way. Namely, deﬁne

τY := {Y ∩ U : U ⊆ X is an open subset}.

Then τY is a topology on Y, called the subspace topology. The subset Y ⊆ X equipped with the subspace topology is called a subspace. Notice that X if Y itself is open in X, then its subspace topology consists of open subsets of X contained in Y, and, similarly, X if Y is closed, then closed subsets with respect to the subspace topology are closed subsets of X contained in Y. n n n Example 1.17 Z ⊆ R is a closed and discrete subspace. Every ﬁnite subspace in R is closed and discrete. 1.2 Topological Spaces 23

n+1 n Example 1.18 Being a subset in R , the n-dimensional sphere S is a subspace. Denote by τSn the subspace topology. On the other hand, the stereographic atlas A , or, equivalently, the atlas of orthogonal projections, induces its own atlas topology τA . We will show in the next section (see Example 1.21) that, actually, τSn = τA . The notion of topology, allows us to talk about continuity. Before giving the main deﬁnition, we m n recall that, given open subsets U ⊆ R and V ⊆ R , the notion of continuous map F : U → V can be expressed purely in terms of the (subspace) topologies of U and V . Namely, a map F : U → V is continuous if and only if, for every open subset V ⊆ V , the preimage F−1(V) ⊆ U is an open subset. This suggests the following Deﬁnition 1.2.2 — Continuous Map. A map F : X → Y between topological spaces is continuous if, for every open subset V ⊆ Y, the preimage F−1(V) ⊆ X is an open subset in X.

Let (X,τX ) and (Y,τY ) be topological spaces and let F : X → Y be a map. We write F : (X,τX ) → Y, or F : X → (Y,τY ), or F : (X,τX ) → (Y,τY ), if we want to insist that X and Y are topological spaces with the topologies τX and τY respectively (and not different ones). For instance, every map F : X → (Y,τcon) to a concrete space is continuous. Every map F : (X,τdis) → Y from a discrete space is continuous. The inclusion iA : A ,→ X of a subspace −1 A ⊆ X is also continuous. Indeed, for any subset B ⊆ X, the preimage iY (B) is simply A ∩ B, so, if −1 B is open, iY (B) is open (with respect to the subspace topology). For any topological space (X,τ), the identity map id : (X,τ) → (X,τ) is continuous. However, if the domain and the codomain are equipped with different topologies, say τ and τ0, then id : (X,τ) → (X,τ0) needs not to be continuous. For instance, if X possesses more than one element, then the concrete topology τcon and the discrete topology τdis on X are different, and, while the map id : (X,τdis) → (X,τcon) is continuous, the map id : (X,τcon) → (X,τdis) is not. Finally, it is easy to see that the composition of continuous maps is continuous. It follows that the restriction F : A → Y of a continuous map F : X → Y to a subspace A ⊆ X is also continuous. Indeed, it is the composition of the continuous map iA : A ,→ X followed by F : X → Y.

Example 1.19 Let G : X → Y be a (not necessarily continuous) map between topological spaces, let B ⊆ Y be a subspace, and suppose that G takes values in B. In particular, we can consider the restriction G : X → B of G : X → Y to B in the codomain, and G : X → B is continuous if and only if so is G : X → Y. Indeed, let G : X → B be continuous, then G : X → Y is continuous as well, because it is the composition of the continuous map G : X → B followed by the inclusion iB : B ,→ Y. Conversely, let G : X → Y be continuous, and let U ⊆ B be an open subset. To avoid confusion, denote GB : X → B the restriction. By deﬁnition of subspace topology, there is an open −1 subset U ⊆ Y such that U = U ∩ B = iB (U ). Hence −1 −1 −1 −1 −1 GB (U) = GB (iB (U )) = (iB ◦ GB) (U ) = G (U ) which is open because G : X → Y is continuous.

Deﬁnition 1.2.3 — Homeomorphism. A map Φ : X → Y between topological spaces is a homeomorphism if it is continuous, bijective, and Φ−1 is also continuous. Two topological spaces X,Y are homeomorphic if there is a homeomorphism Φ : X → Y connecting them.

Notice that, as the example provided by the maps id : (X,τdis) → (X,τcon) and id : (X,τcon) → (X,τdis) shows, a continuous, bijective map needs not to be a homeomorphism, hence we cannot remove the last condition from the deﬁnition. On the other hand, the inverse of a homeomorphism is clearly always a homeomorphism. The identity id : (X,τ) → (X,τ) is a homeomorphism and the composition Ψ ◦ Φ : X → Z of homeomorphisms Φ : X → Y and Ψ : Y → Z is a homeomorphism whose inverse homeomorphism is Φ−1 ◦ Ψ−1 : Z → X. Finally, diffeomorphisms between open n subsets of R are, in particular, homeomorphisms. 24 Chapter 1. Charts, Atlases and Smooth Manifolds

Let Φ : X → Y be a homeomorphism between topological spaces. It immediately follows from the definition that a subset U ⊆ X is open if and only if Φ(U ) ⊆ Y is an open subset. In other words, the homeomorphism Φ identify both X and Y and their topologies. So, homeomorphic topological spaces should be regarded as “the same topological space”. The notion of topology is very general, and a generic topological space may be rather wild. For this reason, one usually considers topological spaces with particularly nice properties only. In this section we define Hausdorff and II-countable topological spaces. Before explaining these notions we stress that we will not use them much in these notes, however, we briefly discuss them here for completeness, especially because they are technically unavoidable to prove important, more advanced properties of smooth manifolds. We begin with the Hausdorff property. Let X be a topological space, and let p ∈ X be a point. First of all, an open neighborhood of p is an open subset U ⊆ X containing p. Now, X is a Hausdorff space if any two different points p,q ∈ X are separated by open subsets, i.e. there exist an open n neighborhood U of p, and an open neighborhood V of q such that U ∩V = ∅. For instance, R is a Hausdorff space, and any discrete space is Hausdorff. Every subspace of a Hausdorff space is Hausdorff (see Exercise 1.8), so the n-dimensional sphere, with the subspace topology, is a Hausdorff space.

Exercise 1.8 Show that every subspace of a Hausdorff space is Hausdorff.

A Hausdorff space enjoys the nice property that its one-point subspaces are closed. Indeed let X be a Hausdorff space, and let p ∈ X be a point. We want to show that X r {p} is open. To do this take q ∈ X r {p}, i.e. q 6= p. From the Hausdorff property, there is, in particular, an open neighborhood Vq of q not containing p, i.e. Vq ⊆ X r {p}. Hence [ X r {p} = Vq q∈Xr{p} is open. Not all topological spaces are Hausdorff. For instance, a concrete space with more than one element is not Hausdorff. We now come to II-countability. First we need to discuss the new notion of basis for a topology. Let (X,τ) be a topological space. A basis for the topology τ is a subfamily B ⊆ τ such that every open subset can be written as a union of elements of B. For instance, the open disks form a basis n for the standard topology of R . The open disks with rational centers and radii do also form a basis. Finally, multi-rectangles

n I1 × ··· × In ∈ R , with I1,...,In ⊆ R open intervals, do also form a basis (and we can even restrict to multi-rectangles with rational vertices). Notice that any basis for a discrete topology should contain all one-point subsets. Let (X,τ) be a topological space, and let B be a basis for τ. It immediately follows from the deﬁnition, that B is an open cover of X, i.e. it consists of open subsets covering X: X = ∪U ∈BU . Additionally, for all U ,V ∈ B, the intersection U ∩V is a union of open subsets in B. Conversely, if X is a set, and B is a family of subsets covering X and such that for all U ,V ∈ B, the intersection U ∩ V is a union of subsets in B, then B is a basis for a, necessarily unique, topology τB. Speciﬁcally, τB consists of subsets of the form [ U U ∈F for some subfamily F ⊆ B. The proof of this fact is easy and it is left as Exercise 1.9. 1.2 Topological Spaces 25

Exercise 1.9 Let X be a set, and let B be a family of subsets covering X and such that for all U ,V ∈ B, the intersection U ∩ V is a union of subsets in B. Show that S τB := { U ∈F U : F ⊆ B}

is a topology and B is a basis for it.

Given a topological space (X,τ), it is often useful to have a basis B for the topology τ. For instance a map F :Y → X from another topological space Y is continuous provided only F−1(B) ⊆Y is open for any B ∈ B (show it as an exercise!). A topological space (X,τ) is II-countable if there is a countable basis for the topology τ. For n instance, R is II-countable (open disks with rational centers and radii are countably many), and any concrete space is II-countable. Every subspace of a II-countable space is II-countable (see Exercise 1.10), so the n-dimensional sphere, with the subspace topology, is II-countable.

Exercise 1.10 Show that every subspace of a II-countable space is II-countable. (Hint: let (X,τ) be a topological space and let Y ⊆ X be a subspace. Denote by τY the subspace topology of Y. Show, preliminarily, that, given a basis B for τ, the family of subsets BY := {B ∩Y : B ∈ B} is

a basis for τY . Now, choose B to be countable.)

Not all topological spaces are II-countable. For instance, a discrete space with uncountably many points is not II-countable. We conclude this section discussing products of topological spaces. Let X1,...,Xk be topological spaces. In the Cartesian product X1 × ··· × Xk consider the following family of subsets

B := {U1 × ··· × Uk : Ui ⊆ Xi is an open subset for all i = 1,...,k}.

It is easy to see that B enjoys the properties of the basis, i.e. the properties stated in Exercise 1.9 (show it as an exercise). It follows that B is a basis for a topology on X1 × ··· × Xk called the product topology. The Cartesian product X1 × ··· × Xk equipped with the product topology is called the product space. Notice that, if, for all i, Bi is a basis for the topology of Xi, then

× B := {B1 × ··· × Bk : Bi ∈ Bi for all i = 1,...,k} (1.2) is a basis for the product topology (can you show it?). The projections onto the factors:

pri : X1 × ··· × Xk → Xi, (p1,..., pk) 7→ pri(p1,..., pk) := pi, i = 1,...,k are continuous map. Indeed, if U ⊆ Xi is an open subset, then

−1 pr (U ) = X1 × ··· × U × ··· × Xk i |{z} i-th place which is an open subset in the product. n Example 1.20 — The Standard Topology on R is the Product Topology. Let n,n1,...,nk be non-negative integers such that n1 + ··· + nk = n. The map

n n n 1 n n−n +1 n 1 n +···+n Φ : R 1 × ··· × R k → R , ((P ,...,P 1 ),...,(P k ,...,P )) 7→ (P ,...,P 1 k ) is a homeomorphism. Indeed, ﬁrst remember that the open multi-rectangles form a basis for the k topology of R (for all k). Hence products of open multi-rectangles form a basis for the topology n n n of R 1 × ··· × R k . Now, the preimage with respect to Φ of an open multi-rectangle in R is the 26 Chapter 1. Charts, Atlases and Smooth Manifolds

product of open multi-rectangles, showing that Φ is continuous. Additionally, Φ is clearly invertible, and the preimage of a product of multi-rectangles with respect to Φ−1 is a multi-rectangle, showing that Φ−1 is continuous as well. In the following, we will often use Φ to identify the topological n1 nk n spaces R × ··· × R and R . The product of Hausdorff spaces is Hausdorff, and the product of II-countable spaces is II- countable. Indeed, let X1,...,Xk be Hausdorff spaces. Take two different points p = (p1,..., pk) and q = (q1,...,qk) in the product space X1 × ··· × Xk. This means that there exists i ∈ {1,...,k} such that pi 6= qi. As Xi is Hausdorff, pi,qi are separated by open subsets Ui,Vi. But then p,q are −1 −1 separated by the open subsets pri (U1),pri (Vi). For the II-countability, it is enough to notice × that, if B1,...,Bk are countable bases of X1,...,Xk then B (see (1.2)) is a countable basis of X1 × ··· × Xk.

1.3 Smooth Manifolds We begin characterizing the atlas topology of a set equipped with an atlas.

Proposition 1.3.1 — Properties of the Atlas Topology. Let (M,A ) be a set equipped with an n-dimensional atlas. The atlas topology τA induced on M by A is the unique topology such that, for every chart (U,ϕ) ∈ A , (1) the coordinate domain U ⊆ M is open, and n (2) the coordinate map ϕ : U → Ub ⊆ R is a homeomorphism. Here both U and Ub are equipped with the subspace topology.

Proof. First of all, we show that the atlas topology τA possesses both Properties (1) and (2) in the statement. Property (1) is clear. For Property (2), we have to show that both ϕ : U → Ub and ϕ−1 : Ub → U are continuous. Equivalently, we have to show that for every open subsets V ⊆ U and Vb ⊆ Ub, both ϕ−1(Vb) and ϕ(V) = (ϕ−1)−1(V) are open. So let Vb ⊆ Ub be an open subset, and consider ϕ−1(Vb) ⊆ U. We have ϕ(ϕ−1(Vb) ∩U) = ϕ(ϕ−1(Vb)) = Vb, which is open. This shows −1 that ϕ (Vb) ∈ τA . So it is an open subset of M contained in the open subspace U, hence it is open in the subspace U. Now, let V ⊆ U be an open subset. Being an open subset in an open subspace, it belongs to τA . In particular, ϕ(V ∩U) = ϕ(V) ⊆ Ub is an open subset. Now we prove uniqueness. Let τ,τ0 be topologies on M with both Properties (1) and (2), and 0 prove that τ = τ . First, for any chart (U,ϕ) ∈ A , denote by τU the subspace topology on U 0 0 induced by τ, and by τU the subspace topology induced by τ . Take an element U ∈ τ. From Property (1), U ∈ τ for every (U,ϕ) ∈ A , hence U ∩ U ∈ τ. So U ∩ U is an open subset of (M,τ) contained in the open subspace U. It follows that U ∩ U ∈ τU . As ϕ : (U,τU ) → Ub is a 0 homeomorphism, ϕ(U ∩ U ) ⊆ Ub is an open subset. But ϕ : (U,τU ) → Ub is a homeomorphism as 0 0 well, hence U ∩ U ∈ τU ⊆ τ . Finally, [ U = U ∩ U (U,ϕ)∈A is the open union of open subsets of (M,τ0). This shows that τ ⊆ τ0. Exchanging the role of τ and 0 0 τ we see that τ ⊆ τ, and this concludes the proof.

Example 1.21 — The Atlas Topology on the Sphere is the Subspace Topology. Consider n n+ the n-dimensional sphere S ⊆ R 1, and its stereographic atlas A . In this example, we use Proposition 1.3.1 to show that the subspace topology τSn and the atlas topology τA coincide. According to Proposition 1.3.1, it is enough to show that τSn possesses Properties (1) and (2) in the statement. So, consider the stereographic charts (U+,ϕ+),(U−,ϕ−). The coordinate domain U+ is n n n U+ = S r {P+} = S ∩ (R r {P+}). 1.3 Smooth Manifolds 27

n n As the second factor is open in R , it follows that U+ is open in the subspace topology of S , and n similarly for U−. For Property (2), notice that ϕ+ : U+ → R is the restriction to the subspace U+ of the continuous map:

n+1 n+1 n 1 n+1 1 n F : R r {t = 1} → R , P = (P ,...,P ) 7→ F(P) = (F (P),...,F (P)) where Pi Fi(P) = , i = 1,...,n. 1 − Pn+1

Hence, ϕ+ is continuous with respect to the subspace topology. It is easy to see that its inverse −1 n n+1 ϕ+ : R → U+ is the restriction to the subspace U+ ⊆ R in the codomain of the continuous map

n n+1 1 n 1 n+1 G : R → R , Q = (Q ,...,Q ) 7→ G(Q) = (G (Q),...,G (Q)), where 2Qi Gi(Q) = , for i = 1,...,n, kQk2 + 1 and kQk2 − 1 Gn+1(Q) = . kQk2 + 1

−1 n Hence, from Example 1.19, ϕ+ : R → U+ is also continuous with respect to the subspace n topology. We conclude that ϕ+ : U+ → R is a homeomorphism with respect to the subspace topology. Similarly for ϕ−. Summarizing, τSn shares both Properties (1) and (2) in the statement n of Proposition 1.3.1 and coincides with τA . It immediately follows that (S ,τA ) is a Hausdorff,

II-countable topological space. Having a smooth structure A on a set M is enough to define a differential calculus on M. However, it is often useful in practice to restrict to the case when the atlas topology τA is particularly nice. Specifically, it is usually required to (M,τA ) to be a Hausdorff and II-countable topological space. Next definition introduces the key object of this chapter and these lecture notes. Definition 1.3.1 — Smooth Manifold. An n-dimensional smooth manifold is a set M equipped with an n-dimensional smooth structure A = Amax such that (M,τA ) is a Hausdorff and II- countable topological space. The non-negative integer n is also called the dimension of M and denoted dimM.

Example 1.22 — Standard Euclidean Spaces and Spheres are Manifolds. The standard n Euclidean space R and the n-dimensional sphere are n-dimensional smooth manifolds. Let (M,A ) be a set equipped with a smooth structure. From now on, by a chart (resp. an atlas) on M we will always mean a chart in A (resp. an atlas contained in A ) unless otherwise stated.

R For every open subset U ⊆ M and every p ∈ U , there is a chart (U,ϕ) around p such that U ⊆ U . In other words, the coordinate domains of a maximal atlas A form a basis for the 0 0 topology τA . Indeed, let (U ,ϕ) be any chart around p, and put U := U ∩ U . Then U is an open subset contained in U0 and (U,ϕ) is a subchart such that U ⊆ U . But, as maximal atlases contain all subcharts of its charts, we have (U,ϕ) ∈ A .

There are simple criteria to check whether (M,A ) is a smooth manifold or not, i.e. whether (M,τA ) is Hausdorff and II-countable or not. 28 Chapter 1. Charts, Atlases and Smooth Manifolds

Proposition 1.3.2 Let (M,A ) be a set equipped with a smooth structure. Then X (M,τA ) is Hausdorff if and only if for any two points p,q ∈ M either p,q belong to the same coordinate domain, or they belong to disjoint coordinated domains; X (M,τA ) is II-countable if and only if it possesses a countable atlas.

Proof. Let p,q be distinct points of M. If M is Hausdorff, then p,q are separated by open subsets U ,V . As coordinate domains form a basis for the topology of M, then p,q are also separated by coordinate domains. Conversely, let p,q belong to disjoint coordinate domains U,V, then U,V ⊆ M are open subsets separating them. On the other hand, if p,q belong to the coordinate domain U of the same chart (U,ϕ), then ϕ(p),ϕ(q) ∈ Ub are distinct point. As Ub is Hausdorff, there are open subsets V,W ⊆ Ub separating them. Hence ϕ−1(V),ϕ−1(W) are open subsets of U (hence of M) separating p,q. For the second part of the statement, let B ⊆ τA be a countable basis. Denote by BA ⊆ B the subfamily consisting of open subsets B ∈ B with the property that there exists (U,ϕ) ∈ A such that B ⊆ U. As the coordinate domains cover M, and every coordinate domain is covered by open subsets in BA , it follows that BA covers M. Additionally, being an open subset in a coordinate domain, every B ∈ BA can be seen itself as the coordinate domain of a (sub)chart (B,ϕ). Choose just one such chart (B,ϕB) for every B and notice that {(B,ϕB) : B ∈ BA } is a countable atlas on M. Conversely, let A0 ⊆ A be a countable atlas. For every (U,ϕ) ∈ A0, let FU be the family of open subsets of U of the form ϕ−1(D), where D ⊆ Ub is an open disk with rational center and radius. It is clear that

[ FU

(U,ϕ)∈A0 is a countable basis for τA0 = τA (can you prove it in details?).

Example 1.23 Let V be an n-dimensional real vector space V. We know that all the charts (V,ϕR) determined by frames R of V belong to the same maximal atlas Amax. Then (V,Amax) is an n-dimensional smooth manifold.

Example 1.24 — Real Projective Spaces are Manifolds. The n-dimensional real projective n space RP equipped with its afﬁne atlas is an n-dimensional smooth manifold. Indeed, let [P¯] = n n n [P¯0 : ··· : P¯ ],[Q¯] = [Q¯0 : ··· : Q¯ ] be distinct points of RP . Then either they belong to the same coordinated domain or they don’t belong simultaneously to any of the afﬁne coordinate domain, in other words, whenever Pi 6= 0 we must have Qi = 0 and vice-versa. In this second case, without loosing much generality, we assume that P¯0,...,P¯k−1 6= 0, P¯k = ··· = P¯n = 0 for some 0 < k ≤ n. Then we have Q¯0 = ··· = Q¯k−1 = 0 and Q¯k,...,Q¯n 6= 0, and we can even choose P¯0 = Q¯k = 1. n Now, let δ,ε be positive real numbers, and, in R , consider the following open subsets: 0 1 n k−1 ¯k−1 Ub0 := {(A ,...,A ) : |A − P | < δ} and

0 1 n 1 Ubk := {(B ,...,B ) : |B | < ε},

0 −1 0 0 −1 0 0 0 and their preimages U0 = ϕ0 (Ub0) ⊆ U0, and Uk = ϕk (Ubk) ⊆ Uk. It is clear that (U0,ϕ0),(Uk,ϕk) ¯ ¯ 0 0 are (sub)charts around [P],[Q]. It remains to show that δ,ε can be chosen so that U0 ∩Uk = ∅. To 0 0 do this recall that a point [P] ∈ U0 ∩Uk has homogeneous coordinates of the form

(1,A1,...,An), with |Ak−1 − P¯k−1| < δ, (1.3) 1.3 Smooth Manifolds 29 but it also has homogeneous coordinates of the form

(B1,..., 1 ,...,Bn) with |B1| < ε. (1.4) |{z} k-th place

It follows that B1 6= 0 and 1 Ak−1 = . (1.5) B1 We conclude noticing that (1.3), (1.4) and (1.5) cannot be simultaneously satisﬁed if we choose 1 n o > max |P¯k−1 − δ|,|P¯k−1 + δ| . ε

II-countability is clear.

0 1 1 − �−1 � �̅ � [ ] ℝ

Figure 1.6: Choosing ε,δ in an appropriate� way.

Finally, we show that products of smooth manifolds are smooth manifolds, and open subsets in a smooth manifold are smooth manifolds in a natural way. This will enrich signiﬁcantly our list of examples. Let M1,...,Mk be smooth manifolds of dimension n1,...,nk and let A1,...,Ak be their smooth structures. Put n := n1 + ··· + nk. First of all, we can construct an n-dimensional atlas on the product M1 ×···×Mk as follows. For every i = 1,...,k, let (Ui,ϕi) ∈ Ai be a chart on Mi. Consider the product U1 × ··· ×Uk ⊆ M1 × ··· × Mk and the map

n ϕ1 × ··· × ϕk : U1 × ··· ×Uk → Ub1 × ··· ×Ubk ⊆ R , (p1,..., pk) 7→ (ϕ1(p1),...,ϕk(pk)).

It is clear that (U1 ×···×Uk,ϕ1 ×···×ϕk) is a chart on M1 ×···×Mk. Every such chart is called a product chart. It is easy to see that the coordinate domains of product charts cover M1 ×···×Mk and every two product charts are compatible (see Exercise 1.11 below). Hence product charts form an × n-dimensional atlas on M1 × ··· × Mk, called the product atlas, and denoted A . Additionally, the atlas topology induced on M1 × ··· × Mk by the product atlas coincides with the product topology.

Exercise 1.11 Prove that product charts on M1 ×···×Mk form an atlas, the product atlas. Prove also that the atlas topology induced by the product atlas coincides with the product topology. (Hint: prove preliminarily that if ϕi : Xi → Yi are homeomorphisms between topological spaces, i = 1,...,k, then the map

ϕ1 × ··· × ϕk : X1 × ··· × Xk → Y1 × ··· ×Yk, (p1,..., pk) 7→ (ϕ1(p1),...,ϕk(pk))

is a homeomorphism between the product spaces. Finally, use this fact and Proposition 1.3.1 to

prove the claim.)

It follows that (M1 × ··· × Mk,τA × ) is a Hausdorff and II-countable topological space. Hence M1 × ··· × Mk is an n-dimensional manifold. 30 Chapter 1. Charts, Atlases and Smooth Manifolds

Example 1.25 — The Torus. The n-dimensional manifold

T n := S1 × ··· × S1 | {z } n times is called the n-dimensional torus. We now come to open subsets in a manifold. So, let M be a manifold, let A be its smooth structure, and let U ⊆ M be an open subset. Every chart (U,ϕ) ∈ A such that U ⊆ U is clearly a chart on U . Additionally, the coordinate domains of such charts cover U . In other words

AU := {(U,ϕ) ∈ A : U ⊆ U } is an atlas on U . It is almost immediate to see that the atlas topology induced by AU coincides with the subspace topology (do you see it?). Hence with the atlas topology, U is Hausdorff and II-countable. We conclude that U is a manifold of the same dimension as M. Every such manifold is called an open submanifold of M.

Example 1.26 — The General Linear Group. Consider the vector space M(n,R) of n × n real matrices. The general linear group GL(n,R) consists, by deﬁnition, of invertible matrices, or, equivalently, non-zero determinant matrices. As the determinant map

det : M(n,R) → R is continuous (even polynomial), and R r {0} ⊆ R is an open subset, it follows that GL(n,R) = −1 det (R r {0}) ⊆ M(n,R) is an open subset, hence an open submanifold of M(n,R). 2. Smooth Maps and Submanifolds

In this chapter, we show how the smooth structure on a smooth manifold allows us to define the notion of smoothness. In particular, we will define smooth real valued functions on a manifold, smooth maps between manifolds, and smooth subsets of a manifold, called submanifolds. These notions are the first building blocks of a differential calculus on manifolds, in particular a coordinate- free calculus.

2.1 Smooth Functions Let M be an n-dimensional manifold. Deﬁnition 2.1.1 — Smooth Function. A smooth function on M is a real valued function f : M → R such that, for every p ∈ M, there exists a chart (U,ϕ) on M such that the real valued − function f ◦ ϕ 1 : Ub → R is smooth.

n Given a real valued function f : M → R, a point p ∈ M, and a chart (U,ϕ = (x1,...,x )) on M − around p, the function f ◦ϕ 1 : Ub → R is called a coordinate representation of f around p, and will be denoted by fbU , or simply fb, if this does not lead to confusion. So Deﬁnition 2.1.1 can be restated saying that a function f : M → R is smooth if it has a smooth coordinate representation around n every point. We will sometimes consider the restriction f : U → R, and write f = f (x1,...,x ) to express that, restricted to U, f can be seen as a function of the coordinates x1,...,xn.

Proposition 2.1.1 A function M → R is smooth if and only if all its coordinate representations are smooth.

Proof. The “if part” of the statement is obvious. Let’s prove the “only if part”. So let f : M → R be a smooth function, and let (V,ψ) be any chart on M. We want to show that the coordinate −1 representation fbV = f ◦ψ : Vb → R is smooth. It is enough to show that fbV is smooth around every point. To do this, take Q ∈ Vb, and let p = ψ−1(Q). As f is smooth, there is a chart (U,ϕ) around p − such that the coordinate representation fb= f ◦ ϕ 1 : Ub → R is smooth. Hence, so is the restriction fb: ϕ(U ∩V) → R of fbto the open subset ϕ(U ∩V). Consider the open subset ψ(U ∩V) ⊆ Vb and 32 Chapter 2. Smooth Maps and Submanifolds notice that the restriction fbV : ψ(V ∩U) → R is given by the following composition (see Figure 2.1) −1 fbV = fb◦ (ϕ ◦ ψ ). −1 As both the transition map ϕ ◦ ψ and the coordinate representation fb are smooth, so is fbV : ψ(V ∩U) → R. From the arbitrariness of Q, the coordinate representation fbV : Vb → R is smooth as well.

p V U

j y

j ∘ y"# % $ ) Q ( f +, ' - ℝ ' +, ℝ

ℝ 0

Figure 2.1: All coordinate representations of a smooth function are smooth.

n Notice that a real function f : U → R on an open subset U of R is smooth in the sense of Deﬁnition 2.1.1 if and only if it is smooth in the standard sense. All constant functions

const : M → R have constant coordinate representations, hence they are smooth.

Exercise 2.1 Let Q(t0,...,tn),R(t0,...,tn) be homogeneous polynomials of the same degree in the real undeterminates t0,...,tn. Assume, additionally, that R(t0,...,tn) does only vanish at the origin (e.g., this is the case if R(t0,...,tn) = (t0)2 + ··· + (tn)2). Prove that the real valued function Q(P0,...,Pn) f : Pn → , [P] = [P0 : ··· : Pn] 7→ f ([P]) := R R R(P0,...,Pn)

is well-deﬁned and smooth.

It is clear that a function f : M → R is smooth if and only if it is locally smooth, i.e., for every point p ∈ M, there exists an open neighborhood U ⊆ M of p such that the restriction f |U : U → R of f to U is smooth, where we interpret U as a manifold with the open submanifold structure. The claim easily follows from the fact that the charts in M whose coordinate domain is contained in U form an atlas on U (check the details as an exercise!). Otherwise stated, it holds the following 2.1 Smooth Functions 33

Lemma 2.1.2 — Gluing Lemma. Let M be a manifold, and let C = {U } be an open cover S of M (i.e. every elements U ∈ C is an open subset U ⊆ M, and M = U ∈C U ). For every U ∈ C , let fU : U → R be a smooth function. If

fV |U ∩V = fU |U ∩V

for every U ,V ∈ C , then there exists a unique smooth function f : M → R such that

f |U = fU

for all U ∈ C . There are a lot of smooth functions on a manifold, as the following theorem shows.

Theorem 2.1.3 — Bump Function Theorem. Let M be a smooth manifold. For every point p ∈ M, and every open neighborhood U ⊆ M of p, there exists a smooth function β : M → R on M such that β = 1 around p (i.e. there is an open neighborhood V ⊆ M of p such that β|V = 1), and the support of β is contained in U .

The statement of Theorem 2.1.3 needs some explanations. First of all, let X be a topological space, and let A ⊆ X be a subset. The closure of A in X is the smallest closed subset A ⊆ X containing A. Equivalently, A is the (closed) intersection of all closed subsets of X containing A. Given a real valued function f : X → R, the support of f is the closure supp f := {p ∈ X : f (p) 6= 0}.

Notice that, if f is a continuous function, then the subset {p ∈ X : f (p) 6= 0} is open (while the support of f is always closed by definition). We omit a full proof of Theorem 2.1.3. However, we stress that the Hausdorff property of M is a key ingredient of the proof (and the theorem does not hold true in general for a set M with a smooth structure A such that (M,τA ) is not Hausdorff). A smooth function β like in the statement of Theorem 2.1.3 is called a bump function (relative to the data (p,U )). Notice that, as a real valued function on M vanishes outside its support, given a bump function relative to the data (p,U ), in particular we have f |MrU = 0. As already mentioned, we will not prove the Bump Function n Theorem in full generality. However, in Example 2.2 we show that it is true when M = R . The general case can be proved from this one (via charts, the Hausdorff property of manifolds, and some little more Topology). Before doing this we present some algebraic structures on the space of smooth functions on a manifold. We begin defining real algebras. Definition 2.1.2 — Algebra. A real, associative, commutative algebra with unit, or simply an R-algebra, or just an algebra, is a real vector space (A,+,·) equipped with an additional interior operation ? : A × A → A such that X ? is R-bilinear, X ? is associative. In particular, (A,+,?) is a ring. Additionally, we require that the ring (A,+,?) is commutative and possesses a unit. A subalgebra of an algebra A is a vector subspace B ⊆ A which is additionally a subring containing 1.

Notice that a subalgebra B of an algebra A is also an algebra with the restricted operations. As for rings and vector spaces, the symbols ?,· are usually omitted in products, and we write aα, and αβ instead of a · α and α ? β, for a ∈ R, and α,β elements in an algebra. As a ﬁrst trivial example, notice that R is an algebra in the obvious way. Complex numbers do also form a real algebra. More generally, for any algebra A, the map i : R → A, a 7→ a · 1, is an injection identifying 34 Chapter 2. Smooth Maps and Submanifolds

R with a subalgebra of A. In the following, we will often identify R with its image i(R) under i. This shows that an algebra A can be also seen as a ring extension R ⊆ A.

Example 2.1 — Algebra of Real Valued Functions. Let X be a set. The space F (X,R) of real valued functions f : X → R is an algebra with the following point-wise operations + : ( f ,g) 7→ f + g, ( f + g)(p) := f (p) + g(p), ? : ( f ,g) 7→ f g, ( f g)(p) := f (p)g(p), · : (a, f ) 7→ a f , (a f )(p) := a f (p), where f ,g ∈ F (X,R), and a ∈ R. The zero in F (X,R) is the constant function whose constant value is 0, the 1 in F (X,R) is the constant function whose constant value is 1, and the inclusion i : R ,→ F (X,R) identiﬁes a real number a with the constant function, also denoted a, whose constant value is a itself.

Proposition 2.1.4 — Algebra of Smooth Functions. Let M be a manifold. Then smooth functions on M form a subalgebra of the algebra F (M,R) of all real valued functions f : M → R. In other words, sum, products, and products by real scalars of smooth functions are smooth functions.

Proof. Left as Exercise 2.2.

Exercise 2.2 Prove Proposition 2.1.4. (Hint: show that, given smooth functions f ,g, a real number a, and a chart (U,ϕ), we have the following relations between coordinate representations:

[f + g = fb+ gb, df · g = fb· gb, ac f = a fb.

Finally, use the fact that sum, products, and products by real scalars of smooth functions on Ub

are smooth functions.)

The algebra of smooth functions on a manifold M will be denoted by C∞(M). n Example 2.2 — Bump Functions on R . Now, we prove the Bump Function Theorem in the n special case M = R . From this case, the general case can be proved with a little more Topology (but we will not take this ﬁnal step). We begin considering the real valued function f : R → R given by

0 if t < 0 f (t) := − 1 , t ∈ R. e t if t ≥ 0

By studying the derivatives of f at the origin, one can prove that f is a smooth function. Second, for every two real numbers r,R such that r < R, we deﬁne a smooth function h : R → R by putting f (R −t) h(t) = , t ∈ . f (R −t) + f (t − r) R

Clearly, h(t) = 1 for all t ≤ r, and h(t) = 0 for all t ≥ R. Additionally, 0 < h(t) < 1 for r < t < R. n n The function h is called a cut-off function. Finally, let P0 ∈ R , and let U ⊆ R be any open neighborhood of P0. There are positive real numbers r,R such that r < R, and, additionally, the 2.2 Smooth Maps 35

1 "

0 ℝ

Figure 2.2: The function e−1/t .

1 ℎ

0 ℝ # $

Figure 2.3: Cut-off function.

closed disks Dr,DR centered at P0 with radii r,R are contained in U . It follows that the smooth function

n β : R → R, P 7→ β(P) := h(kP − P0k)

is a bump function relative to the data (P0,U ). Indeed β|Dr = 1, in particular, f = 1 in an open neighborhood of P0, and suppβ = DR ⊆ U .

2.2 Smooth Maps We can compare two smooth manifolds via smooth maps between them. Let M and N be smooth manifolds, with dimM = m and dimN = n. Deﬁnition 2.2.1 — Smooth Map. A smooth map between M and N is a map F : M → N with the following property: for every p ∈ M, there exist a chart (U,ϕ) on M around p and a chart (V,ψ) on N around F(p), such that (1) F(U) ⊆ V, and (2) the composition ψ ◦ F ◦ ϕ−1 : Ub → Vb is smooth. Given a map F : M → N, a point p ∈ M, a chart (U,ϕ) on M around p, and a chart (V,ψ) on N around F(p) such that F(U) ⊆ V, the composition ψ ◦ F ◦ ϕ−1 : Ub → Vb is called a coordinate representation of F around p, and will be denoted by FbU,V , or simply Fb if this does not lead to confusion. If two such charts (U,ϕ), (V,ψ) exist for every p ∈ M, we say that F has coordinate representations (around every point). So Deﬁnition 2.2.1 can be restated saying that a map F : M → N is smooth if it has a smooth coordinate representation around every point. Let F : M → N be a map having coordinate representations, and let (U,ϕ = (x1,...,xm)) be a chart on 36 Chapter 2. Smooth Maps and Submanifolds

$ &' %

ℝ" U

Figure 2.4: Bump function on n. R

M F N

p V ((,) U

j %& y & P ( !" (&(*)

$ ℝ ℝ'

Figure 2.5: A smooth map between manifolds.

M and (V,ψ = (y1,...,yn)) a chart on N such that F(U) ⊆ V. In this situation, we will sometimes consider the vector valued map ψ ◦ F : U → Vb. Its components will be denoted F1,...,Fn: ψ ◦ F = (F1,...,Fn) and we will write Fa = Fa(x1,...,xm), a = 1,...,n, to express the fact that a n a a the F can be seen as functions of the coordinates x1,...,x . Notice that F = y ◦ F : U → R for all a = 1,...,n, and their coordinate representations Fba with respect to the chart (U,ϕ) are exactly the components of the coordinate representation Fb: Fb = (Fb1,...,Fbn). Accordingly, F is smooth if a and only if it has coordinate representations FbU,V around every point and, additionally, the F are smooth for every (U,ϕ),(V,ψ) as above.

Proposition 2.2.1 A map F : M → N is smooth if and only if it has a coordinate representation around every point p ∈ M, and all its coordinate representations are smooth.

Proof. Left as Exercise 2.3. 2.2 Smooth Maps 37

Exercise 2.3 Prove Proposition 2.2.1.

We now provide a list of examples of smooth maps. A map F : U → V between open subsets m n of standard Euclidean spaces U ⊆ R and V ⊆ R is smooth in the sense of Deﬁnition 2.2.1 if and only if it is smooth in the standard sense. All constant maps const : M → N have constant coordinate representations, hence they are smooth. A map f : M → R is smooth if and only if it is a smooth function in the sense of Deﬁnition 2.1.1. Finally, the identity id : M → M is a smooth map. More generally, the inclusion iU : U ,→ M of an open submanifold U ⊆ M has coordinate representations, and they can be chosen to be simply the identity (choose in U and in M the same chart). Hence iU is a smooth map. n Example 2.3 Consider the n-dimensional projective space RP , and let

n+1 n π : R r {0} → RP , P 7→ π(P) := [P]

0 n be the canonical projection. Then π is a smooth map. To see this, ﬁx a point P0 = (P0 ,...,P0 ) ∈ n+1 i i n+1 R r{0}. Then there exists i ∈ {0,...,n} such that P0 6= 0. Denote Vi := {t 6= 0} ⊆ R r{0}. n+1 n+1 It is an open subset and (Vi,id) is a subchart of the chart (R r {0},id) on R r {0} around P0. n Additionally, π(Vi) ⊆ Ui, the coordinate domain of the i-th afﬁne chart (Ui,ϕi) on RP . This shows that π has coordinate representations around every point P0. Now, we compute

−1 n πb = ϕi ◦ π ◦ id = ϕi ◦ π : Vbi = Vi → R .

0 n For every P = (P ,...,P ) ∈ Vi we have ! P0 cPi Pn π(P) = ϕ (π(P)) = ϕ ([P]) = ,..., ,..., . b i i Pi Pi Pi

So πb is a smooth map for all i (it has smooth components). We conclude that π is a smooth map.

n n+1 Exercise 2.4 Prove that the inclusion iSn : S ,→ R of the n-dimensional sphere in the (n + 1)-dimensional standard Euclidean space is a smooth map.

Exercise 2.5 Let m,n be non-negative integers, and let Q0(t0,...,tm),...,Qn(t0,...,tm) be homogeneous polynomials in the undeterminates t0,...,tm, all of the same degree, and such that they do only vanish simultaneously at the origin. Prove that the map

m n 0 n F : RP → RP , [P] 7→ F([P]) := [Q (P) : ··· : Q (P)]

is well-deﬁned and smooth.

Example 2.4 — Projections onto Factors are Smooth. Let M1,...,Mk be smooth manifolds of dimensions n1,...,nk respectively. Consider the product manifold M1 ×···×Mk. The projection onto the i-th factor:

pri : M1 × ··· × Mk → Mi, (p1,..., pk) 7→ pri(p1,..., pk) := pi is a smooth map for every i = {1,...,k}. Indeed, let (p¯1,..., p¯k) ∈ M1 × ··· × Mk, and, for every j = {1,...,k}, let (Uj,ϕ j) be a chart on Mj aroundp ¯ j. Consider the product chart

(U1 × ··· ×Uk,ϕ1 × ··· × ϕk) 38 Chapter 2. Smooth Maps and Submanifolds

Clearly pri(U1 × ··· ×Uk) ⊆ Ui, showing that pri has coordinate representation

prb i : Ub1 × ··· ×Ubk → Ubi. 1 n j Additionally, denote by (t( j),...,t( j)) the standard coordinates on Ubj. Then prb i is given by

1 n1 1 nk 1 ni prb i(t(1),...,t(1),...,t(k),...,t(k)) = (t(i),...,t(i)), which is smooth. The proof of the following lemma should be clear (is it?).

Lemma 2.2.2 — Gluing Lemma for Smooth Maps. Let M be a manifold, and let C = {U } be an open cover of M. For every U let FU : U → N be a smooth map. If

FU |U ∩V = FV |U ∩V

for every U ,V ∈ C , then there exists a unique smooth map F : M → N such that

F|U = FU

for all U ∈ C . Let M,N be manifolds, and let F : M → N be a map. In order to check whether or not F is smooth we have to check, ﬁrst of all, whether or not F has coordinate representations. Notice that continuous maps F : M → N do always have coordinate representations around every point. Indeed, let p ∈ M, and let (V,ψ) be a chart on N around F(p). The preimage F−1(V) ⊆ M is an open neighborhood of p. Hence there is a chart (U,ϕ) around p such that U ⊆ F−1(V) (the coordinate domains of charts in the maximal atlas form a basis for the topology). It follows that F(U) ⊆ V. Actually, a map F : M → N is smooth if and only if it is continuous, and its coordinate representations (they exist by continuity) are smooth. The only non-obvious thing in this claim is that a smooth map is continuous. This is the content of the next

Proposition 2.2.3 Smooth maps are continuous.

Proof. Let F : M → N be a smooth map between manifolds, and let V ⊆ N be an open subset. We want to show that the preimage F−1(V ) ⊆ M is an open subset. So, let p ∈ F−1(V ) be a point, and let (U,ϕ) be a chart on M around p, and (V,ψ) a chart on N around F(p) such that F(U) ⊆ V (they exist because F is smooth). Consider also the coordinate representation −1 Fb = ψ ◦ F ◦ ϕ : Ub → Vb. As p is arbitrary, it is enough to show that ϕ(U ∩ F−1(V )) ⊆ Ub is an open subset. But −1 −1 ϕ(U ∩ F (V )) = Fb (ψ(V ∩ V )). (do you see it? If no, prove it as Exercise 2.6, see also Figure 2.6). As V ⊆ N is an open subset, ψ(V ∩ V ) ⊆ Vb is an open subset, and since Fb is smooth, hence continuous, Fb−1(ψ(V ∩ V )) ⊆ Ub −1 is an open subset as well. So ϕ(U ∩ F (V )) is an open subset and this concludes the proof.

Exercise 2.6 Complete the proof of Proposition 2.2.3 showing that for any smooth map F : M → N, any chart (U,ϕ) on M, any chart (V,ψ) on N such that F(U) ⊆ V, and any open subset V (intersecting V) we have

−1 −1 ϕ(U ∩ F (V )) = Fb (ψ(V ∩ V )). 2.2 Smooth Maps 39

M N

F -1(V ) F V p V ((,) U

j %& y & P ( !" (&(*)

$ ℝ ℝ'

Figure 2.6: Smooth maps are continuous.

Proposition 2.2.4 The composition of smooth maps is smooth.

Proof. Let F : M → N, and G : N → Q be smooth maps between manifolds. We want to show that G ◦ F : M → Q is a smooth map. So take a point p ∈ M. As G is smooth, there is a chart (V,ψ) on N around F(p), and a chart (W, χ) on Q around G(F(p)) such that G(V) ⊆ W and the coordinate representation Gb : Vb → Wb is smooth. Since F is continuous, F−1(V) ⊆ M is an open neighborhood of p. Hence there is a chart (U,ϕ) on M around p such that U ⊆ F−1(V), i.e. F(U) ⊆ V. Denote by F : U → V the coordinate representation of F with respect to (U,ϕ) and (V,ψ). All coordinate b b b representations of a smooth map are smooth, so Fb is smooth. Collecting all previous remarks we get M N Q

F G

−1 p � (�) U V W

�(�) �(�(�))

� P

�෠(�) �෠(�෠(�)) �෠ ෠ �

� � � ℝ �̂ ℝ ෠ ℝ ̂ Figure 2.7: The composition of� smooth maps is smooth. � that Gb◦ Fb : Vb → Wb is a smooth coordinate representation of G ◦ F around p. From the arbitrariness 40 Chapter 2. Smooth Maps and Submanifolds of p, we get that G ◦ F is smooth.

Exercise 2.7 Let M,M1,...,Mk be manifolds. Consider a map

F : M → M1 × ··· × Mk.

The i-th component of F is, by deﬁnition, the composition Fi = pri ◦ F : M → Mi of F followed by the projection pri : M1 × ··· × Mk → Mi onto the i-th factor, i = 1,...,k. Prove that F is

smooth if and only if so are its components.

Let M,N be smooth manifolds. Definition 2.2.2 — Diffeomorphism. A diffeomorphism between M and N is a map Φ : M → N such that (1) Φ is smooth, (2) Φ is invertible, (3) Φ−1 is smooth. The manifolds M and N are diffeomorphic if there is a diffeomorphism Φ : M → N connecting them. n A map Φ : U → V between open subsets of R is a diffeomorphism in the sense of Definition 2.2.2 if and only if it is a diffeomorphism in the standard sense. Notice that Property (3) cannot be dropped from Definition 2.2.2. For instance F : R → R, t 7→ t3 is smooth and invertible, but − / its inverse F 1 : R → R, s 7→ s1 3 is not smooth. Every diffeomorphism is a homeomorphism. Hence diffeomorphic manifolds are also homeomorphic. The identity is a diffeomorphism, the composition of diffeomorphisms is a diffeomorphism, and the inverse of a diffeomorphism is a diffeomorphism. It follows that being diffeomorphic is an equivalence relation. Additionally, diffeomorphisms from a manifold M to itself form a group under composition, called the group of diffeomorphisms of M, and denoted Diffeo(M).

Example 2.5 Consider the homeomorphism

n n n Φ : R 1 × ··· × R k → R of Example 1.20. It is easy to see that it is actually a diffeomorphism.

Example 2.6 The map

R : Sn → Sn, P 7→ −P

−1 is a diffeomorphism with R = R (do you see it?).

Example 2.7 Let M be a smooth manifold, and let k be a positive integer. Consider the product

M×k := M × ··· × M. | {z } k times

For every permutations σ ∈ Sk, the map

×k ×k Φσ : M → M , (p1,..., pk) 7→ Φσ (p1,..., pk) := (pσ(1),..., pσ(k))

−1 is a diffeomorphism with Φσ = Φσ −1 (do you see it?). 2.2 Smooth Maps 41

Exercise 2.8 Consider the correspondence

2P0P1 (P1)2 − (P0)2 Φ : P1 → S1, [P0 : P1] 7→ Φ([P0 : P1]) := , . R (P1)2 + (P0)2 (P1)2 + (P0)2

Prove that it is a well-deﬁned diffeomorphism. (Hint: consider the afﬁne charts (U0,ϕ0),(U1,ϕ1) 1 on the projective line RP and the stereographic charts (U+,ϕ+),(U−,ϕ−) on the circle. Prove that Φ(U0) ⊆ U+ and Φ(U1) ⊆ U−, and, more precisely,

−1 (ϕ+ ◦ ϕ0)([P]) if [P] ∈ U0 Φ([P]) = −1 . (2.1) (ϕ− ◦ ϕ1)([P]) if [P] ∈ U1

Use (2.1) to complete the proof.)

Proposition 2.2.5 Let M be an n-dimensional manifold and let (U,ϕ) be a chart on M. Then the coordinate map

ϕ : U → Ub

is a diffeomorphism (between the open submanifold U of M, and the open submanifold Ub of n the Euclidean space R ).

Proof. Left as Exercise 2.9.

Exercise 2.9 Prove Proposition 2.2.5.

Diffeomorphisms identify the smooth structures in the following sense. Let Φ : M → N be a diffeomorphism. Given a chart (U,ϕ) on M, the pair (Φ(U),ϕ ◦Φ−1) is a chart on N. Additionally, the correspondence (U,ϕ) 7→ (Φ(U),ϕ ◦ Φ−1) is a bijection between the smooth structures on M and N. Accordingly, two diffeomorphic manifolds should be identified. Our next aim is to show that a smooth map F : M → N determines an algebra homomorphism F∗ : C∞(N) → C∞(M). We begin defining algebra homomorphisms. So let A,B be (associative, commutative) R-algebras (with unit). Definition 2.2.3 — Algebra Homomorphism. An algebra homomorphism between A and B is an R-linear map H : A → B such that (1) H preserves the product, i.e. for every α,β ∈ A

H(αβ) = H(α)H(β).

(2) H preserves the unit, i.e.

H(1) = 1.

An algebra isomorphism is an invertible algebra homomorphism.

Notice that an algebra homomorphism is, in particular, a ring homomorphism. The identity id : A → A is an algebra homomorphism. More generally, the inclusion of a subalgebra is an algebra homomorphism. The composition of algebra homomorphisms is an algebra homomorphism and the inverse of an algebra isomorphism is an algebra isomorphism. Finally, the composition of algebra isomorphisms is an algebra isomorphism. 42 Chapter 2. Smooth Maps and Submanifolds

Example 2.8 — Pull-Back of Real Valued Functions. Let X,Y be sets and let F : X → Y be a map. Consider the algebras of real-valued functions F (X,R),F (Y,R). It is easy to see that the map

∗ ∗ F : F (Y,R) → F (X,R), g 7→ F (g) := g ◦ F

is an algebra homomorphism, called the pull-back along F. It is clear that the pull-back along the identity id : X → X is the identity id : F (X,R) → F (X,R). If Z is another set and G : Y → Z is another map. Then

(G ◦ F)∗ = F∗ ◦ G∗.

∗ It follows that the pull-back Φ : F (Y,R) → F (X,R) along a bijection Φ : X → Y is an algebra − ∗ isomorphism with inverse given by the pull-back (Φ 1) : F (X,R) → F (Y,R) along the inverse map Φ−1 : Y → X, i.e.

(Φ∗)−1 = (Φ−1)∗.

We leave to the reader to check all the claims in this example as Exercise 2.10.

Exercise 2.10 Prove all unproved claims in Example 2.8.

Now, let F : M → N be a smooth map between smooth manifolds. It follows from Proposition ∗ 2.2.4 that the pull-back F : F (N,R) → F (M,R) takes smooth functions to smooth functions: F∗(C∞(N)) ⊆ C∞(M). The restriction F∗ : C∞(N) → C∞(M) is also an algebra homomorphism, and we also call it the pull-back along F. Analogous remarks as for maps of sets hold in the case of smooth maps of manifolds: the pull-back along the identity id : M → M is the identity id : C∞(M) → C∞(M). If Q is another smooth manifold and G : N → Q is another smooth map, then (G ◦ F)∗ = F∗ ◦ G∗. Finally, the pull-back Φ∗ : C∞(N) → C∞(M) along a diffeomorphism Φ : M → N is an algebra isomorphism with inverse given by the pull-back (Φ−1)∗ : C∞(M) → C∞(N) along the inverse diffeomorphism Φ−1 : N → M.

R We actually have a much stronger result that we state without proof in the following Theorem 2.2.6 Let M,N be smooth manifolds. (1) A map F : M → N is smooth if and only if the pull-back F∗ takes smooth functions to smooth functions. (2) Every algebra homomorphism H : C∞(N) → C∞(M) is the pull-back along a unique smooth map F : M → N, i.e. for every algebra homomorphism H : C∞(N) → C∞(M) there is a unique smooth map F : M → N such that H = F∗. In other words, the correspondence F 7→ F∗ is a(n identity preserving and composition reversing) bijection between smooth maps M → N and algebra homomorphisms C∞(N) → C∞(M).

Proof. Omitted.

2.3 Submanifolds In this section we present submanifolds. They are particularly nice subsets of a manifold M. They are so nice that they inherit a smooth structure from that of M. We will make a systematic use of 2.3 Submanifolds 43 the Implicit Function Theorem. Therefore, we begin recalling the appropriate version of it. First of all, we deﬁne regular systems. So, let d,n be non-negative integers such that d ≤ n. Consider the n n n standard Euclidean space R with standard coordinates (t1,...,t ), let U ⊆ R be an open subset, and let

1 n−d n−d F = (F ,...,F ) : U → R be a smooth map. The zero locus of F is the closed subset

Z(F) := {P ∈ U : F(P) = 0} ⊆ U .

In other words, Z(F) is the space of solutions of the system

 1 1 n  F (t ,...,t ) = 0  . SF : . .  Fn−d(t1,...,tn) = 0

Denote by M(n − d,n;R) the real vector space of real (n − d) × n matrices and recall that the Jacobian matrix of F is the matrix-valued smooth map

∂Fa a=1,...,n−d JF : U → M(n − d,n;R), P 7→ JF (P) := i (P) . ∂t i=1,...,n Here the index a ranges over the rows, and the index i ranges over the columns. We say that 0 is a regular value of the smooth map F, or that the system SF is regular, if

rankJF (P) = n − d for all P ∈ Z(F).

n n−d Example 2.9 Let F : R → R be a surjective afﬁne map. Then F is of the form

n F(P) = A · P + P0, P ∈ R ,

n−d for some P0 ∈ R , and some (n − d) × n matrix A such that rankA = n − d. Clearly, JF = A constantly. In particular,

SF : {A · P + P0 = 0

is a regular system. This shows that every reduced linear system is regular. d n−d 1 n−d Example 2.10 Let U ⊆ R and V ⊆ R be open subsets, let f = ( f ,..., f ) : U → V be a n−d smooth map, and let F : U ×V → R be the smooth map given by

n−d F : U ×V → R , (Q,R) 7→ F(Q,R) := R − f (Q).

 d+1 1 1 d  t − f (t ,...,t ) = 0  . SF : . ,  tn − f n−d(t1,...,td) = 0 and the zero locus of F is

n do Z(F) = (Q, f (Q)) : Q ∈ U ⊆ R ⊆ U ×V : 44 Chapter 2. Smooth Maps and Submanifolds the graph of the smooth map f . The Jacobian matrix of F is

 1 1  − ∂ f ··· − ∂ f ··· ∂t1 ∂td 1 0  ......  JF =  ...... ,  n−d n−d  − ∂ f ··· − ∂ f ··· ∂t1 ∂td 0 1 n−d whose rank is n−d at all points of U ×R , in particular all points of Z(F). Hence SF is a regular system. n+1 Example 2.11 Let F : R r {0} → R be the smooth map given by 2 n+1 F(P) := kPk − 1, P ∈ R . In other words F(t1,...,tn) = (t1)2 + ··· + (tn+1)2 − 1. The Jacobian matrix of F is the single row 1 n+1 JF = (2t ,...,2t ) whose rank is 0 at the origin and 1 elsewhere. As Z(F) = Sn is the n-dimensional sphere, and 0 ∈/ Sn, we conclude that 0 is a regular value of F, and the system 1 2 n+1 2 SF : (t ) + ··· + (t ) − 1 = 0 is regular. 2 2 Example 2.12 On R with coordinates (x,y) consider the smooth map F : R → R given by F(x,y) := xy. The Jacobian matrix of F is the row

JF = (y,x) whose rank is 0 at the origin and 1 elsewhere. However, the origin is in Z(F), so the system

SF : {xy = 0 is not regular.

Notice that, if rankJF (P0) = n−d for some P0 ∈ U , then there exists an (n−d)×(n−d) minor M of JF which is non-zero at P0. Then, by continuity, M is non-zero in a full open neighborhood V ⊆ U of P0 (in particular, rankJF (P) = n − d for all P ∈ V ). It follows from this discussion that, if SF is a regular system, then, for every P0 ∈ Z(F) there exists an (n − d) × (n − d) minor M of JF which is non zero in an open neighborhood V ⊆ U of P0. Let M be the minor identiﬁed by (all the rows and) the columns

i1,...,in−d 1 n−d i i of JF . In this situation, we denote by (v ,...,v ) the coordinates (t 1 ,...,t n−d ) respectively, and by (u1,...,ud) the remaining coordinates, in their order. So, the n-tuple (u1,...,ud,v1,...,vn−d) differs from (t1,...,tn) only by a permutation. Denote it σ. In the following, we also denote d n−d n−d d u = (u1,...,u ), and v = (v1,...,v ). Given open subsets V ⊆ R and U ⊆ R , we denote by n U ×σ V ⊆ R the open subset deﬁned by n U ×σ V := {P ∈ R : u(P) ∈ U, and v(P) ∈ V}. 2.3 Submanifolds 45

n Proposition 2.3.1 Let U ,V ⊆ R be open subsets and let Φ : V → U be a diffeomorphism. m Additionally, let F : U → R be a smooth map. Then 0 is a regular value of F if and only if it is a regular value of F ◦ Φ.

l Exercise 2.11 Prove Proposition 2.3.1 (Hint: First of all, let W ⊆ R be an open subset and let G : W → U be a smooth map. Use the chain rule for the derivatives of a composition of smooth maps to show that:

JF◦G(Q) = JF (G(Q)) · JG(Q). (2.2)

for all Q ∈ W (see also Chapter 4). Now apply (2.2) to the case F = Φ−1 and G = Φ to show that the Jacobian matrix JΦ of Φ is invertible (the inverse being JΦ−1 ◦ Φ). Finally, apply (2.2) to the case G = Φ.)

We are now ready to state the Implicit Function Theorem.

Theorem 2.3.2 — Implicit Function Theorem. Let d,n be non-negative integers such that n d ≤ n, let U ⊆ R be an open subset, and let

n−d F : U → R

be a smooth map. If 0 is a regular valued of F, then, locally, around every point, Z(F) is the graph of a smooth map, i.e., for every point P0 ∈ Z(F), there are d (1) an open neighborhood U ⊆ R of u(P0), n−d (2) an open neighborhood V ⊆ R of v(P0), and (3) a smooth map f = ( f 1,..., f n−d) : U → V, such that U ×σ V ⊆ U , and, in U ×σ V, the system SF is equivalent to the system

 1 1 1 d  v = f (u ,...,u )  . . , (2.3)  vn−d = f n−d(u1,...,ud)

i.e. Z(F) ∩ U ×σ V is exactly the space of solutions of (2.3) (here (u,v) are the tuples of coordinates deﬁned in the discussion preceding Exercise 2.11).

R In the situation described in the statement of the Implicit Function Theorem we also say that, locally, around the point P0 ∈ Z(F), we can solve the system SF with respect to the variables (v1,...,vn−d). This can be done even globally, if F is, for instance, an afﬁne map (see Example 2.9), and, in this case, the proof of the Implicit Function Theorem boils down to the standard Gauss elimination method.

n+1 Example 2.13 Let F : R r {0} be as in Example 2.11, and let 1 n+1 n P0 = (P0 ,...,P0 ) ∈ Z(F) = S . Then 1 n+1 JF (P0) = (2P0 ,...,2P0 ) 6= 0 i i and there is i ∈ {1,...,n + 1} such that P0 6= 0. Assume for simplicity that P0 > 0. In this case, we can choose v = ti and u = (t1,...,tbi,...,tn+1) (where, as usual, a hat denotes omission). Additionally, we can choose

46 Chapter 2. Smooth Maps and Submanifolds

#$, … , #'

( = /(#) *(,) ($, … , (")' 01

ℝ"

Figure 2.8: The Implicit Function Theorem.

n (1) U ⊆ R the unit open disk, (2) V ⊆ R the open interval (0,+∞), and (3) f (u) = p1 − kuk2, so that ( ) j 2 i U ×σ V = ∑(t ) < 1, and t > 0 j6=i and

1 n n i Z(F) ∩U ×σ V = {P = (P ,...,P ) ∈ S : P > 0} is the space of solution of the system ( q r v = 1 − kuk2 ⇔ ti = 1 − ∑(t j)2 . j6=i

Yet in other words, (locally around P0) we were able to solve the equation (t1)2 + ··· + (tn+1)2 − 1 = 0

i i with respect to t . The case P0 < 0 can be discussed in a similar (and obvious) way (do you see it?).

n n−d Exercise 2.12 Given the open subset U ⊆ R and the smooth map F : U → R , prove that 0 is a regular value of F, and, for every point P0 ∈ Z(F), ﬁnd U,V, f as in the statement of the Implicit Function Theorem, in the following two cases: 3 3 X in R with coordinates (x,y,z), U = R r {x = y = 0}, and F : U → R given by p 2 F(x,y,z) = x2 + y2 − R + z2 − r2,

where r,R are positive real numbers with r < R; 2.3 Submanifolds 47

4 4 4 2 X in R with coordinates (w,x,y,z), U = R , and F : R → R given by F(w,x,y,z) = (w2 − x2 − 1,y2 + z2 − 1),

so that w2 − x2 − 1 = 0 S : . F y2 + z2 − 1 = 0

We finally come to the main definition of this section. Let M be an n-dimensional manifold. Definition 2.3.1 — Submanifold. A d-dimensional submanifold of M is a subset S ⊆ M such that, for every point p0 ∈ S, there is a chart (U,ϕ) on M around p0 such that ϕ(S ∩U) is the n−d solution space of a regular system, i.e. there is a smooth map F : Ub → R such that (1) 0 is a regular value of F, and (2) ϕ(S ∩U) = Z(F).

A submanifold is a smooth subset in the sense that, in local coordinates, it looks like the solution space of a regular system, hence, by the Implicit Function Theorem, as the graph of a smooth map.

Proposition 2.3.3 Let S ⊆ M be a submanifold. Then, for every point p0 ∈ S, and every chart (V,ψ) around p0, there is a subchart (V0,ψ) around p0, such that ψ(S ∩V0) is the solution space of a regular system.

Proof. Let (U,ϕ) be a chart around p0 such that ϕ(S ∩U) is the solution space of a regular system. Put U0 = U ∩V. Then ϕ(S ∩U0) is the solution space of a regular system (deﬁned on the open subset ϕ(U0) ⊆ Ub). We want to show that ψ(S ∩U0) is also the solution space of a regular system. To do this, notice that −1 (1) ϕ(U0) and ψ(U0) are related by the transition map ϕ ◦ ψ : ψ(U0) → ϕ(U0), and −1 (2) the transition map ϕ ◦ ψ identiﬁes ψ(S ∩U0) and ϕ(S ∩U0). Finally, use Proposition 2.3.1.

Example 2.14 A d-dimensional afﬁne subspace A in a vector space V is a d-dimensional submanifold. Indeed, in linear coordinates, A can be always written as the solution space of a reduced linear system (see also Example 2.9).

Example 2.15 — Graph of a Smooth Map. Let F : M → N be a smooth map between manifolds, dimM = n. By deﬁnition, the graph of F is the subset graphF ⊆ M × N in the product manifold deﬁned by

graphF := {(p,F(p)) : p ∈ M}.

Then graphF ⊆ M ×N is an n-dimensional submanifold. Indeed, pick a point p ∈ M, and let (U,ϕ) be a chart on M around p, and (V,ψ) a chart on N around F(p) such that F(U) ⊆ V. Denote by (U ×V,ϕ × ψ) the product chart on M × N. It is easy to see that ϕ × ψ(graphF) is the graph of the coordinate representation Fb : Ub → Vb, hence it is the solution space of a regular system as in

Example 2.10.

Example 2.16 — Spheres are Submanifolds. Example 2.11 shows that the n-dimensional n+1 sphere is an n-dimensional submanifold of R . 3 Example 2.17 In the 3-dimensional projective space RP consider the subset

0 1 2 3 3 0 3 1 2 S := [P : P : P : P ] ∈ RP : P P − P P = 0 . 48 Chapter 2. Smooth Maps and Submanifolds

We want to show that S ⊆ RP3 is a 2-dimensional submanifold. To do this, consider the afﬁne 3 3 charts (Ui,ϕi) on RP , i = 0,...,3, and compute ϕi(S ∩Ui) ⊆ R . For instance, if i = 0,

1 2 3 3 1 2 S ∩U0 = [1 : Q : Q : Q ] : Q − Q Q = 0

3 and ϕ0(S ∩U0) is the solution space of the regular system on R

S : t3 −t1t2 = 0 .

Similarly for i = 1,2,3.

Example 2.18 — “Trivial” Submanifolds. Let M be an n-dimensional manifold. It is easy to see that X 0-dimensional submanifolds of M are exactly discrete subspaces, and X n-dimensional submanifolds are exactly open submanifolds (do you see it?). There are two useful characterizations of submanifolds. We summarize them in the following

Proposition 2.3.4 Let M be an n-dimensional manifold and let S ⊆ M be a subset. The following three conditions are equivalent: (1) S is a d-dimensional submanifold; d (2) for every P0 ∈ S there is a chart (W,ψ) around P0 such that Wb = U ×V, with U ⊆ R and n−d V ⊆ R open subsets, and ψ(S ∩W) is the graph of a smooth map f : U → V; 1 d 1 n−d (3) for every P0 ∈ S there is a chart (W0, χ = (x ,...,x ,y ,...,y )) around P0 such that d n−d Wb0 = U0 ×V0, with U0 ⊆ R and V0 ⊆ R open subsets, and S ∩W0 is the zero locus of a 1 n−d the y , i.e. S ∩W0 = {p ∈ U0 : y (p) = ··· = y (p) = 0} (see Figure 2.10).

Proof.

(1) ⇒ (2) It follows immediately from the Implicit Function Theorem, restricting to suitable subcharts.

d n−d (2) ⇒ (3) First of all, let U ⊆ R be an open subset, and let f : U → R be a smooth map. The map

n−d n−d Φ f : U × R → U × R , (P,Q) 7→ (P,Q − f (P)) is a diffeomorphism with inverse Φ− f . Additionally, Φ f maps the graph of f to the graph of the n−d zero map 0 : U → R . Now, let S, P0, and (W,ψ) be as in the statement. Then, composing with Φ f we get a new chart (W, χ := Φ f ◦ ψ) such that χ(W ∩ S) is the graph of the zero section. We can always restrict to a subchart (W0, χ) around P0 whose coordinate domain is of the form Wb0 = U0 ×V0 (see Figure 2.9), and this concludes the proof.

(3) ⇒ (1) χ(S ∩W0) is given by the vanishing of the last n − d coordinates, and this condition is a regular system. Let M be an n-dimensional manifold and let S ⊆ M be a d-dimensional submanifold in M.

Deﬁnition 2.3.2 A chart (W0, χ) on M around a point p ∈ S like in item (3) of Proposition 2.3.4 is said to be adapted to S.

2.3 Submanifolds 49

ℝ)*( ℝ)*( $% % Φ" Φ"($) graph f V P

( Φ"(+) ( ℝ $%, ℝ U

Figure 2.9: The diffeomorphism Φ f .

Exercise 2.13 Let M,N be manifolds, and let Φ : M → N be a diffeomorphism. Show that a

subset S ⊆ M is a d-dimensional submanifold if and only if so is Φ(S) ⊆ N.

Our next aim is to show that submanifolds in a manifold M inherit from M a manifold structure. This is explained in the next theorem and its proof.

Theorem 2.3.5 — Submanifold Chart Theorem. Let M be an n-dimensional manifold and let S ⊆ M be a d-dimensional submanifold in M. There exists a unique smooth structure AS on S such that (1) (S,AS) is a d-dimensional manifold, (2) the inclusion iS : S ,→ M is a smooth map, and (3) given a map between manifolds F : N → M taking values in S, i.e. F(N) ⊆ S, we have that F : N → M is smooth if and only if F : N → S is smooth.

Proof. We begin with the existence. First of all, we consider two auxiliary smooth maps:

n d 1 n 1 d prd : R → R , P = (P ,...,P ) 7→ prd(P) := (P ,...,P ), and

d n 1 d 1 d ind : R → R , Q = (Q ,...,Q ) 7→ ind(Q) := (Q ,...,Q ,0,...,0).

In the following it will be sometimes useful to keep in mind that X prd maps open subsets to open subsets, and X ind is a homeomorphism onto its image (equipped with the subspace topology) (do you see it?). Now, let p ∈ S and let (U,ϕ) be a chart on M around p adapted to S. According d n−d n to the deﬁnition of adapted chart we have, in particular, Ub = UbS ×V ⊆ R × R = R , for some d n−d open subsets UbS ⊆ R and V ⊆ R . Put US := S ∩U. The composition

ϕS := prd ◦ ϕ : US → UbS is inverted by

−1 ϕ ◦ ind : UbS → US.

Hence (US,ϕS) is a chart on S around p. We will refer to any such chart as a submanifold chart.

50 Chapter 2. Smooth Maps and Submanifolds M

%&$ % U p j ℝ ℝ !" !- P S ℝ$ j-

in$ pr$ '( ℝ$ !"-

Figure 2.10: The construction of a submanifold chart.

Submanifold charts cover S. The next step is proving that they actually form an atlas. We have to prove that any two submanifold charts are compatible. So let (U,ϕ),(V,ψ) be adapted charts on M around p, and let (US,ϕS),(VS,ψS) be the induced submanifold charts. First of all ϕS(US ∩VS) ⊆ UbS is an open subset, indeed ind maps it to ϕ(U ∩V) ∩ UbS × {0} which is open in the subspace UbS × {0} ⊆ Ub. Similarly, ψS(US ∩VS) ⊆ VbS is an open subset. The transition map −1 ψS ◦ ϕS : ϕS(US ∩VS) → ψS(US ∩VS) can be written as the following composition of smooth maps: −1 ψS ◦ ϕS = prd ◦ (ψ ◦ ϕ) ◦ ind (see Figure 2.11), hence it is smooth. Exchanging the role of (U,ϕ) and (V,ψ) shows that the −1 inverse ϕS ◦ψS is also smooth, so that the transition map is a diffeomorphism and (US,ϕS),(VS,ψS) are compatible charts. The atlas {(US,ϕS)} on S consisting of submanifold charts will be referred to as the submanifold atlas, and it is contained in a unique smooth structure denoted AS. In order to prove that (S,AS) is a smooth manifold, we have to show that the atlas topology τAS induced by the maximal atlas AS, or, equivalently, the submanifold atlas, is Hausdorff and II-countable. Notice that it is enough to prove that τAS coincides with the subspace topology τS. To do this we use Proposition 1.3.1. We have to show that for every submanifold chart (US,ϕS), US ⊆ S is an open subset with respect to the subspace topology, which is obvious because US = S ∩U and U ⊆ M is an open subset, and, additionally, that ϕS : US → UbS is a homeomorphism when US ⊆ M is equipped with the subspace topology. For the continuity of ϕS, write ϕS as the following continuous composition of continuous maps:

ϕS = prd ◦ ϕ ◦ iS : US → UbS. −1 −1 For the continuity of the inverse ϕS , notice that ϕS : UbS → US is the restriction to the subspace US ⊆ U in the codomain, of the continuous composition of continuous maps −1 ϕ ◦ ind : UbS → US, 2.3 Submanifolds 51

U M V S

%&$ %&$ ℝ j ℝ " . ! + &0 * . ∘ j ℝ$ ℝ$

in$ pr$

ℝ$ ℝ$ " *+ !) &0 ) .) ∘ j)

Figure 2.11: Submanifold charts are compatible.

−1 hence ϕS is continuous with respect to the subspace topology (see Example 1.19). We conclude that τAS = τS, in particular (S,τAS ) is Hausdorff and II-countable, hence it is a smooth (d-dimensional) manifold.

Now we have to show that the manifold (S,AS) enjoys Properties (2) and (3) in the statement. For the smoothness of iS : S ,→ M notice that, if (US,ϕS) is a submanifold chart induced by an adapted chart (U,ϕ), then iS(US) ⊆ U showing that iS has coordinate representations. Additionally, the coordinate representation biS : UbS → Ub with respect to the charts (US,ϕS) and (U,ϕ) is simply ind which is smooth. So iS is smooth. Finally, take a map between manifolds F : N → M such that F(N) ⊆ S. For the sake of clarity, we denote by FS : N → S the restriction of F to S in the codomain. We have to show that F is smooth if and only if so is FS. If FS is smooth, then F = iS ◦FS is the smooth composition of smooth maps, hence it is smooth. Conversely, let F be smooth. Take a point q ∈ N, let (U,ϕ) be an adapted chart around F(q), and let (US,ϕS) be the induced submanifold chart. As F is a continuous map, F−1(U) ⊆ N is an open subset, hence there is a chart (V,ψ) on N around q such that V ⊆ F−1(U), i.e. F(V) ⊆ U. But F takes values in S, hence FS(V) = F(V) ⊆ U ∩ S = US, showing that FS has coordinate representations. It remains to show that the coordinate representation FbS : Vb → UbS with respect to the charts (V,ψ),(US,ϕS) is smooth. It is easy to see that, actually, FbS = prd ◦Fb, in particular, it is a smooth composition of smooth maps. This concludes the proof of Property (3) for the smooth structure AS.

We conclude the proof of the theorem showing that AS is the only smooth structure with Properties (1), (2) and (3) in the statement. So, suppose A 0,A 00 are smooth structures on S with those properties. We denote by S0 the manifold (S,A 0), and by S00 the manifold (S,A 00). We want to show that S0 = S00, i.e. A 0 = A 00. To do this, it is enough to prove that the identity map id : S → S 0 00 0 induces a diffeomorphism id : S → S (do you see it?). So, denote by iS0 : S ,→ M the inclusion. 0 00 As S has Property (2), iS0 is smooth. Additionally, it takes values in S. As S has Property (3), it 00 follows that the restriction of iS0 to S in the codomain is also smooth. But the latter restriction is 52 Chapter 2. Smooth Maps and Submanifolds

M U 0(2)

N ", '4 F S &'% 0 (") - ℝ ℝ& 0, V "# q 0/ % ! ℝ -, ./ in% pr% 0/ , ℝ% "#,

Figure 2.12: Smoothness of a map taking values in a submanifold. exactly id : S0 → S00. Exchanging the role of S0 and S00 we see that the inverse map id : S00 → S0 is 0 00 also smooth, hence id : S → S is a diffeomorphism as claimed, and this concludes the proof.

Deﬁnition 2.3.3 — Submanifold Atlas. A chart (US,ϕS) on S as in the proof of Theorem 2.3.5 is called a submanifold chart, and the atlas {(US,ϕS)} is a submanifold atlas.

Example 2.19 Let M be a manifold and let U ⊆ M be an open submanifold. In particular U is a submanifold. Actually, the submanifold atlas on U agrees with the atlas AU constructed at the end of the previous chapter. n+1 n n+1 Example 2.20 In R consider the n-dimensional sphere S ⊆ R . We already explained n n+1 that S is an n-dimensional submanifold. So, it inherits a smooth structure ASn from R , and n (S ,ASn ) is an n-dimensional manifold. We want to show that ASn is not a new smooth structure on Sn but it coincides with the smooth structure A containing the stereographic atlas (or, equivalently, the atlas consisting of orthogonal projections). We do this via Theorem 2.3.5. We already know n n+1 n that iSn : (S ,A ) → R is smooth (Exercise 2.4). It remains to show that (S ,A ) has Property n+ n (3) in the statement of Theorem 2.3.5. So, let F : N → R 1 be a smooth map taking values in S . n n We want to show that the restriction FSn : N → S of F to S in the codomain is smooth. We use a n trick: we interpret F as a map with values in R r {0}:

n+1 F : N → R r {0}.

As such it is still smooth. Now, there is a smooth projection P π : n+1 {0} → (Sn,A ), P 7→ π(P) := R r kPk

(show that π is actually smooth as an exercise). Conclude noticing that FSn = π ◦ F, i.e. it is the smooth composition of smooth maps. 2.3 Submanifolds 53

Exercise 2.14 Show that a submanifold T ⊆ S in a submanifold S ⊆ M is a submanifold in M, and that S and M induce the same smooth structure on T.(Hint: use adapted charts on S, and Proposition 2.3.3, to show that there are charts in M where T ⊆ M looks like the solution space

of a regular system.)

We conclude this chapter discussing brieﬂy smooth functions on submanifolds. So let S ⊆ M be a submanifold in a manifold, and, as usual, denote by iS : S ,→ M the (smooth) inclusion. For ∞ every smooth function f ∈ C (M), its restriction f |S to S is a smooth function on S, indeed

∗ f |S = iS( f ) = f ◦ iS :

∞ the pull-back of f along the smooth map iS. However, in general, not all smooth functions g ∈ C (S) are restrictions of smooth functions on M as the following counter-example shows.

Example 2.21 In M = R with coordinate t, consider the open submanifold S = R r {0}. The function 1 : S → t R is a smooth function, but it is not the restriction of a smooth function on R. However, every smooth function g on S is locally the restriction of a smooth function on M in the sense of the following

Lemma 2.3.6 — Local Extension Lemma. Let M be a manifold, and let S ⊆ M be a submani- ∞ fold. For every smooth function g ∈ C (S) and every point p0 ∈ S, there exist ∞ (1) a smooth function ge∈ C (M), and (2) an open neighborhood V ⊆ M of p0, such that ge and g agree on S ∩V, i.e. g|S∩V = ge|S∩V . Proof. The statement is a corollary of the Bump Function Theorem (Theorem 2.1.3). Choose an adapted chart (U,ϕ) around p0, and let (US,ϕS) be the induced submanifold chart. Let gb: UbS → R be the coordinate representation of g with respect to the chart (US,ϕS). The composition

0 ge := gb◦ prd ◦ ϕ : U → R U g 0 ∈ C∞(U) g 0| = g| is a smooth function on , i.e. e . It is easy to see that e US US , however, in general, ge is not the restriction to U of a smooth function on M. We can cure this problem locally, using ∞ a bump function. Se let β ∈ C (M) be a smooth function such that suppβ ⊆ U and β|V = 1 for some open neighborhood V ⊆ U of p. Consider the function ge: M → R deﬁned by β(p)g 0(p) if p ∈ U g(p) := e . e 0 if p ∈ M r suppβ

Clearly, ge is well-deﬁned, and it is smooth by the Gluing Lemma. Finally, for every p in S ∩V, we have

0 ge(p) = β(p)ge (p) = g(p), as claimed.

3. Tangent Vectors

In this chapter we begin developing a differential calculus on manifolds. The “smoothness” of manifolds allows us to “take derivatives”. In particular, we can define tangent vectors and tangent spaces to manifolds. Tangent vectors at some point p of a manifold M can be informally understood as infinitesimal displacements from p. Similarly, the tangent space at p should be understood as the “first order, linear approximation” to M. As such, tangent spaces carry natural structures of vector spaces. We can also define the tangent map at p to a smooth map F: it is the “first order, linear approximation” to F.

3.1 Tangent Spaces In order to motivate Deﬁnition 3.1.2 of tangent vector, we begin with a familiar situation: let M be n a submanifold in the standard Euclidean space R , and consider a parameterized curve in M, i.e. a smooth map γ : I → M from an open interval I ⊆ R to M. We denote by t the standard coordinate n on I, and interpret t as the “time”. As γ takes values in R , we can understand γ as a vector valued 1 n map: γ = (γ ,...,γ ). Its velocity at time t0 ∈ I is the vector d dγ 1 dγ n | γ := (t ),..., (t ) . dt t0 dt 0 dt 0

More importantly, it can be well interpreted as a tangent vector to M at the point p = γ (t0) (see Figure 3.1). Our intuition confirms that every tangent vector to M at p is the velocity of some curve “passing through” p at some time. We would like to extend this picture to a generic manifold M. First we need to define curves in M. It is clear how to do this: Definition 3.1.1 — Smooth Curve. A smooth curve (or, simply, a curve) in M is a smooth map γ : I → M from some open interval I ⊆ R to M. Given a curve γ : I → M, we will usually denote by t the standard coordinate on I and interpret it as the “time”. The next step is defining the velocity of γ at the time t0 ∈ I. However, this is not n as easy as in the case when M is a submanifold in R , because we don’t know (yet) how to take derivatives of a map with values in a manifold. Notice that, however defined, the velocity of γ

56 Chapter 3. Tangent Vectors

ℝ"

n Figure 3.1: Velocity of a curve in a submanifold of R . should measure how fast does the point γ (t) ∈ M changes when t changes in I. In order to go back to a more familiar situation, we pick a “test function” f ∈ C∞(M) and, instead of measuring directly how fast does γ (t) change at the time t0, we measure ﬁrst how fast does the real number f (γ (t)) change at the time t0. This can be easily done taking the derivative at t0 of the real valued function f ◦ γ : I → R: d | f ◦ γ. dt t0 In order to get rid of the dependence on the test function f , we repeat this for every function f ∈ C∞(M). In this way we get a map d γ˙(t ) : C∞(M) → , f 7→ γ˙(t )( f ) := | f ◦ γ. (3.1) 0 R 0 dt t0 This map measures well enough how fast does γ (t) change. Hence we call it the velocity of γ at the time t0 and denote it also by d d dγ | γ (t) or | γ or (t ). dt t=t0 dt t0 dt 0

The main properties of γ˙(t0) are summarized in the next

∞ Exercise 3.1 Show that γ˙(t0) : C (M) → R is R-linear and satisﬁes the following identity: for all f ,g ∈ C∞(M)

γ˙(t0)( f g) = f (γ (t0))γ˙(t0)(g) + g(γ (t0))γ˙(t0)( f ).

We interpret γ˙(t0) as a tangent vector to M at the point γ (t0). Even more, given a manifold M, and a point p ∈ M, we axiomatize the properties of γ˙(t0) stated in Exercise 3.1 and give the following 3.1 Tangent Spaces 57

Deﬁnition 3.1.2 — Tangent Vector. A tangent vector to M at the point p is a linear map

∞ v : C (M) → R

satisfying the following Leibniz rule at the point p: for all f ,g ∈ C∞(M),

v( f g) = f (p)v(g) + g(p)v( f ). (3.2)

The set of all tangent vectors to M at p is called the tangent space to M at p and denoted by TpM.

R Given a tangent vector v to M at the point p, we will always assume p as part of the defining data of v. From this point of view, a tangent vector should be better defined as a pair (p,v) where p ∈ M is a point, and v is as in Definition 3.1.2. Doing this we avoid anomalies like the zero map 0 : C∞(M) → R being a tangent vector at all points. We will explicitly understand tangent vectors as pairs (p,v) when defining the tangent bundle later on in this chapter.

Later on in this chapter, we will actually show that every tangent vector in the sense of Deﬁnition 3.1.2 is the velocity of some curve at some time. We now give another example of a tangent vector besides the velocity of a curve.

Example 3.1 — Coordinate Vectors. Let M be an n-dimensional manifold and let p ∈ M be a point. Fix a chart (U,ϕ = (x1,...,xn)) on M around p, and, for each i = 1,...,n, deﬁne the map

∂ ∞ ∂ ∂ fb |p : C (M) → R, f 7→ |p f := (ϕ(p)), ∂xi ∂xi ∂ti

−1 ∂ where, as usual, fb= f ◦ ϕ : Ub → R is the coordinate representation of f . The map ∂xi |p is called the i-th coordinate vector and it is indeed a tangent vector to M at p. The proof is left as Exercise

3.2.

Exercise 3.2 ∂ Show that the i-th coordinate vector ∂xi |p is a tangent vector to M at p. In the next lemma, we present two properties of tangent vectors following from the Leibniz rule (and the linearity).

Lemma 3.1.1 — Elementary Properties of Tangent Vectors. Let M be a manifold, let p ∈ M be a point, and let v be a tangent vector to M at p, then: ∞ X v(c) = 0 for every constant function c = const ∈ C (M); ∞ X v is a local operator, i.e. v( f ) = v(g) for every two functions f ,g ∈ C (M) coinciding on an open neighborhood of p. Yet in other words, v( f ) does only depend on the values of f around p.

Proof. First of all,

v(1) = v(1 · 1) = 2 · v(1), hence v(1) = 0, where, in the second step, we used the Leibniz rule. Now, let c ∈ C∞(M) be a constant function. Then c identiﬁes with a real number, also denoted by c ∈ R, and c = c · 1, so that

v(c) = v(c · 1) = c · v(1) = 0, as claimed, where, in the second step, we used the R-linearity of v. 58 Chapter 3. Tangent Vectors

For the second part of the statement, let f ,g ∈ C∞(M) be functions that agree around p, i.e. f |U = g|U for some open neighborhood U ⊆ M of p, then h := f − g vanishes around p, i.e. h|U = 0. If we prove that v vanishes on h, we have done by linearity. Indeed, if v(h) = 0, then v( f ) − v(g) = v( f − g) = v(h) = 0.

So it remains to prove that v vanishes on every function h such that h|U = 0. To do this, consider a bump function β ∈ C∞(M) relative to the data (p,U ), i.e. β = 1 around p, and suppβ ⊆ U , and denote η = 1 − β. It follows that h = η · h (do you see it?), so that

v(h) = v(η · h) = η(p)v(h) + h(p)v(η) = 0 because h(p) = η(p) = 0. This concludes the proof.

We conclude this section remarking that the tangent space TpM to a manifold M at a point p ∈ M is a vector space. First notice that, by deﬁnition of tangent vectors, TpM is a subset in the ∞ ∞ ∞ space HomR(C (M),R) of R-linear maps C (M) → R (the dual space of the vector space C (M)): ∞ TpM ⊆ HomR(C (M),R).

An easy check, that we leave as Exercise 3.3, reveals that TpM is actually a vector subspace. In particular, it is a vector space.

Exercise 3.3 Show that the tangent space to M at the point p is a vector subspace in the space ∞ ∞ HomR(C (M),R) dual to the vector space C (M).

Our intuition suggests that the tangent space TpM should share the same dimension as M. This is indeed true, according to the following

Theorem 3.1.2 — Tangent Vector Theorem. Let M be an n-dimensional manifold, let p ∈ M be a point, and let (U,ϕ = (x1,...,xn)) be a chart on M around p. Then the coordinate vectors

∂ ∂ | ,..., | ∂x1 p ∂xn p

form a frame in TpM (called the coordinate frame). In particular, TpM is ﬁnitely generated and dimTpM = dimM = n.

We postpone a detailed proof of the Tangent Vector Theorem to the next section. For now, we n only prove it in the case M = R . The proof relies on the following n n Lemma 3.1.3 — Hadamard Lemma. Let f ∈ C∞(R ) be a smooth function on R , and let 1 n n ∞ n P0 = (P0 ,...,P0 ) ∈ R be a point. Then there exist smooth functions g1,...,gn ∈ C (R ) such that n i i f = f (P0) + ∑(t − P0) · gi, (3.3) i=1 and, additionally,

∂ f g (P ) = (P ), i = 1,...,n. (3.4) i 0 ∂ti 0

n Proof. Let P ∈ R be an arbitrary point, and, for every ε ∈ R, denote 1 n P(ε) = P0 + ε(P − P0) = (P (ε),...,P (ε)). 3.1 Tangent Spaces 59

So, when P = P0, P(ε) = P0 constantly, otherwise, it is the generic point on the line through P and P0. In any case, P(0) = P0, and P(1) = P. Compute

f (P) − f (P0) = f (P(1)) − f (P(0)) Z 1 d = f (P(ε))dε 0 dε n Z 1 ∂ f dPi(ε) = ∑ i (P(ε)) dε i=1 0 ∂t dε n Z 1 i i ∂ f = ∑ P − P0 · i (P(ε))dε. i=1 0 ∂t

∞ n Now, deﬁne gi ∈ C (R ) by putting Z 1 ∂ f gi(P) := i (P(ε))dε. 0 ∂t So n i i f (P) = f (P0) + ∑ P − P0 · gi(P), i=1 and (3.3) follows from the arbitrariness of P. Finally, we check (3.4). Remember that when P = P0, we have P(ε) = P0 constantly, so, for all i ∈ {1,...,n}, Z 1 ∂ f ∂ f Z 1 ∂ f gi(P0) = i (P0)dε = i (P0) dε = i (P0). 0 ∂t ∂t 0 ∂t

n We are now ready to prove the Tangent Vector Theorem in the case M = R . In other words we are proving the following

n Lemma 3.1.4 — Tangent Vector Theorem for R . Consider the n-dimensional standard Eu- n 1 n n clidean space R with standard coordinates (t ,...,t ), and let P0 ∈ R be a point. Then the coordinate vectors ∂ ∂ | ,..., | (3.5) ∂t1 P0 ∂tn P0

n form a frame of TP0 R .

Proof. Notice, preliminarly, that the coordinate vectors (3.5) are exactly the partial derivatives (of n 1 n smooth functions R → R) at P0. Now, let a ,...,a ∈ R be such that the linear combination n i ∂ v := ∑ a i |P0 i=1 ∂t vanishes. In particular, v(t j) = 0 for all j ∈ {1,...,n}, i.e.

n n i ∂ j i j j 0 = ∑ a i |P0 t = ∑ a δi = a i=1 ∂t i=1

j ∂ for all j, where δi is the Kronecker delta. This shows that the ∂ti |P0 are independent. Finally, we n n use the Hadamard Lemma (Lemma 3.1.3) to show that they generate TP0 R . So let v ∈ TP0 R be any 60 Chapter 3. Tangent Vectors

n tangent vector, and let f ∈ C∞(R ) be an arbitrary smooth function. We compute v( f ) by writing f in the Hadamard form (3.3):

n ! i i v( f ) = v f (P0) + ∑(t − P0) · gi (Hadamard Lemma) i=1 n i i = v( f (P0)) + ∑ v (t − P0) · gi) (v is R-linear) i=1 n i i = ∑ v (t − P0) · gi) (v vanishes on constants) i=1 n i i i i = ∑ v(t − P0) · gi(P0) + (P0 − P0) · v(gi) (Leibniz rule at the point P0) i=1 n i ∂ = ∑ v(t ) i |P0 f , (Equation 3.4) i=1 ∂t and, from the arbitrariness of f ,

n i ∂ v = ∑ v(t ) i |P0 i=1 ∂t

is indeed a linear combination of the coordinate vectors.

3.2 Tangent Maps Before proving the Tangent Vector Theorem we need to elaborate a little bit more on tangent vectors. In particular, we have to study how do tangent vectors interact with smooth maps, and deﬁne the tangent map to a smooth map. So, let M,N be smooth manifolds, let F : M → N be a smooth map, and let p ∈ M be a point. Deﬁnition 3.2.1 — Tangent Map. The tangent map to F : M → N at the point p ∈ M is the linear map

∗ dpF : TpM → TF(p)N, v 7→ dpF(v) := v ◦ F . (3.6)

We leave as Exercise 3.4 to prove that the composition of the pull-back F∗ : C∞(N) → C∞(M) followed by a tangent vector v : C∞(M) → R to M at the point p is indeed a tangent vector to N at the point F(p), and that dpF is indeed a linear map.

Exercise 3.4 Prove that the tangent map dpF in (3.6) is well-deﬁned (i.e. dpF(v) ∈ TF(p)N for every v ∈ TpN) and R-linear.

Example 3.2 Let M be a manifold, let γ : I → M be a smooth curve, and let t0 ∈ I. It immediately follows from the deﬁnition of velocity γ˙(t0) of γ at the time t0 (3.1), that γ˙(t0) is the image of the d coordinate tangent vector dt |t0 ∈ Tt0 I under the tangent map dt0 γ : d γ˙(t ) = d γ | 0 t0 dt t0

(do you see it?). 3.2 Tangent Maps 61

Proposition 3.2.1 — Chain Rule. Let M,N,Q be smooth manifolds, and let F : M → N and G : N → Q be smooth maps. For every point p ∈ M, (1) the tangent map to the identity idM : M → M is the identity:

dpidM = idTpM;

(2) the tangent map to the composition G◦F : M → Q is the composition of the tangent maps:

dp(G ◦ F) = dF(p)G ◦ dpF, (3.7)

(Equation (3.7) is also known as the chain rule); (3) the tangent map to a diffeomorphism Φ : M → N is an isomorphism whose inverse is the tangent map to Φ−1 :

−1 −1 (dpΦ) = dΦ(p)Φ .

Proof. Left as Exercise 3.5.

Exercise 3.5 Prove Proposition 3.2.1.

R It follows from Proposition 3.2.1 and the Tangent Vector Theorem (Theorem 3.1.2, not proved yet in full generality) that two diffeomorphic manifolds M,N share the same dimension: dimM = dimN. In particular, there are no diffeomorphisms between open subsets in Euclidean spaces of different dimensions.

We now look at the tangent spaces to a submanifold S ⊆ M in a manifold M. As usual, denote by iS : S ,→ M the inclusion and recall that it is a smooth map. Let p ∈ S be a point. Our intuition suggests that the tangent space to S at p is a subspace of the tangent space to M at the same point p, and, additionally, if S = U ⊆ M is an open submanifold, that U and M share the same tangent space. Our intuition is correct, as rigorously stated in the next

Proposition 3.2.2 Let S ⊆ M be a submanifold in a manifold M, and let p ∈ S be a point. Then the tangent map to the inclusion iS : S ,→ M at the point p is injective (hence a monomorphism of vector spaces). If, additionally, S = U ⊆ M is an open submanifold, then dpiU is also surjective, hence an isomorphism.

Proof. Consider the tangent map

dpiS : TpS → TpM.

It is enough to show that kerdpiS = {0}. So let v ∈ TpS be such that dpiS(v) = 0. We have to show that v(g) = 0 for all smooth functions g on S. So, take g ∈ C∞(S) and use the Local Extension ∞ Lemma (Lemma 2.3.6) to pick a smooth function ge∈ C (M) such that g and ge agree on an open neighborhood of p in S. As tangent vectors are local operators (Lemma 3.1.1, Point (2)), we have

∗ v(g) = v(ge|S) = v(iS(ge)) = dpiS(v)(ge) = 0.

This concludes the proof of the ﬁrst part of the statement. For the second part, let S = U ⊆ M be an open submanifold. We have to show that, for any v ∈ TpM, there exists w ∈ TpU such that ∞ ∞ ∞ v = dpiU (w). We deﬁne w : C (U ) → R as follows. Let g ∈ C (U ), and let ge ∈ C (M) be a 62 Chapter 3. Tangent Vectors smooth function on M that coincides with g around p (it exists in view of the Local Extension Lemma). Put

w(g) := v(ge).

First of all, as tangent vectors are local operators, v(ge) does only depend on the values of ge around ∞ p, hence w(g) is well-deﬁned. Secondly, if f ,g are functions on S, and fe,ge∈ C (M) coincide with them around p, then a fe+ bge, fege coincide with a f + bg, f g around p for all a,b ∈ R. So

w(a f + bg) = v(a fe+ bge) = av( fe) + bv(ge) = aw( f ) + bw(g), and

w( f g) = v( fege) = fe(p)v(ge) + ge(p)v( fe) = f (p)w(g) + g(p)w( f ). This shows that w is a tangent vector. Finally, let h ∈ C∞(M) and compute

∗ dpiU (w)(h) = w(iU (h)) = w(h|U ) = v(h), and, from the arbitrariness of h, dpiU (w) = v. This concludes the proof.

In the following, we will use dpiS to identify TpS with its image (unless otherwise stated). In other words, we will interpret TpS as a subspace in TpM, and write TpS ⊆ TpM. In particular, if U ⊆ M is an open submanifold, then TpU = TpM. So a tangent vector v ∈ TpM can be safely applied to any smooth function f deﬁned only locally around p, i.e. any smooth function f ∈ C∞(U ) on some open neighborhood U ⊆ M of p. In the following, we will do this without further comments. We are ﬁnally ready to prove the Tangent Vector Theorem (Theorem 3.1.2).

Proof of the Tangent Vector Theorem 3.1.2. Let M be an n-dimensional manifold, let p ∈ M be a point, and let (U,ϕ = (x1,...,xn)) be a chart on M around p. The claim is that the coordinate vectors ∂ ∂ | ,..., | (3.8) ∂x1 p ∂xn p form a frame of TpM. To prove this, denote by P the image ϕ(p), and consider the chain of isomorphisms

d i d ϕ dpi p U p Ub n TpMTo pU / TPUb / TPR . (3.9)

It easily follows from the deﬁnition of coordinate vectors that (3.9) identify

∂ ∂ ∂ ∂ | ,..., | and | ,..., | ∂x1 p ∂xn p ∂t1 P ∂tn P

n (do you see it?). As the latter is a frame of TPR (Lemma 3.1.4), the former must be a frame of TpM as claimed. Let M be an n-dimensional manifold, let p ∈ M be a point, and let (U,ϕ = (x1,...,xn)) be a chart on M around p. In the following, the frame (3.8) of TpM will be referred to as the coordinate frame. In view of the Tangent Vector Theorem any tangent vector v ∈ TpM can be uniquely written in the form n i ∂ v = ∑ v i |p (3.10) i=1 ∂x 3.2 Tangent Maps 63 for some real numbers vi. Notice that, from the deﬁnition of coordinate vector,

∂ | x j = δ j, i, j = 1,...,n. (3.11) ∂xi p i

1 n It follows that the components v ,...,v of a tangent vector v ∈ TpM in the coordinate frame are given by the action of v on the coordinates x1,...,xn, i.e.

v j = v(x j), j = 1,...,n. (3.12)

Indeed

n n j i ∂ j i j j v(x ) = ∑ v i |px = ∑ v δi = v , i=1 ∂x i=1 where we used (3.11). Now on we adopt the Einstein summation convention: we will understand the sum over pairs of upper-lower repeated indices and simply write, e.g.,

∂ v = vi | , ∂xi p instead of (3.10).

Example 3.3 — Coordinate Expression for the Velocity of a Curve. let γ : I → M be a curve, and let t0 ∈ I. Denote p0 = γ(t0). As γ is smooth we can choose an open subinterval J ⊆ I 1 n containing t0, and a chart (U,ϕ = (x ,...,x )) on M around p0, such that γ(J) ⊆ U. Denote by i ∗ i γ = γ (x ) : J → R the components of (the coordinate representation of) γ with respect to the chart (U,ϕ). Then, for every t ∈ J,

∂ γ˙(t) = vi | , ∂xi γ(t)

i for some v ∈ R given by

d dγi γ˙(t)(xi) = xi(γ(t)) = (t). dt dt Summarizing

dγi ∂ γ˙(t) = (t) | . (3.13) dt ∂xi γ(t)

Clearly, if we change coordinates around p, the coordinate frame changes. We want to find out how. So let (U˜ ,ϕ˜ = (x˜1,...xñ)) be another chart on M around p. We denote by t1,...,tn the standard coordinates on ϕ(U), and, to avoid confusion, by t˜1,...,tñ the standard coordinates on ϕ˜(U˜ ). We also put P = ϕ(p). The transition matrix

∂ ∂ ∂ ∂ ∂(ϕ˜ ◦ ϕ−1) j | x˜j = | x˜j ◦ϕ−1 = | x˜j ◦ϕ˜ −1 ◦ϕ˜ ◦ϕ−1 = | t˜j ◦(ϕ˜ ◦ϕ−1) = (P). ∂xi p ∂ti P ∂ti P ∂ti P ∂ti

This shows that MR,R˜ is exactly the Jacobian matrix at the point P = ϕ(p) of the transition map

ϕ˜ ◦ ϕ−1 : ϕ(U ∩U˜ ) → ϕ˜(U ∩U˜ ).

Next we compute the representative matrix of the tangent map with respect to two coordinate frames. Namely, let M,N be manifolds with dimM = m,dimN = n, let F : M → N be a smooth map, and let p ∈ M be a point. Choose a chart (U,ϕ = (x1,...,xm)) on M around p and a chart (V,ψ = (y1,...,yn)) on N around F(p) such that F(U) ⊆ V. Put P = ϕ(p). The restriction F : U → V is completely determined by functions Fa := ya ◦ F = F∗(ya) ∈ C∞(U), a = 1,...,n. The representative matrix

dpF a a=1,...,n MR,S = (bi )i=1,...,m ∈ M(n,m;R) of the tangent map dpF : TpM → TF(p)N with respect to the coordinate frames ∂ ∂ ∂ ∂ R = |p,..., |p and S = | ,..., | ∂x1 ∂xm ∂y1 F(p) ∂yn F(p) is given by a ∂ b = the a-th component of dpF |p in the frame S i ∂xi ∂ ∂ ∂ = d F | ya = | F∗(ya) = | Fa, p ∂xi p ∂xi p ∂xi p and using the deﬁnition of coordinate vector we ﬁnd

∂ ∂ ∂Fba ba = | Fa = | Fa ◦ ϕ−1 = (P), i ∂xi p ∂ti P ∂ti where the Fba are the components of the coordinate representation Fb : Ub → Vb of F. This shows that dpF MR,S is exactly the Jacobian matrix at the point P = ϕ(p) of Fb. We conclude this section discussing two further applications of the Tangent Vector Theorem. We begin showing that every tangent vector is the velocity of a suitable (non-unique) curve.

Proposition 3.2.3 Let M be a manifold, and let p ∈ M. For every tangent vector v ∈ TpM there exists a curve γ : I → M, such that 0 ∈ I, γ (0) = p, and γ˙(0) = v.

Proof. Let n = dimM, choose a chart (U,ϕ = (x1,...,xn)) around p, and put P = ϕ(p). Then ∂ v = vi | ∂xi p n n n for some n-tuple V := (v1,...,v ) ∈ R . Consider the following curve in R n γb : R → R , t 7→ γb(t) := P +tV. 3.2 Tangent Maps 65

U M v p j

V % P !" %&

0 ℝ$ I

Figure 3.2: Every tangent vector is the velocity of a curve.

−1 The pre-image γb (Ub) is an open neighborhood of 0 in R. In particular, there is an open interval I ⊆ R containing 0, such that γb(I) ⊆ Ub. Denote again by γb : I → Ub the restriction, and let 1 n (γ ,...,γ ) be its components. The curve γb is the coordinate representation of a unique curve −1 γ = ϕ ◦ γb : I → M (taking values in U). The curve γ is the one we are looking for. To show this, we compute the velocity γ˙(0). From (3.13),

dγ i ∂ ∂ γ˙(0) = (0) | = vi | = v. dt ∂xi γ(0) ∂xi p

This concludes the proof.

Exercise 3.6 Let F : M → N be a smooth map between manifolds, and let γ : I → M be a curve in M. Show that, for every t0 ∈ I, the tangent map to F transforms the velocity of a curve γ into the velocity of the transformed curve F ◦ γ, i.e.

dγ d(F ◦ γ ) d F (t ) = (t ), (3.14) γ (t0) dt 0 dt 0

Notice that, in view of Proposition 3.2.3, Equation (3.14) can be understood as an alternative deﬁnition of the tangent map. Our second application of the Tangent Vector Theorem is a description of the tangent spaces to a vector space. So, let V be an n-dimensional real vector space. In particular, V is an n-dimensional manifold. We begin noticing that there are two (a priori different) ways to understand the velocity of a curve γ : I → V in V at time t0 ∈ I. First of all, as the vector γ˙(t0) in the tangent space Tγ (t0)V. 0 Secondly, as a vector in V, denoted γ (t0), as follows. Choose a frame R = (e1,...,en) in V. Then, i i for all t ∈ I, γ (t) = v (t)ei, where the v : I → R are smooth functions. Put

dvi γ 0(t ) = (t )e ∈ V. 0 dt 0 i

0 It is easy to see that γ (t0) is actually independent of the choice of R (do you see it?). 66 Chapter 3. Tangent Vectors

Exercise 3.7 Let V be a ﬁnite dimensional real vector space, let v1,...,vk be (non-necessarily independent) vectors in V, and let f1,... fk : I → R be smooth functions on some open interval I ⊆ R. Show that the map

γ : I → V, t 7→ γ(t) := f1(t)v1 + ··· + fk(t)vk,

is a smooth curve in V, and that d f d f γ0(t) = 1 (t)v + ··· + k (t)v dt 1 dt k

for all t ∈ I. We are now ready to describe the tangent spaces to V.

Proposition 3.2.4 — Tangent Space to a Vector Space. For any v0 ∈ V, there is a canonical ∼ vector space isomorphism V = Tv0V given by d V → T V, v 7→ v↑ := | (v +tv). (3.15) v0 dt t=0 0

Additionally, for any curve γ : I → V, and any t0 ∈ I, we have

0 ↑ γ (t0) = γ˙(t0),

∼ 0 i.e. the isomorphism V = Tv0V identiﬁes the “two velocities” γ (t0) and γ˙(t0) of γ.

Probably, the deﬁnition in (3.15) requires some explanations. For any v0,v ∈ V, we are considering the smooth curve Γ : R → V, t 7→ Γ(t) := v0 +tv in V (do you see that Γ is smooth?), ↑ and v ∈ Tv0V is the velocity of Γ at time 0.

↑ Proof of Proposition 3.2.4. We have to show that the map V → Tv0V, v 7→ v is linear and bijective. We take care of both things at once. Choose a frame R = (e1,...,en) in V, and let

1 n (V,ϕR = (x ,...,x )) be the associated chart on V. Recall that, for any v ∈ V, xi(v) is, by deﬁnition, the i-th component vi of v in the frame R. Let ∂ ∂ RT = |v ,..., |v ∂x1 0 ∂xn 0

n be the coordinate frame in Tv0V induced by the chart (V,ϕR). Finally, let ϕRT : Tv0V → R be the ↑ coordinate isomorphism induced by RT . We want to compute ϕRT (v ). To do this, remember that the i-th component of a velocity vector in the coordinate frame is the time derivative of the i-th component of the coordinate representation of the curve (see (3.13)). As v↑ is the velocity at time 0 ↑ of the curve Γ, we have, in particular, that the coordinate vector of v in the coordinate frame RT is

dΓ1 dΓn ϕ (v↑) = (0),..., (0) RT dt dt

1 n n where ϕR ◦ Γ = (Γ ,...,Γ ) : R → R is the coordinate representation of Γ. But

i i i Γ = v0 +tv , 3.2 Tangent Maps 67 hence

dΓi (0) = vi dt for all i = {1,...,n}. We conclude that

↑ 1 n ϕRT (v ) = v ,...,v = ϕR(v).

↑ This shows that the map V → Tv0V, v 7→ v , is given by the composition of the vector space isomorphism ϕ : V → n, followed by the vector space isomorphism ϕ−1 : n → T V: R R RT R v0

v↑ = (ϕ−1 ◦ ϕ )(v), for all v ∈ V. RT R

Hence, it is a vector space isomorphism as well.

!" + (! V !

#$%&

Figure 3.3: The tangent space to a vector space V is canonically isomorphic to V.

i For the second part of the proposition, let γ : I → V, t 7→ γ (t) = v (t)ei be a curve in V, and let 0 t0 ∈ I. By deﬁnition of γ (t0), we have

dv1 dvn ϕ (γ 0(t )) = (t ),..., (t ) . R 0 dt 0 dt 0

On the other hand, the coordinate representation of γ with respect to the chart (V,ϕR) is ϕR ◦ γ = n n (v1,...,v ) : I → R , and, from (3.13) again,

dvi ∂ γ˙(t ) = (t ) | 0 dt 0 ∂xi γ(t0) so that

dv1 dvn ϕ (γ˙(t )) = (t ),..., (t ) = ϕ (γ 0(t )) RT 0 dt 0 dt 0 R 0

0 ↑ showing that γ (t0) = γ˙(t0) as claimed.

In the following, given a curve γ : I → V in a ﬁnite dimensional real vector space V, we will 0 always interpret its velocity γ˙(t0) at time t0 ∈ I as a vector in V, identifying γ˙(t0) and γ (t0) via the ∼ isomorphism V = Tv0V, unless otherwise stated. 68 Chapter 3. Tangent Vectors

3.3 The Tangent Bundle Let M be an n-dimensional manifold. Tangent vectors to M can be organized into a new manifold called the tangent bundle to M, and denoted TM. In this section we construct TM, including its

smooth structure, and discuss its ﬁrst properties. First of all, we put TM := ∏ TpM = {(p,v) : p ∈ M and v ∈ TpM}. p∈M

When it is clear what is p, a point (p,v) in TM will be also denoted simply by v (see also the remark following Deﬁnition 3.1.2). The set TM comes with a natural surjection τ : TM → M, (p,v) 7→ p, and the pair (TM,τ) is called the tangent bundle to M. Sometimes we understand τ and call TM itself the tangent bundle. The smooth structure on M induces an atlas on TM as we now show. Begin with a chart (U,ϕ = (x1,...,xn)) on M, and deﬁne a chart (TU,Tϕ) on TM as follows.

First of all, put −1 ∏ TU := τ (U) = TpM = {(p,v) ∈ TM : p ∈ U}. p∈U

As the tangent spaces to U identify canonically with the tangent spaces to M at points of U, TU is exactly the tangent bundle to U. Next deﬁne a map:

n 1 n Tϕ : TU → Ub × R , (p,v) 7→ Tϕ(p,v) := (ϕ(p);v(x ),...,v(x )).

Notice that the last n entries of Tϕ(p,v) are nothing but the components of v in the coordinate frame

∂ ∂ Rp = |p,..., |p ∂x1 ∂xn

of TpM. It is clear that (TU,Tϕ) is a 2n-dimensional chart on TM. Every such chart on TM is called a standard charts. Let (TU,Tϕ) be the standard chart on TM induced by a chart (U,ϕ = (x1,...,xn)) on M. By an abuse of notation, the ﬁrst n components of Tϕ are denoted again by (x1,...,xn). The last n components are denoted (x˙1,...,x˙n), and we write

(TU,Tϕ = (x1,...,xn;x ˙1,...,x˙n)).

Proposition 3.3.1 — Smooth Structure on the Tangent Bundle. Standard charts on TM form an atlas. With the associated smooth structure, TM is a smooth 2n-dimensional manifold, and τ : TM → M is a smooth map.

Proof. As the charts on M cover M, the standard charts on TM cover TM. Next we show that any two standard charts are compatible. So, take two charts (U,ϕ = (x1,...,xn)),(U˜ ,ϕ˜ = (x˜1,...,x˜n)), and the associated standard charts (TU,Tϕ),(TU˜ ,Tϕ˜). If U ∩U˜ = ∅, then TU ∩ TU˜ = ∅, and (TU,Tϕ),(TU˜ ,Tϕ˜) are compatible. If U ∩U˜ 6= ∅, then TU ∩ TU˜ = T(U ∩U˜ ) and we have to show that the transition map

Tϕ˜ ◦ (Tϕ)−1 : Tϕ(TU ∩ TU˜ ) → Tϕ˜(TU ∩ TU˜ )

n is a diffeomorphism between open subsets of R2 . To do this, ﬁrst notice that

n Tϕ(TU ∩ TU˜ ) = Tϕ(T(U ∩U˜ )) = ϕ(U ∩U˜ ) × R 3.3 The Tangent Bundle 69 is indeed an open subset, and similarly for Tϕ˜(TU ∩ TU˜ ). Now, check the smoothness of Tϕ˜ ◦ − n n (Tϕ) 1: take a point (P;v1,...,v ) ∈ ϕ(U ∩U˜ ) × R and compute

(Tϕ˜ ◦ (Tϕ)−1)(P;v1,...,vn) −1 i ∂ = Tϕ˜ ϕ (P);v | −1 ∂xi ϕ (P) −1 i ∂ 1 i ∂ n = (ϕ˜ ◦ ϕ )(P);v | −1 x˜ ,...,v | −1 x˜ ∂xi ϕ (P) ∂xi ϕ (P) ∂(ϕ˜ ◦ ϕ−1)1 ∂(ϕ˜ ◦ ϕ−1)n = (ϕ˜ ◦ ϕ−1)(P);vi (P),...,vi (P) . ∂ti ∂ti

As (U,ϕ),(U˜ ,ϕ˜) are compatible, the first n entries depend smoothly on P. The last n entries depend smoothly on P, and linearly, hence smoothly, on v1,...,vn. This shows that Tϕ˜ ◦(Tϕ)−1 is smooth. Changing the roles of (U,ϕ) and (U˜ ,ϕ˜) reveals that Tϕ ◦ (Tϕ˜)−1 is smooth as well. We conclude that both transition maps Tϕ˜ ◦ (Tϕ)−1 and Tϕ ◦ (Tϕ˜)−1 are diffeomorphisms, and this concludes the proof of the first part of the statement. The atlas on TM consisting of standard charts will be called the standard atlas. For the second part of the statement, we have to show that the atlas topology induced by the standard atlas is Hausdorff and II-countable. We use the criterion provided by Proposition 1.3.2. For the Hausdorff property, take (p,v),(q,w) ∈ TM. If p = q then p,q belong to the same coordinate domain U, hence (p,v),(q,w) belong to the same standard coordinate domain TU. If p 6= q, we can find disjoint coordinate domains U,V such that p ∈ U and q ∈ V. It follows that TU,TV are disjoint standard coordinate domains such that (p,v) ∈ TU and (q,w) ∈ TV. For II-countability, we cover M by countably many charts (U,ϕ). Hence TM is covered by the standard charts (TU,Tϕ), and they are countably many. We leave it to the reader to prove the very last part of the statement (about the smoothness of τ : TM → M) as Exercise 3.8.

Exercise 3.8 Complete the proof of Proposition 3.3.1 showing that the natural projection

τ : TM → M is smooth. We now come to a special class of maps M → TM that will play an important role in the next chapter: sections of TM. Deﬁnition 3.3.1 — Section of the Tangent Bundle. A section of TM is a map s : M → TM that inverts τ : TM → M on the right, i.e. τ ◦s = idM. In other words, a section is the assignment of a tangent vector s(p) to M at the point p, for every point p ∈ M.

There is a natural algebraic structure on the space of sections of TM: they form a module over the algebra C∞(M). More generally, let A be a real algebra. Deﬁnition 3.3.2 — Module. a module over A, or simply an A-module, is a set M equipped with two operations +,· such that (1) + : M × M → M is an interior operation and (M ,+) is an abelian group, (2) · : A × M → M is an exterior operation and X α · (β · µ) = (α · β) · µ, X (α + β) · µ = α · µ + β · µ, X α · (µ + ν) = α · µ + α · ν, X 1 · µ = µ, for all α,β ∈ A, and µ,ν ∈ M . A submodule of an A-module M is a subset N ⊆ M which is stable under both operations in M . 70 Chapter 3. Tangent Vectors

% ' &

%' "(') "(!) ( "

! '

Figure 3.4: A section of the tangent bundle.

In other words, a module satisfies all the axioms of a vector space except that the ring of scalar is now a generic algebra, not necessarily a field. We usually write αµ, instead of α · µ, for α ∈ A and µ ∈ M . As a first trivial example, notice that any real vector space is an R-module. Conversely, every module over a real algebra is also a real vector space. A submodule N of a module M is also a module with the restricted operations. In many respects, the theory of modules is similar to the theory of vector spaces. For instance, linearly independent systems and system of generators are defined exactly as for vector spaces, except that now linear combinations are taken with coefficients in the algebra rather than just in the field. A basis of a module is a system of independent generators. The main difference between modules and vector spaces is that a module may not possess a basis. Linear maps between modules over the same algebra A are defined exactly as for vector spaces, except that now they are compatible with the multiplication by a scalar in the algebra rather than just in the field. In order to distinguish explicitly this kind of linearity from R-linearity, we call it A-linearity, and A-linear maps between A-modules are also called A-module homomorphisms. An invertible A-linear map is called an A-module isomorphism, and its inverse is also an A-module isomorphism. The following example is our main example of a module.

Example 3.4 — Module of Sections of TM. Let M be a manifold. The space of sections of the tangent bundle TM is a C∞(M)-module with the following point-wise operations:

+ : (s1,s2) 7→ s1 + s2, (s1 + s2)(p) := s1(p) + s2(p), · : ( f ,s) 7→ f s, ( f s)(p) := f (p)s(p),

∞ where f ∈ C (M), and s,s1,s2 are sections. The zero section is the section mapping a point p to the zero tangent vector 0 ∈ TpM, for all p. We are mainly interested in smooth sections. Discussing the smoothness of a section is easier than discussing the smoothness of a generic map. Indeed, let M be a manifold, and let s be a section of the tangent bundle TM. For every chart (U,ϕ) on M, the section s maps U to TU. This shows that all sections have coordinate representations. Now, look at the restriction

s : U → TU. 3.3 The Tangent Bundle 71

As already remarked in the discussion following Deﬁnition 2.2.1, s is smooth provided only the pull-backs s∗(xi), s∗(x˙i) of the coordinates on TU are smooth, i = 1,...,n. Now, by deﬁnition of section, for every p ∈ U,

s∗(xi)(p) = xi(s(p)) = xi(p), which is automatically smooth in its argument p. We conclude that the section s is smooth provided only the functions s∗(x˙i) on U are smooth for some family of charts (U,ϕ) covering M. In the following, we denote si := s∗(x˙i) and call the si the components of the section s in the chart (U,ϕ). We can then rephrase the preceding discussion saying that a section s of TM is smooth if and only if its components si are smooth in every chart of an atlas.

Proposition 3.3.2 Smooth sections of TM form a submodule in the module of all sections.

Proof. Left as Exercise 3.9.

Exercise 3.9 Prove Proposition 3.3.2. (Hint: prove that, for every three sections s,s1,s2, any smooth function f , and any chart (U,ϕ) on M the following identities hold:

i i i (s1 + s2) = s1 + s2 i i ( f s) = f |U s

for all i ∈ {1,...,n}. Conclude that, if s,s1,s2 are smooth, then so are s1 + s2 and f s.)

We denote by Γ(TM) the C∞(M)-module of smooth sections of the tangent bundle TM. In the following, by “section” we will always mean “smooth section”, unless otherwise stated. We conclude this chapter discussing the tangent bundle to a submanifold. So let M be an n-dimensional manifold, and let S ⊆ M be a d-dimensional submanifold. As TpS ⊆ TpM for all p ∈ S, the tangent bundle to S can be seen as a subset in the tangent bundle to TM: TS ⊆ TM.

Proposition 3.3.3 Let M be an n-dimensional manifold, and let S ⊆ M be a d-dimensional submanifold. Then the tangent bundle to S is a 2d-dimensional submanifold in the tangent bundle to M (and the submanifold atlas is compatible with the standard atlas on TS).

1 d 1 n−d Proof. Let p0 ∈ S, and let (U,ϕ = (y ,...,y ;z ,...,z )) be an adapted chart on M around p0. This means that

S ∩U = {p ∈ U : za(p) = 0, for all a = 1,...,n − d}.

1 d Consider also the submanifold chart (US,ϕS = (yS,...,yS)). As we know, the coordinate representation of the inclusion iS : S ,→ M with respect to the charts (US,ϕS) and (U,ϕ) is

1 d 1 d 1 d biS = ind : UbS → Ub, (t ,...,t ) 7→ ind(t ,...,t ) = (t ,...,t ,0,...,0).

It follows that the image of a tangent vector

i ∂ v = v i |p ∈ TpS ∂yS to S at a point p ∈ US under the tangent map dpiS : TpS → TpM is

∂ d i (v) = vi | ∈ T M. p S ∂yi p p 72 Chapter 3. Tangent Vectors

We conclude that a tangent vector

∂ ∂ w = vi | + wa | ∈ T M, p ∈ U , ∂yi p ∂za p p S is in TpS if and only if

wa = z˙a(w) = 0, for all a ∈ {1,...,n − d}.

In other words,

TS ∩ TU = {(p,w) ∈ TU : za(p,w) = z˙a(p,w) = 0, for all a = 1,...,d}, showing that (TU,Tϕ) is an adapted chart to TS (up to a permutation of the coordinates). The last part of the statement follows from the easy remark that, for every chart (U,ϕ) on M adapted to S, the submanifold chart induced on TS by (TU,Tϕ), agrees with (TUS,TϕS): the standard chart induced by the submanifold chart (US,ϕS) (do you see it?). So TS can be covered by charts that are simultaneously in the submanifold atlas and the standard atlas. 4. Full Rank Smooth Maps

By definition, the rank of a smooth map F : M → N between manifolds is the rank of the tangent map. So the rank is a non-negative integer valued function on M, bounded from the above by the minimum of the dimensions of M and N. Different smooth maps F,G : M → N may look very different, even locally, depending on how does their rank vary on M. However, all constant rank smooth maps, with fixed rank, say r, look locally the same. This is the main content of the Rank Theorem, which is a nice and important example of a “local normal form theorem” or “local classification theorem” in Differential Geometry. In this chapter, we begin defining the simplest possible examples of constant rank maps: immersions, submersions, and local diffeomorphisms. Thereafter, we state the Rank Theorem, and discuss a couple of consequences.

4.1 Full Rank Maps Let M,N be smooth manifolds, with dimM = m and dimN = n, and let F : M → N be a smooth map. Deﬁnition 4.1.1 — Rank of a Smooth Map. The rank of F is the non-negative integer valued map:

rankF : M → N0, p 7→ rankp F := rankdpF = dim(imdpF).

X F is an immersion if m ≤ n, and rankp F = m for all p ∈ M (in other words, the tangent map dpF is injective for all p ∈ M). X F is a submersion if m ≥ n, and rankp F = n for all p ∈ M (in other words, the tangent map dpF is surjective for all p ∈ M). X F is a local diffeomorphism if m = n, and rankp F = m = n for all p ∈ M (in other words, the tangent map dpF is an isomorphism for all p ∈ M).

Let F : M → N be a smooth map between manifolds as above, and let p ∈ M. Choose (U,ϕ = (x1,...,xm)), a chart on M around p, and (V,ψ = (y1,...,yn)), a chart on N around F(p), such that F(U) ⊆ V. In particular, we can consider the coordinate representation Fb : Ub → Vb. Put 74 Chapter 4. Full Rank Smooth Maps

P = ϕ(p), and recall that the Jacobian matrix

!a=1,...,m ∂Fa J (P) = b (P) Fb ∂ti i=1,...,m is exactly the representative matrix of the tangent map dpF : TpM → TF(p)N in the coordinate frames ∂ ∂ ∂ ∂ | ,..., | and | ,..., | . ∂x1 p ∂xm p ∂y1 F(p) ∂yn F(p)

It follows that

rank F = rankJ (P). p Fb We now provide a list of examples. 2 3 Example 4.1 Consider the standard Euclidean spaces R and R with standard coordinates (u,v), and (x,y,z) respectively, and the smooth map

2 3 2 2 F : R → R , (u,v) 7→ F(u,v) := (u ,uv,v ).

The Jacobian matrix of F is

 ∂F1 ∂F1    ∂u ∂v 2u 0 J =  ∂F2 ∂F2  = v u . F  ∂u ∂v    ∂F3 ∂F3 0 2v ∂u ∂v We conclude that 0 if P = (0,0) rank F = rankJ (P) = . P F 2 if P 6= (0,0)

In particular, F is neither an immersion nor a submersion (but F : R2 r {(0,0)} → R3 is an immersion - do you see it?).

Example 4.2 Let n,m be non-negative integers with n ≤ m. The smooth map

n m 1 n 1 n inn : R → R , P = (P ,...,P ) 7→ inn(P) := (P ,...,P ,0,...,0) is an immersion. Indeed, its Jacobian matrix is

 1 ··· 0   . .. .   . . .     0 ··· 1  Jin =  , n  0 ··· 0     . .. .   . . .  0 ··· 0 whose rank is constant and equal to n. 2 Example 4.3 Consider the standard Euclidean space R with coordinates (x,y). The curve

2 γ : R → R , t 7→ (sint,cost) 4.1 Full Rank Maps 75 is an immersion. Indeed, its Jacobian matrix is

Jγ = (−cost,sint).

Whose rank is constant and equal to 1. Alternatively, we could argue as follows. For any t0 ∈ R, the d 2 tangent space Tt0 R is spanned by dt |t0 , whose image under the tangent map dt0 γ : Tt0 R → Tγ(t0)R is d ∂ ∂ d γ | = γ˙(t ) = −cost | + sint | 6= 0. t0 dt t0 0 0 ∂x γ(t0) 0 ∂y γ(t0)

It follows that dt0 γ is injective for all t0 ∈ R, hence γ is an immersion. More generally, a curve γ : I → M in a manifold M is an immersion if and only if the velocity γ˙(t0) is non-zero for all t0 ∈ I.

Example 4.4 In view of Proposition 3.2.2, the inclusion iS : S ,→ M of a submanifold S ⊆ M in a manifold M is always an immersion.

Example 4.5 Let n,m be non-negative integer with n ≤ m. The smooth map

m n 1 m 1 n prn : R → R , P = (P ,...,P ) 7→ prn(P) := (P ,...,P ) is a submersion. Indeed, its Jacobian matrix is

 1 ··· 0 0 ··· 0  J =  ...... , prn  ......  0 ··· 1 0 ··· 0 whose rank is constant and equal to n. n+1 0 n Example 4.6 Consider the standard Euclidean space R with coordinates (t ,...,t ). We already know that the canonical projection

n+1 n π : R r {0} → RP , P 7→ π(P) = [P], is smooth (see Example 2.3). We want to show that it is a submersion. To do this, pick a point 0 n n+1 i i P = (P ,...,P ) ∈ R r {0}, and let i ∈ {0,...,n} be such that P 6= 0. So (Vi := {t 6= 0},id) is a chart around P, and π(Vi) ⊆ Ui, where, as usual, we denote by (Ui,ϕi) the i-th afﬁne chart on n RP . Recall from Example 2.3 that the coordinate representation of π with respect to these charts is ! t0 tbi tn π : V → n, (t0,...,tn) 7→ π(t0,...,tn) = ,..., ,..., . b i R b ti ti ti

The Jacobian matrix of πb is  1 t0  ti ··· 0 − (ti)2 0 ··· 0  ......   ......   . . . . .   1 ti−1   0 ··· ti − (ti)2 0 ··· 0  Jπ =  i+1 , b  0 ··· 0 − t 1 ··· 0   (ti)2 ti     ......   ......  tn 1 0 ··· 0 − (ti)2 0 ··· ti whose rank is constant and equal to n. In particular, rankP π = n. So, from the arbitrariness of P, π is a submersion. 76 Chapter 4. Full Rank Smooth Maps

n m Example 4.7 Let U ⊆ R be an open subset, and let F : U → R be a smooth map, with m ≤ n. Assume that 0 is a regular value of F. We already remarked that, in this case, rankJF = m in an open neighborhood of any point P ∈ Z(F) (see the discussion following Example 2.12). Hence m rankF = m in an open neighborhood of V ⊆ U of Z(F), and F|V : V → R is a submersion. m Conversely, if there exists an open neighborhood V ⊆ U of Z(F) such that F|V : V → R is a submersion, then 0 is a regular value of F (do you see it?).

Example 4.8 Every diffeomorphism is a local diffeomorphism (see Proposition 3.2.1). 2 Example 4.9 Consider the curve γ : R → R from Example 4.3. Obviously, it takes values in the 0 circle S1 ⊆ R2 and can be restricted to S1 in the codomain. Denote by γ : R → S1 the restriction. From the Submanifold Chart Theorem (Theorem 2.3.5), γ0 is a smooth map. Additionally, γ is the 0 1 2 composition of γ followed by the inclusion iS1 : S ,→ R . Hence, for any t ∈ R, 0 0 dt γ = dt (iS1 ◦ γ ) = dγ(t)iS1 ◦ dt γ , 0 where we used the chain rule (3.7). As dt γ is injective, dt γ is injective as well, and, from 0 0 the arbitrariness of t, γ is also an immersion. Finally, dimR = dimS1 = 1, hence γ is a local diffeomorphism. n+1 n Example 4.10 Consider the canonical projection π : R r {0} → RP , and restrict it to the n n+ 0 n n 0 sphere S ⊆ R 1 r {0} to get a new smooth map π : S → RP . We want to show that π is a n n 0 local diffeomorphism. As dimS = dimRP = n, it is enough to check that π is a submersion. To do this, consider the projection P Π : n+1 {0} → Sn, P 7→ Π(P) := . R r kPk We already mentioned in Example 2.20 that Π is smooth. Additionally, π is the composition of Π 0 n+ followed by π (do you see it?). Hence, for any P ∈ R 1 r {0}, 0 dPπ = dΠ(P)π ◦ dPΠ. 0 As dPπ is surjective, dΠ(P)π is surjective as well. Finally, from the surjectivity of Π, every point in n 0 S is of the form Π(P), and π is a submersion (hence a local diffeomorphism).

Exercise 4.1 Let M1,...,Mk be manifolds, and consider the product manifold M1 × ··· × Mk. Show that the projection

pri : M1 × ··· × Mk → Mi, (p1,..., pk) 7→ pri(p1,..., pk) := pi,

onto the i-th factor is a submersion.

4.2 Rank Theorem Theorem 4.2.1 — Rank Theorem. Let M,N be smooth manifolds, dimM = m, dimN = n, let F : M → N be a smooth map, and let r ≤ min{m,n} be a non-negative integer. The following conditions are equivalent: (1) the rank of F is constant and equal to r; (2) for every point p ∈ M, there exist a chart (U,ϕ) in M, centered at p (recall that this means that p ∈ U, and ϕ(p) = 0), and a chart (V,ψ) in N, centered at F(p), such that F(U) ⊆ V, and the coordinate representation Fb : Ub → Vb of F reads

1 m 1 r 1 m Fb(t ,...,t ) = (t ,...,t ,0,...,0) = (inr ◦ prr)(t ,...,t ). 4.2 Rank Theorem 77

In particular, X if m ≤ n, then F is an immersion if and only if, around every point of M, it has a coordinate representation Fb such that

1 m 1 m 1 m Fb(t ,...,t ) = (t ,...,t ,0,...,0) = inm(t ,...,t );

X if m ≥ n, then F is a submersion if and only if, around every point of M, it has a coordinate representation Fb such that

1 m 1 n 1 m Fb(t ,...,t ) = (t ,...,t ) = prn(t ,...,t );

X if m = n, then F is a local diffeomorphism if and only if, around every point of M, it as a coordinate representation Fb such that

1 m 1 m 1 m Fb(t ,...,t ) = (t ,...,t ) = id(t ,...,t ).

Proof.

(1) ⇒ (2) Omitted. (2) ⇒ (1) Left as Exercise 4.2.

Exercise 4.2 Prove the implication (2) ⇒ (1) in the statement of the Rank Theorem (Theorem

4.2.1).

In practice, the Rank Theorem says that the local structure of a smooth map F : M → N of constant rank r does only depend on dimM,dimN and r. Yet in other words, all smooth maps F of constant rank r between an m-dimensional and an n-dimensional manifold do locally look the same. For this reason we also say that the Rank Theorem is a local normal form theorem. We now discuss some consequences of the rank theorem.

Proposition 4.2.2 Let M,N be smooth manifolds, dimM = m, dimN = n, and let F : M → N be a smooth map such that rankF is constant and equal to r ≤ min{m,n}. Then X for every point p ∈ M, there is an open neighborhood U ⊆ M of p, such that F(U) ⊆ N is an r-dimensional submanifold, and the restriction F : U → F(U) is a submersion; −1 X for every point q ∈ F(M), the preimage Sq := F (q) is an (m −r)-dimensional submanifold, and, for every point p ∈ Sq, the tangent space TpSq ⊆ TpM is the kernel of the tangent map dpF : TpM → TqN.

Proof. Let p ∈ M, and let (U,ϕ) and (V,ψ) be charts as in the statement of the Rank Theorem. As prr maps open subsets to open subsets, and inr is a homeomorphism onto its image,

r W := (prr ◦ Fb)(Ub) = prr(Ub) ⊆ R

r r 0 n−r is an open neighborhood of 0 in R , and there exist open subsets W ⊆ W ⊆ R and W ⊆ R , such 0 −1 0 −1 that 0 ∈ W ×W ⊆ Vb. Consider the subcharts (V0 := ψ (W ×W ),ψ), and (U0 := F (V0)∩U,ϕ). We have

1 n 0 r+1 n ψ(F(U) ∩V0) = ψ(F(U0)) = W × {0} = (Q ,...,Q ) ∈ W ×W : Q = ··· = Q = 0 .

Hence (V0,ψ) is an adapted chart to F(U), around F(p). We conclude that F(U) is an r- dimensional submanifold. Finally, the adapted chart (V0,ψ) induces a submanifold chart (W0,ψ0 := 78 Chapter 4. Full Rank Smooth Maps prr ◦ ψ) with W0 = V0 ∩ F(U) and Wb0 = W. The coordinate representation Fb : Ub0 → Wb0 of F : U → F(U) with respect to the charts (U0,ϕ), (W0,ψ0) is given by

1 m 1 r 1 m Fb(t ,...,t ) = (t ,...,t ) = prr(t ,...,t ),

showing that F : U0 → W0 is a submersion.

ℝ7'% U 5

M p 12 1 % 9 ℝ

4 &'% ℝ in% ∘ pr% !

" ,′ N 9 % 4(6) ℝ "#

in% pr% ,9 V !-(/) ℝ%

Figure 4.1: Locally, the image of a constant rank map is a submanifold.

For the second item in the statement, let q ∈ N be a point in the image of F, and let p ∈ Sq. This simply means that F(p) = q. Take charts (U,ϕ) and (V,ψ) as in the statement of the Rank Theorem. Then

−1 −1 ϕ(Sq ∩U) = ϕ(F (q) ∩U) = Fb (0) = Z(Fb).

But Z(Fb) is the solution space of the following regular system or r equations

 1  t = 0  . . .  tr = 0

It follows that Sq is an (m − r)-dimensional submanifold. It remains to compute the tangent space TpSq. First of all, notice that TpSq and kerdpF are both (m − r)-dimensional subspaces in TpM. Indeed, TpSq is the tangent space to an (m − r)-dimensional submanifold, and kerdpF is the kernel of a rank r linear map. So, in order to prove that TpSq = kerdpF, it is enough to check that TpSq ⊆ kerdpF. To do this, pick a tangent vector v ∈ TpSq. In view of Proposition 3.2.3, v is the velocity of some curve γ : I → Sq in Sq at some time t0 ∈ I: v = γ˙(t0). Finally, use Exercise 3.6 to compute

dγ d(F ◦ γ) d F(v) = d (F) (t ) = (t ). p p dt 0 dt 0

4.2 Rank Theorem 79

ℝ2'% U 0 M p () % ℝ

/ &'% ℝ in% ∘ pr% !

N ℝ% 1 "#

Figure 4.2: Fibers of a constant rank map are submanifolds.

But (F ◦ γ)(t) = q constantly on I, so dpF(v) is the velocity of a constant curve, which is clearly 0 (do you see it?). n+1 Example 4.11 — Tangent Spaces to the Sphere. Consider the standard Euclidean space R with standard coordinates t1,...,tn+1, and the smooth map

n+1 1 n+1 1 2 n+1 2 F : R r {0} → R, P = (P ,...,P ) 7→ F(P) := (P ) + ··· + (P ) .

n+ n+ The rank of F at the point P = (P1,...,P 1) ∈ R 1 r {0} is

1 n+1 rankP F = rankJF (P) = rank 2P ,...,2P = 1, constantly. So F is a constant rank map (actually a submersion). The n-dimensional sphere n n+ n − S ⊆ R 1 can be seen as the preimage of 1 under F: S = F 1(1). It now follows from Proposition n 1 n+1 n 4.2.2 that TPS = kerdPF for all P = (P ,...,P ) ∈ S . So, a tangent vector

∂ V = V i | ∂ti P

n+ n n to R 1 at some point P ∈ S is tangent to S if and only if ∂F d d 0 = d F(V) =V i (P) | = 2(V 1P1 +···+V n+1Pn+1) | ⇔ V 1P1 +···+V n+1Pn+1 = 0, P ∂ti dt F(P) dt 1 i.e.

(V 1,...,V n+1)⊥(P1,...,Pn+1),

in agreement with our usual understanding from elementary Euclidean Geometry. 80 Chapter 4. Full Rank Smooth Maps

ℝ%'( % "#$

P !

Figure 4.3: The tangent space to the sphere at the point P is “orthogonal” to P.

Example 4.12 — The Orthogonal Group. The orthogonal group is the following subgroup of the general linear group:

t O(n) := X ∈ GL(n,R) : X X = In . t where the “(−) ” denotes transposition and In is the identity matrix. In other words O(n) is the ﬁber over In of the map: t F : GL(n,R) → M(n,R), X 7→ X X, −1 i.e. O(n) = F (In). It is clear that F is a smooth map (do you see it?). If we manage to prove that the rank of F is constant, it will follow from Corollary 4.2.2 that O(n) is a submanifold in GL(n,R). Now let A ∈ GL(n,R) be any invertible matrix, and consider the maps:

ΦA : GL(n,R) → GL(n,R), X 7→ XA and

t ΨA : M(n,R) → M(n,R), Y 7→ A YA.

They are smooth for every A (do you see it?). Additionally, they are inverted by the maps ΦA−1 and ΨA−1 respectively. Hence they are diffeomorphisms. A direct computation shows that the following diagram

ΦA GL(n,R) / GL(n,R)

F F ΨA M(n,R) / M(n,R) commutes, i.e. F ◦ ΦA = ΨA ◦ F. It then follows from the chain rule that

dXAF ◦ dX ΦA = dXt X ΨA ◦ dX F.

As dX ΦA and dXt X ΨA are both isomorphisms, we clearly have

rankX F = rankdX F = rankdXAF = rankXA F. 4.3 Embeddings 81

0 − 0 Finally, let X ∈ GL(n,R) be any other invertible matrix. Choosing, A = X 1X , we immediately get

rankX F = rankXA F = rankX0 F.

Hence F has constant rank. One can actually show that

n(n + 1) rankF = . 2

2 We conclude that O(n) ⊆ GL(n,R) is a submanifold of dimension n −n(n +1)/2 = n(n −1)/2.

Exercise 4.3 — The Special Linear and the Special Orthogonal Groups. Prove that (1) the special linear group

SL(n,R) := {X ∈ GL(n,R) : detX = 1}

is an (n2 − 1)-dimensional submanifold in GL(n,R), and (2) the special orthogonal group

SO(n) := O(n) ∩ SL(n,R) = {X ∈ O(n) : detX = 1}

is an open submanifold in O(n). (Hint: for point 1) proceed as in Example 4.12. For point 2) notice that, since detX = ±1 for

all X ∈ O(n), we have SO(n) = {X ∈ O(n) : detX > 0}).

We conclude this section discussing one more corollary of the Rank Theorem.

Proposition 4.2.3 — Inverse Function Theorem. Let M,N be manifolds, and let F : M → N be a smooth map. Then F is a local diffeomorphism if and only if, for every point p ∈ M, there exist an open neighborhood U ⊆ M of p, and an open neighborhood U0 ⊆ N of F(p), such that F(U) ⊆ U0 and the restriction F : U → U0 is a diffeomorphism.

Proof. The “if part” of the statement is obvious. Let’s discuss the “only if” part. So let F : M → N be a local diffeomorphism, and let p ∈ M be a point. Choose charts (U,ϕ), (V,ψ) as in the statement of the Rank Theorem. Then Fb : Ub → Vb acts as the identity. In particular, Fb is injective, and Ub = Fb(Ub) ⊆ Vb. Put U0 := ψ−1(Ub). It is clear that F(U) ⊆ U0, and F : U → U0 is a diffeomorphism as claimed (do you see it?). Notice that the Inverse Function Theorem motivates the terminology “local diffeomorphism”.

Corollary 4.2.4 Let M,N be manifolds, and let F : M → N be a map. Then F is a diffeomorphism if and only if it is a bijective local diffeomorphism.

Proof. Left as Exercise 4.4.

Exercise 4.4 Prove Corollary 4.2.4.

4.3 Embeddings If we specialize Proposition 4.2.2 to an immersion i : N → M, we see that, restricting i to a sufﬁciently small open subset U ⊆ N we can make i(U) to be a submanifold. In general, we cannot choose U = N as the following counter-example shows. 82 Chapter 4. Full Rank Smooth Maps

2 Example 4.13 — The Eight Figure. Consider R with standard coordinates x,y and the following curve:

2 i : R → R , t 7→ i(t) := (sin2t,sint). The image of i is called the eight figure (for obvious reasons). The eight figure is not a submanifold. Indeed, around the origin, it is neither of the form y = f (x) nor of the form x = f (y) (for some smooth function f ). Indeed, every vertical (or horizontal) line through a non-zero point in an open neighborhood U ⊆ i(N) of 0, intersects U in at least two points (do you see it?). Notice that the issue is still there if we restrict i to the interval (−π,π) turning it in an injective immersion 2 i : (−π,π) → R . Indeed i(−π,π) = i(R) which is again the eight figure.

Figure 4.4: The eight ﬁgure.

However, for every positive real number ε < π, i(−π + ε,π − ε) is a submanifold and i : (−π + ε,π − ε) → i(−π + ε,π − ε) is a diffeomorphism. In the next proposition, given an injective immersion i : N → M, we ﬁnd conditions on i that guarantee that i(N) is a submanifold. Before stating the result, we need some preliminary remarks. So, let i : N → M be an injective immersion. In particular, the restriction of i to its image i(N) in the codomain is a bijection i : N → i(N). On i(N) there are at least two topologies: the subspace topology τsub induced be the inclusion i(N) ⊆ M, and the topology τi induced from that of N via the bijection i : N → i(N), i.e.

τi := {i(U) ⊆ i(N) : U ⊆ N is an open subset}.

In other words, τi is the unique topology on i(N) such that i : N → (i(N),τi) is a homeomorphism. In general, τsub 6= τi. For instance, when i is the injective immersion 2 i : (−π,π) → R from Example 4.13, then, for every positive real number ε < π, any open neighborhood of 0 in (i(N),τsub) does necessarily intersect nontrivially i(π − ε,π) and i(−π,−π + ε). On the other hand, i(−π/4,π/4) ∈ τi, but it does not intersect, for instance, i(3π/4,π), nor i(−π,−3π/4). Next proposition shows that this is precisely the reason why the eight ﬁgure is not a submanifold. 4.3 Embeddings 83

Proposition 4.3.1 Let i : N → M be an injective immersion. Then the following two conditions are equivalent: (1) i(N) ⊆ M is a submanifold of the same dimension as N; (2) τsub = τi, or, equivalently, i : N → (i(N),τsub) is a homeomorphism. Additionally, if (1), or, equivalently, (2), holds true, then i : N → i(N) is a diffeomorphism.

Proof.

(1) ⇒ (2) If i(N) is a submanifold of the same dimension as N, then i : N → i(N) is a bijective local diffeomorphism (do you see it?), hence a diffeomorphism. In particular, it is a homeomorphism.

(2) ⇒ (1) Assume that i : N → (i(N),τsub) is a homeomorphism, pick a point q ∈ i(N) and let p ∈ N be the unique point such that q = i(p). From Proposition 4.2.2, there is an open neighborhood U ⊆ M of p such that i(U) ⊆ M is a submanifold of the same dimension as N. On the other hand, we have i(U) ∈ τi = τsub. Hence there is an open subset U ⊆ M such that i(U) = i(N) ∩ U . Summarizing, we have showed that, for any q ∈ i(N) there exists an open neighborhood U ⊆ M of q such that i(N) ∩ U is a submanifold. This easily implies that i(N) is itself a submanifold. We leave the details as Exercise 4.5.

Exercise 4.5 Conclude the proof of Proposition 4.3.1, showing that a subset S ⊆ M in a manifold M is a submanifold if an only if it is locally so, i.e., for every point q ∈ S, there exists an open

neighborhood U ⊆ M of q such that S ∩ U is a submanifold.

Deﬁnition 4.3.1 — Embedding. An embedding is an injective immersion i : N → M satisfying one, hence both, of the equivalent conditions in Proposition 4.3.1.

Example 4.14 — Embeddings and Submanifolds. Let M be a manifold. The inclusion iS : S ,→ M of a submanifold S ⊆ M is an embedding, and every embedding i : N → M can be written as the composition iS ◦ Φ of a diffeomorphism Φ : N → S onto some submanifold S ⊆ M, followed by the inclusion iS : S ,→ M.

Example 4.15 Let d,n be non-negative integers such that d ≤ n. The smooth map

d n 1 d 1 d ind : R → R , P = (P ,...,P ) 7→ ind(P) = (P ,...,P ,0,...,0) is an embedding. Indeed it is an injective immersion, and its image is the d-dimensional submanifold d d n−d n R × {0} ⊆ R × R = R . An important class of embeddings is provided by sections of surjective submersions, as explained in Proposition 4.3.2 below. Before stating it, we give a deﬁnition. Let M,N be smooth manifolds, and let π : M → N and i : N → M be smooth maps such that π ◦i = idN. In this situation, i is called a (smooth) section of π (cf. Section 3.3).

Proposition 4.3.2 Let π : M → N be a smooth map of manifolds, and let i : N → M be a section of π. Then i is an embedding, and there is an open neighborhood U of i(N) in M such that π : U → N is a surjective submersion.

Proof. From the section property π ◦ i = idN, it immediately follows that π is surjective and i is injective. Now, let p ∈ N. The chain rule at p reads

di(p)π ◦ dpi = dpidN = idTpN, 84 Chapter 4. Full Rank Smooth Maps so that dpi is injective and di(p)π is surjective. From the arbitrariness of p, i is an immersion, hence an injective immersion, and rankp π = dimN for all p ∈ i(N), hence for all p in an open neighborhood U ⊆ M of i(N) in M. So, π : U → N is a surjective submersion. It remains to prove that i : N → (i(N),τsub) is a homeomorphism. It is enough to prove that the inverse map −1 −1 i : (i(N),τsub) → N is continuous. But, from the section property π ◦ i = idM again, i is the restriction of π to the subspace i(N), hence it is indeed continuous. This concludes the proof.

Example 4.16 Let M,N be manifolds, and let F : M → N be a smooth map. Consider the map

ΓF : M → M × N, p 7→ ΓF (p) := (p,F(p)).

From Exercise 2.7, ΓF is a smooth map. More precisely, it is a section of the projection

prM : M × N → M, (p,q) 7→ prM(p,q) := p onto the ﬁrst factor. So, from Proposition 4.3.2, ΓF is an embedding. In particular, ΓF (M) is a submanifold of the same dimension as M in the product M × N. As ΓF (M) = graphF, the latter fact is not new (see Example 2.15). The new fact is that ΓF (M) is, additionally, diffeomorphic to

M. 1 1 Example 4.17 — The Segre Embedding. We want to show that the product RP ×RP of two copies of the projective line RP1 can be embedded into the 3-dimensional projective space RP3. More precisely, there is an embedding, called the Segre embedding,

1 1 3 σ : RP × RP → RP , given by

0 1 0 1 0 0 0 1 1 0 1 1 0 1 0 1 1 σ s : s , t : t := s t : s t : s t : s t , s : s , t : t ∈ RP . (4.1) To see this, first notice that σ is well-defined by (4.1) (do you see it?). Second, we show that σ is a smooth map. Denote by (U0,ϕ0),(U1,ϕ1) the usual affine charts 1 3 on RP , and by (Vi,ψi) the affine charts on RP , i = 0,...,3. It is clear that

σ(U0 ×U0) ⊆ V0, σ(U0 ×U1) ⊆ V1, σ(U1 ×U0) ⊆ V2, σ(U1 ×U1) ⊆ V3, showing that σ has coordinate representations around every point. Let’s compute the ﬁrst one:

3 σb : Ub0 ×Ub0 = R × R → Vb0 = R . (4.2) For any (P,Q) ∈ R × R we have −1 σb(P,Q) = (ψ0 ◦ σ ◦ (ϕ0 × ϕ0) )(P,Q) = (ψ0 ◦ σ)([1 : P],[1 : Q])

= ψ0 ([1 : Q : P : PQ]) = (Q,P,PQ), which depends smoothly on (P,Q). Similarly, all other coordinate representations are smooth (check it as an exercise). Third, we show that σ is an immersion. Denote by x,y the standard coordinates on Ub0 ×Ub0 = R × R. Then, the Jacobian matrix of the coordinate representation (4.2) is

1 1  ∂σb ∂σb    ∂x ∂y 0 1 2 2  ∂σb ∂σb  = 1 0 ,  ∂x ∂y    3 3 ∂σb ∂σb y x ∂x ∂y 4.3 Embeddings 85 whose rank is constant and equal to 2. This shows that rankp σ = 2 for all p ∈ U0 ×U0. Similarly the rank of σ is 2 on any other point (check it as an exercise), and σ is an immersion. Fourth, we show that σ is injective. So let

p = s0 : s1,t0 : t1 andp ˜ = s˜0 :s ˜1,t˜0 : t˜1 be two points in RP1 × RP1 such that σ(p) = σ(p˜). This means that there exists a non-zero scalar λ ∈ R such that

sit j = λs˜it˜j, for all i, j ∈ {0,1}.

Now, let j be such that t j 6= 0. Then

λt˜j si = s˜i, for all i ∈ {0,1}. t j Hence, s0 : s1 = s˜0 :s ˜1. Similarly, t0 : t1 = t˜0 : t˜1, so that p = p˜. We conclude proving that σ(RP1 × RP1) ⊆ RP3 is a 2-dimensional submanifold. To do this, we show that it coincides with the submanifold S from Example 2.17. So let q = P0 : ··· : P3 be a point in RP3. We claim that q is in the image of σ if and only if

P0P3 − P1P2 = 0. (4.3)

The “only if part” of the claim follows from an easy computation that we leave as an exercise. For the if part, notice that the condition (4.3) can be re-written in the form

P0 P1 det = 0. (4.4) P2 P3

Let (P0,P1) 6= (0,0). Then, from (4.4), there exists a scalar a ∈ R such that

(P2,P3) = (aP0,aP1), so that

q = P0 : ··· : P3 = P0 : P1 : aP0 : aP1 = σ [1 : a],P0 : P1.

The case (P2,P3) 6= (0,0) can be discussed in a similar way and we leave the obvious details to the reader. Summarizing, we have proved that σ is an embedding. More generally, for any two non-negative n m n+m+ integers n,m there is an embedding, also called the Segre embedding, σ : RP × RP → RP 1. We will not discuss the details of the latter construction.

Vector Fields and Differential II Forms

5 Vector Fields ...... 89 5.1 Vector Fields 5.2 Vector Fields and Fields of Vectors 5.3 Vector Fields and Smooth Maps

6 Flows and Symmetries ...... 103 6.1 Integral Curves of a Vector Field 6.2 Flow of a Vector Field 6.3 Symmetries and Inﬁnitesimal Symmetries

7 Covectors and Differential 1-Forms .. 127 7.1 Covectors and the Cotangent Bundle 7.2 Differential 1-Forms and Fields of Covectors 7.3 Differential 1-Forms and Smooth Maps

8 Differential Forms and Cartan Calculus 143 8.1 Algebraic Preliminaries: Alternating Forms 8.2 Higher Degree Differential Forms 8.3 Cartan Calculus

Index ...... 173

5. Vector Fields

In this chapter, we continue developing of a differential calculus on manifolds, defining vector fields. There are two alternative (but equivalent) definitions of a vector field on a manifold. Geometrically, a vector field on a manifold M is the assignment of a tangent vector Xp to M at the point p, for every point p ∈ M, what we also call a field of vectors. Additionally, we require that Xp depends smoothly on p. This can be formalized rigorously via sections of the tangent bundle. There is another possible algebraic definition of a vector field, which is particularly efficient, and that we adopt as the primary definition: vector fields on M are derivations of the algebra C∞(M) of smooth functions on M. This latter definition can be easily transported to other areas of Geometry like Algebraic Geometry or Supergeometry. We will provide both definitions and discuss their equivalence. We also discuss how do vector fields interact with smooth maps. In the next chapter we will discuss two important (and interrelated) interpretations of vector fields: vector fields as ordinary differential equations, and vector fields as infinitesimal diffeomorphisms.

5.1 Vector Fields The most intuitive definition of a vector field on a manifold M is the geometric one: a vector field is a field of vectors, i.e. the assignment of a tangent vector Xp ∈ TpM for each p ∈ M. We will always require that Xp depends smoothly on p. This can be formalized noticing that a field of vectors is the same as a section s : M → TM of the tangent bundle (see Section 3.3), for which the smoothness is a well-defined property. However, in these lecture notes, we will adopt a different, more algebraic, definition, similar in spirit to the very definition of a tangent vector. This choice is, partly, a matter of personal taste, but, for the remaining part, is motivated by the efficiency of Algebra in several different situations in Differential Geometry. We begin with a purely algebraic definition: let A be a real algebra. Definition 5.1.1 — Derivation. A derivation of A is an R-linear map

X : A → A 90 Chapter 5. Vector Fields

Figure 5.1: A ﬁeld of vectors on a manifold M.

satisfying the following Leibniz rule: for all α,β ∈ A

X(αβ) = αX(β) + βX(α). Now, let M be a manifold. Definition 5.1.2 — Vector Field. A vector field on M is a derivation of the algebra C∞(M) of smooth functions on M. The set of all vector fields on M is denoted by X(M).

In this section we elaborate on Definition 5.1.2, discussing its algebraic consequences for the space of vector fields. In the next section, we show that vector fields on M are equivalent to (smooth) fields of vectors, i.e. (smooth) sections of the tangent bundle TM.

Example 5.1 — Coordinate Vector Fields. Let M be an n-dimensional manifold and let (U,ϕ = (x1,...,xn)) be a chart on M. For each i = 1,...,n, deﬁne the map

∂ ∂ ∂ fb : C∞(U) → C∞(U), f 7→ f := ◦ ϕ, ∂xi ∂xi ∂ti where, as usual, fb= f ◦ ϕ−1 is the coordinate representation of f . Notice that, for each f ∈ C∞(U), ∂ ∂ fb the coordinate representation of ∂xi f is exactly ∂ti :

d∂ ∂ fb f = . (5.1) ∂xi ∂ti

∂ The map ∂xi is called the i-th coordinate vector ﬁeld and it is indeed a vector ﬁeld on U (beware, not on M!). The proof of this claim is left as Exercise 5.1.

Exercise 5.1 ∂ Show that the i-th coordinate vector field ∂xi is a vector field on U. We now show that the space X(M) of vector fields on M does actually carry some algebraic structures. We begin defining the sum of two vector fields and the product of a vector field by a smooth function. So let X,Y ∈ X(M) be two vector fields, and let f ∈ C∞(M) be a smooth function. We define new maps:

X +Y : C∞(M) → C∞(M), g 7→ (X +Y)(g) := X(g) +Y(g), 5.1 Vector Fields 91 and

f · X : C∞(M) → C∞(M), g 7→ ( f · X)(g) := f · X(g)

Exercise 5.2 Show that (1) for every X,Y ∈ X(M), X +Y is a vector ﬁeld; (2) for every X ∈ X(M), and every f ∈ C∞(M), f · X is a vector ﬁeld; (3) (X(M),+,·) is a C∞(M)-module.

We conclude this section discussing one more algebraic structure on X(M): vector fields form a Lie algebra. We begin with the definition of Lie algebra. Definition 5.1.3 — Lie Algebra. A real Lie algebra, or simply a Lie algebra, is a real vector space g equipped with one more interior operation

[−,−] : g × g → g, (v,w) 7→ [v,w],

called the Lie bracket, with the following properties: X [−,−] is R-bilinear; X [−,−] is skew-symmetric, i.e. [v,w] = −[w,v] for all v,w ∈ g; X [−,−] satisﬁes the following Jacobi identity: [u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0 for all u,v,w ∈ g.

A Lie subalgebra of a Lie algebra g is a vector subspace h ⊆ g which is preserved by the Lie bracket, i.e. [v,w] ∈ h, for all v,w ∈ h.

Notice that a Lie subalgebra h of a Lie algebra g is also a Lie algebra with the restricted Lie bracket. The following example is our main example of a Lie algebra (besides vector ﬁelds).

Example 5.2 — Lie Algebra of Endomorphisms. Let V be a real vector space. The real vector space EndR(V) of (R-linear) endomorphisms of V is a Lie algebra with the following Lie bracket: [−,−] : (h,l) 7→ [h,l] := h ◦ l − l ◦ h.

Prove this claim as an exercise! The Lie bracket in EndR(V) is called the commutator, and two endomorphisms h,l ∈ EndR(V) are said to commute if their commutator vanishes: [h,l] = 0.

Proposition 5.1.1 — Lie Algebra of Vector Fields. Let M be a manifold. Vector fields on M ∞ form a Lie subalgebra of the Lie algebra EndR(C (M)) of R-linear endomorphisms of the real vector space C∞(M). In other words, the commutator of two vector fields is a vector field.

Proof. Left as Exercise 5.3.

Exercise 5.3 Prove Proposition 5.1.1.

In particular, the space X(M) of vector ﬁelds, with the commutator, is a Lie algebra. Notice that the commutator of vector ﬁelds is R-bilinear, but it is not C∞(M)-bilinear (in general). Actually, the C∞(M)-module structure and the Lie algebra structure on X(M) interact in a more complicated (but still nice) way, as explained in the next 92 Chapter 5. Vector Fields

Proposition 5.1.2 Let X,Y ∈ X(M) be vector ﬁelds and let f ∈ C∞(M) be a smooth function. Then

[X, fY] = f [X,Y] + X( f )Y.

Proof. A simple computation left as Exercise 5.4.

Exercise 5.4 Prove Proposition 5.1.2.

5.2 Vector Fields and Fields of Vectors Let M be a manifold. In this section, we prove that vector fields on M are equivalent to fields of vectors. For a precise statement see Proposition 5.2.1. We begin noticing that a vector field X ∈ X(M) determines a field of vectors. Indeed, for each point p ∈ M, we can consider the map

∞ AseXp : C (M) → R, f 7→ Xp( f ) := X( f )(p). (5.2)

It can be proved that Xp is a tangent vector to M at p (see Exercise 5.5), called the value at p of the vector ﬁeld X.

Exercise 5.5 Prove that, for every p ∈ M, the map Xp deﬁned in (5.2) is a tangent vector to M at the point p. Prove also that, for every X,Y ∈ X(M), and every f ∈ C∞(M),

(X +Y)p = Xp +Yp,

( f X)p = f (p)Xp.

Finally, let (U,ϕ = (x1,...,xn)) be a chart around p. Prove that the value of the coordinate ∂ ∂ vector ﬁeld ∂xi at p, is exactly the coordinate tangent vector ∂xi |p.

The assignment p 7→ Xp is a ﬁeld of vectors that can be interpreted as a(n a priori non-necessarily smooth) section sX of the tangent bundle TM:

sX : M → TM, p 7→ sX (p) := Xp.

We are now ready to state the main result of this section.

Proposition 5.2.1 The map

X(M) → Γ(TM), X 7→ sX (5.3)

is a well-deﬁned C∞(M)-module isomorphism.

Proof. We have to check several things. First of all, we have to check that, for every X ∈ X(M), 1 n the section sX is actually smooth. It is enough to check that, for every chart (U,ϕ = (x ,...,x )) i on M, the components sX of the section sX in the chart (U,ϕ) (as deﬁned at the end of Section i 3.3) are smooth functions on U. To do this, we ﬁx an arbitrary point p0 in U and show that sX is i smooth around p0 (the smoothness of sX then follows from the arbitrariness of p0). So, let p ∈ U and compute

i i i i sX (p) = x˙ (sX (p)) = x˙ (Xp) = i-th component of Xp in the coordinate frame = Xp(x ). 5.2 Vector Fields and Fields of Vectors 93

i ∞ i i In order to continue our computation, we need to choose a function xe ∈ C (M) such that xe = x in an open neighborhood V ⊆ U of p0. Such function exists in view of the Local Extension Lemma (Lemma 2.3.6), and, from the locality of tangent vectors, we have that i i i i sX (p) = Xp(x ) = Xp(xe) = X(xe)(p) i for all p ∈ V. As X maps smooth functions to smooth functions, we conclude that sX |V is smooth, and (5.3) is well-defined. We leave it as Exercise 5.6 (Point (1)) to check that the map (5.3) is C∞(M)-linear. It remains to check that (5.3) is invertible (the C∞(M)-linearity of the inverse then follows from general properties of module isomorphisms). To do this we define its inverse. Namely, given a section s ∈ Γ(TM) of the tangent bundle, we construct a vector field Xs ∈ X(M) in such a way that

sXs = s and XsX = X, (5.4) for all s ∈ Γ(TM), and all X ∈ X(M), i.e. in such a way that the map Γ(TM) → X(M), s 7→ Xs inverts (5.3). For every f ∈ C∞(M) we deﬁne

Xs( f ) : M → R, p 7→ Xs( f )(p) := s(p)( f ). We have to show that ∞ ∞ X Xs( f ) ∈ C (M) for all s ∈ Γ(TM) and all f ∈ C (M), ∞ ∞ ∞ X Xs : C (M) → C (M) is a derivation of C (M), hence a vector ﬁeld, and X the identities (5.4) hold for all s ∈ Γ(TM) and all X ∈ X(M). We only discuss the ﬁrst point and leave it to the reader to prove the remaining two points as ∞ Exercise 5.6 (Points (2) and (3)). Let s ∈ Γ(TM) and f ∈ C (M). In order to prove that Xs( f ) is smooth, we prove that the restriction Xs( f )|U is smooth for every chart (U,ϕ). So, let p ∈ U, and compute

Xs( f )(p) = s(p)( f ) = s(p)( f |U ) i ∂ i ∂ i ∂ = s (p) i |p f |U = s i f |U = s i ( f |U )(p), ∂x ∂x p ∂x i i ∂ where, in the ﬁrst step of the second line, we used Exercise 5.5. As the s are smooth, s ∂xi is a vector ﬁeld on U, and the last expression depends smoothly on p. This concludes the proof.

Exercise 5.6 Complete the proof of Proposition 5.2.1 showing that ∞ (1) the map X 7→ sX is C (M)-linear, ∞ ∞ (2) for every s ∈ Γ(TM), the map Xs : C (M) → C (M) is a derivation, and (3) the map s 7→ Xs inverts X 7→ sX .

Proposition 5.2.1 has some important consequences. First of all, a vector ﬁeld X on M can be “restricted” to an open submanifold U ⊆ M. Namely, X corresponds to a section s : M → TM of TM. Restricting s to U in the domain and to TU in the codomain, we get a section s|U : U → TU of TU . Finally, s|U corresponds to a vector ﬁeld on U that we denote X|U , and call the restriction of X to U . Notice that, by construction, for every point p ∈ U , the value of X|U at p agrees with the value of X at the same point p:

(X|U )p = Xp, for all p ∈ U . ∞ Hence, we also have X|U ( f |U ) = X( f )|U for all X ∈ X(M), and all f ∈ C (M). Now, let X ∈

Lemma 5.2.2 — Gluing Lemma for Vector Fields. Let M be a manifold, and let C = {U } be an open cover of M. For every U ∈ C , let XU be a vector ﬁeld on U . If

XV |U ∩V = XU |U ∩V

for every U ,V ∈ C , then there exists a unique vector ﬁeld X on M such that

X|U = XU

for all U ∈ C .

Proof. Left as Exercise 5.7.

Exercise 5.7 Prove the Gluing Lemma for Vector Fields (Lemma 5.2.2).

A third consequence of Proposition 5.2.1 is that, for every chart (U,ϕ), the C∞(U)-module of vector fields on U possesses a finite frame consisting of the coordinate vector fields, according to the following

Proposition 5.2.3 Let M be an n-dimensional manifold, and let (U,ϕ = (x1,...,xn)) be a chart on M. Then the coordinate vector ﬁelds ∂ ∂ ,..., ∂x1 ∂xn

form a frame in X(U) (called the coordinate frame), i.e. they are independent and generate X(U). In particular, every vector ﬁeld X ∈ X(U) can be uniquely written in the form

∂ X = Xi ∂xi for some smooth functions Xi ∈ C∞(U).

Proof. We ﬁrst prove that the coordinate vector ﬁelds are independent. So let f 1,..., f n be smooth functions on U such that ∂ f i = 0. ∂xi

i ∂ ∞ j This means that f ∂xi g = 0 for all g ∈ C (U). In particular, we can chose g = x , j ∈ {1,...,n}. ∂ j j From (5.1), ∂xi x = δi , hence ∂ 0 = f i x j = f iδ j = f j. ∂xi i It remains to prove that every vector ﬁeld X ∈ X(U) can be written in the form

∂ X = Xi ∂xi i ∞ for some smooth functions X ∈ C (U). To do this, consider the section sX ∈ Γ(TU) corresponding i i to U, and put X = sX . This means that, for every p ∈ U i ∂ i ∂ Xp = X (p) i |p = X i , ∂x ∂x p 5.2 Vector Fields and Fields of Vectors 95

i ∂ where we used Exercise 5.5. As X and X ∂xi share the same values (in points of U), they must coincide. We can combine the Gluing Lemma 5.2.2 with Proposition 5.2.3, to ﬁnd that, for every vector ﬁeld X on M, (1) X is completely determined by its restrictions X|U to some coordinate domains U covering M, 1 n (2) for every chart (U,ϕ = (x ,...,x )), the restriction X|U can be uniquely written in the form

∂ X| = Xi U ∂xi for some smooth functions Xi ∈ C∞(U), called the components of X in the chart (U,ϕ). The proof of Proposition 5.2.3 then shows that the components Xi of a vector ﬁeld X ∈ X(M) in the chart (U,ϕ = (x1,...,xn)), coincide with the components of the corresponding section, and they are given by

Xi = X(xi).

Now, let X be a vector ﬁeld on M. How do the components Xi of X in a chart change under a change of coordinates? To answer this question take two charts (U,ϕ = (x1,...,xn)) and (U,ϕ˜ = 1 n (x˜ ,...,x˜ )) with the same coordinate domain U. Then X|U can be written either as

∂ ∂ X| = Xi , or as X| = X˜ i U ∂xi U ∂x˜i for some smooth functions Xi,X˜ i ∈ C∞(U), and we want to ﬁnd the relationship between those. We have ∂ X˜ i = X(x˜i) = X j x˜i. ∂x j In its turn, the matrix

i=1,...,n ∂ i j x˜ ∂x j=1,...,n is essentially the Jacobian matrix of the transition map ϕ˜ ◦ ϕ−1 : Ub → Ub (up to a composition with ϕ). Indeed, from (5.1),

∂ d∂ ∂xb˜i ∂(ϕ˜ ◦ ϕ−1)i x˜i = x˜i ◦ ϕ = ◦ ϕ = ◦ ϕ. ∂x j ∂x j ∂t j ∂t j where a hat “(d−)” denotes the coordinate representation with respect to the chart (U,ϕ) (not ∂ i (U,ϕe)). Yet in other words, the coordinate representation of ∂x j x˜ with respect to the chart (U,ϕ) ∂(ϕ˜◦ϕ−1)i is ∂t j :

d∂ ∂(ϕ˜ ◦ ϕ−1)i x˜i = . ∂x j ∂t j 96 Chapter 5. Vector Fields

Exercise 5.8 Consider the 2-dimensional standard Euclidean space R2 with standard coordinates x,y, and the polar chart (U,ψ = (r,φ)) on it. Here U = R2 r {y = 0, x ≥ 0}, Ub = (0,+∞) × (0,2π) and p ψ : U → Ub, (x,y) 7→ ψ(x,y) = (r,φ) := x2 + y2,φ(x,y) ,

where  arccos √ x if y ≥ 0  2 2 φ(x,y) := x +y . −arccos √ x if y < 0  x2+y2

The inverse map is

−1 −1 ψ : Ub → U, (r,φ) 7→ ψ (r,φ) = (r cosφ,r sinφ).

Consider the following vector ﬁelds on R2 ∂ ∂ ∂ ∂ X = x + y and Y = x − y . ∂x ∂y ∂y ∂x Prove that ∂ ∂ X| = r and Y| = . U ∂r U ∂φ

2 Exercise 5.9 Consider the real projective plane RP , and denote (Ui,ϕi = (xi,yi)) the standard afﬁne charts on it, i = 0,1,2. Prove that there exists a unique vector ﬁeld X ∈ X(RP2) such that

2 ∂ ∂ X|U0 = 1 + x0 + x0y0 , ∂x0 ∂y0 2 ∂ ∂ X|U1 = − 1 + x1 − x1y1 , ∂x1 ∂y1 ∂ ∂ X|U2 = x2 − y2 . ∂y2 ∂x2

5.3 Vector Fields and Smooth Maps In this section we discuss how do vector fields interact with smooth maps. Unlike for smooth functions, there is no way to transport a vector field from a manifold to another manifold along a smooth map, i.e. there is no analogue of the pull-back construction for vector fields, unless we restrict to diffeomorphisms. However, there is a way to define the compatibility of two vector fields, one on a manifold M, and the other on a manifold N, with respect to a smooth map F : M → N. So let M,N be manifolds, and let F : M → N be a smooth map. Definition 5.3.1 A vector field X on M and a vector field Y on N are F-related if

X ◦ F∗ = F∗ ◦Y. Before providing examples, we remark that the F-relatedness of two vector fields can be 5.3 Vector Fields and Smooth Maps 97 equivalently defined interpreting vector fields as fields of vectors, according to the following

Proposition 5.3.1 A vector ﬁeld X on M and a vector ﬁeld Y on N are F-related if and only if, for every p ∈ M,

(dpF)(Xp) = YF(p).

Proof. Let X ∈ X(M), and Y ∈ X(N). Then X and Y are F-related if and only if X(F∗( f )) = F∗(Y( f )) for all f ∈ C∞(N). In turn, this is equivalent to

X(F∗( f ))(p) = F∗(Y( f ))(p), for all f ∈ C∞(N), and all p ∈ M.

But

∗ ∗ X(F ( f ))(p) = Xp(F ( f )) = dpF(Xp)( f ), and

∗ F (Y( f ))(p) = Y( f )(F(p)) = YF(p)( f ).

This concludes the proof. 2 Example 5.3 In the standard Euclidean plane R with standard coordinates x,y, consider the following curve :

2 γ : R → R , t 7→ γ(t) := (cost,sint).

Consider also the vector ﬁelds

d ∂ ∂ 2 X = ∈ X(R) and Y = x − y ∈ X(R ). dt ∂y ∂x

Then X and Y are γ-related. Indeed, for every smooth function f = f (x,y) on R2 d d X(γ∗( f )) = f ◦ γ = f (cost,sint) dt dt ∂ f d cost ∂ f d sint = (cost,sint) + (cost,sint) ∂x dt ∂y dt ∂ f ∂ f = −sint (cost,sint) + cost (cost,sint) ∂x ∂y ∂ ∂ = x − y f ◦ γ = γ∗(Y( f )). ∂y ∂x

Example 5.4 Let d,n be non-negative integers, with d ≤ n, and consider the standard Euclidean d n d d n−d spaces R and R , with coordinates (t1,...,t ) and (s1,...,s ,r1,...,r ) respectively. Consider also the usual projection

n d 1 n 1 d prd : R → R , P = (P ,...,P ) 7→ prd(P) = (P ,...,P ).

The vector ﬁelds ∂ ∂ X = pr∗(Xi) +Y a d ∂si ∂ra

98 Chapter 5. Vector Fields

ℝ#$" ℝ#

' ℝ"

pr"

( ℝ"

Figure 5.2: The vector ﬁeld X and Y from Example 5.4 are prd-related.

n on R , and ∂ Y = Xi ∂ti d i i 1 d ∞ d a on R , are prd-related, for every choice of the X = X (t ,...,t ) ∈ C (R ), and the Y = a d n−d n Y (s1,...,s ,r1,...,r ) ∈ C∞(R ) (Exercise 5.10).

Exercise 5.10 Show that the vector fields X and Y from Example 5.4 are prd-related. The case when F is the inclusion of a submanifold is particularly relevant. So let M be a manifold, and let S ⊆ M be a submanifold. As usual, denote by iS : S ,→ M the inclusion. Definition 5.3.2 A vector field X on M is tangent to S if there exists a vector field Y on S such that Y and X are iS-related. In this case, Y is denoted by X|S and called the restriction of X to S.

Next proposition shows that Deﬁnition 5.3.2 agrees with our intuition on a vector ﬁeld X ∈ X(M) being tangent to a submanifold S ⊆ M.

Proposition 5.3.2 A vector ﬁeld X on M is tangent to a submanifold S ⊆ M if and only if, for every p ∈ S, Xp ∈ TpS, and, in this case, (X|S)p = Xp.

Proof. First assume that X is tangent to S, and let X|S be its restriction. Then, from Proposition 5.3.1,

Xp = XiS(p) = dpiS(X|S)p. for all p ∈ S. This shows that Xp ∈ TpS. Conversely, assume that Xp ∈ TpS for all p ∈ S. Think of X as a section sX of TM. Then the hypothesis says that sX |S takes values in TS. In particular, sX can be restricted to S in the domain and to TS in the codomain, to get a section sX |S : S → TS of TS. As TS ⊆ TM is a submanifold (Proposition 3.3.3), sX |S is a smooth section. Denote by Y ∈ X(S) the associated vector ﬁeld. By construction Yp = dpiS(Yp) = Xp for all p ∈ S, so, from Proposition 5.3.1 again, Y and X are iS-related.

5.3 Vector Fields and Smooth Maps 99

M X S

X|S

Figure 5.3: A vector ﬁeld tangent to a submanifold.

It immediately follows from the last part of Proposition 5.3.2 that, if the vector field X ∈ X(M) is tangent to a submanifold S ⊆ M, then there is a unique restriction of X to S, i.e. a unique vector field X|S on S such that X|S and X are iS-related. 3 Example 5.5 Consider the standard 3-dimensional Euclidean space R with coordinates (x,y,z), and the vector field ∂ ∂ X = y − z ∈ X(M). ∂z ∂y Use Proposition 5.3.2 to show that X is tangent to the 2-dimensional sphere S2 ⊆ R3. So let 2 2 2 2 2 P = (x0,y0,z0) ∈ S , i.e. x0 + y0 + z0 = 1, and check that XP ∈ TPS . We have ∂ ∂ ∂ ∂ XP = y − z = y0 |P − z0 |P, ∂z ∂y P ∂z ∂y 2 which is in TPS if and only if its component vector (0,−z0,y0) is orthogonal to P = (x0,y0,z0) (Example 4.11). This is indeed the case as

0 · x0 − z0 · y0 + y0 · z0 = 0.

Exercise 5.11 Consider the vector ﬁeld X from Example 5.5. As X is tangent to the sphere S2, 2 2 it can be restricted to S , and then further restricted to U+ = S r {P+}. Denote by (A,B) the stereographic coordinates on U+. Prove that

∂ B2 − A2 + 1 ∂ X| = AB + . U+ ∂A 2 ∂B

Next proposition explores the relationship between F-relatedness and the algebraic structures on vector ﬁelds.

Proposition 5.3.3 Let M,N be manifolds, let F : M → N be a smooth map, let X,X1,X2 be vector ﬁelds on M, and let Y,Y1,Y2 be vector ﬁelds on N. If X,X1,X2 are F-related to Y,Y1,Y2 respectively, then X X1 + X2 and Y1 +Y2 are F-related, 100 Chapter 5. Vector Fields

∂ B2−A2+1 ∂ Figure 5.4: The vector ﬁeld AB ∂A + 2 ∂B from Exercise 5.11.

∞ ∗ X for every f ∈ C (N), F ( f )X and fY are F-related, and X [X1,X2] and [Y1,Y2] are F-related.

Proof. Left as Exercise 5.12

Exercise 5.12 Prove Proposition 5.3.3.

Example 5.6 Let M be a manifold, let S ⊆ M be a submanifold, and let X1,X2,X be vector ﬁelds on M that are tangent to S. According to our notation, denote by X1|S,X2|S,X|S the restrictions to S of X1,X2,X respectively. It immediately follows from from Proposition 5.3.3 that X X1 + X2 is tangent to S, and (X1 + X2)|S = X1|S + X2|S, ∞ X f X is tangent to S for every f ∈ C (M), and ( f X)|S = f |SX|S, X [X1,X2] tangent to S, and [X1,X2]|S = [X1|S,X2|S] (do you see it?).

Example 5.7 Let M be an n-dimensional manifold, and let U ⊆ M be an open submanifold. As “vector fields can be restricted to open submanifolds”, every vector field X ∈ X(M) is tangent to U , and, given two vector fields X,Y ∈ X(M), the restriction [X,Y]|U of their commutator, is the 1 n commutator [X|U ,Y|U ] of the restrictions. In particular, if (U,ϕ = (x ,...,x )) is a chart on M, and ∂ ∂ X| = Xi , and Y| = Y i , U ∂xi U ∂xi then ∂ [X,Y]| = [X| ,Y| ] = Zi , U U U ∂xi where the components Zi are given by ∂ ∂ Zi = [X| ,Y| ](xi) = X| (Y| (xi))−Y| (X| (xi)) = X| (Y i)−Y| (Xi) = X j Y i −Y j Xi. U U U U U U U U ∂x j ∂x j 5.3 Vector Fields and Smooth Maps 101

Summarizing ∂ ∂ ∂ ∂ [X,Y]| = X| (Y i) −Y| (Xi) = X j Y i −Y j Xi . U U U ∂xi ∂x j ∂x j ∂xi In particular ∂ ∂ , = 0, for all i, j ∈ {1,...,n}. ∂xi ∂x j

Finally, we show that vector ﬁelds can be pulled-back along diffeomorphisms.

Proposition 5.3.4 Let M,N be manifolds, and let Y be a vector ﬁeld on N. Then there exists a unique vector ﬁeld Φ∗(Y) on M, called the pull-back of Y along Φ, such that Φ∗(Y) and Y are Φ-related. The pull-back Φ∗(Y) is given by

Φ∗(Y) = Φ∗ ◦Y ◦ (Φ−1)∗

or, equivalently,

∗ −1 Φ (Y)p = (dΦ(p)Φ )(YΦ(p)), for all p ∈ M. (5.5)

∞ Additionally, for any three vector ﬁelds Y,Y1,Y2 on N and any function f ∈ C (N), we have ∗ ∗ ∗ X Φ (Y1 +Y2) = Φ (Y1) + Φ (Y2), ∗ ∗ ∗ X Φ ( fY) = Φ ( f )Φ (Y), ∗ ∗ ∗ X Φ ([Y1,Y2]) = [Φ (Y1),Φ (Y2)]. Finally, if Ψ : N → Q is another diffeomorphism, and Z ∈ X(Q), then

∗ ∗ ∗ ∗ idQ(Z) = Z, and (Ψ ◦ Φ) (Z) = Φ (Ψ (Z)).

Proof. Left as Exercise 5.13

Exercise 5.13 Prove Proposition 5.3.4.

1 n Example 5.8 Let M be an n-dimensional manifold, and let (U,ϕ = (x ,...,x )) be a chart on M. Then, every function f ∈ C∞(U) can be seen as the pull-back ϕ∗( fb) of its coordinate representation fbalong the diffeomorphism ϕ : U → Ub. It follows that Identity (5.1) can be rephrased as follows: ∂ ∂ = ϕ∗ (5.6) ∂xi ∂ti

(do you see it?).

1 n R Let M be a manifold, and let (U,ϕ = (x ,...,x )) be a chart on it. Identity (5.6) basically says that, if we identify U and U via , then the coordinate vector ﬁelds ∂ identify with b ϕ ∂xi the standard partial derivatives ∂ . In particular, the ∂ inherits all standard properties of ∂ti ∂xi standard derivatives, and, to stress this, in the following, for every function g on U, we simply write ∂g ∂ ∂ 2g ∂ ∂ for g, or for g, etc. ∂xi ∂xi ∂xi∂x j ∂xi ∂x j Let’s make one example, take another manifold N, a chart (V,ψ = (y1,...,yn)) on N, and a smooth map F : M → N such that F(U) ⊆ V. We already know that the restriction F : U → V 102 Chapter 5. Vector Fields

is completely determined by its components Fa = F∗(ya). Take a function f ∈ C∞(N), and its pull-back F∗( f ) = f ◦ F. It is easy to see using (5.6) that:

∂( f ◦ F) ∂Fa ∂ f = · ◦ F (5.7) ∂xi ∂xi ∂ya

(check it as an exercise) which is the chain rule for coordinate vector ﬁelds. In the following, we will apply this and similar formulas without further comments. 6. Flows and Symmetries

In this chapter, we provide two more interpretations of vector fields. First of all, a vector field can be interpreted as an ordinary differential equation (ODE), more precisely, a system of explicit and autonomous ordinary differential questions. The solutions of such system are the so called integral curves of the vector field, and the Theorem of Existence, Uniqueness, and Smooth Dependence on Initial Data of Solutions of a System of ODEs from Calculus guarantees that integral curves are particularly well-behaved. Secondly, a vector field can be seen as an infinitesimal diffeomorphism. Points of a manifold move along the integral curves of a vector field, and the vector field says what is the velocity of this displacement. Hence a vector field can be seen as an infinitesimal displacement of every point of the manifold. This displacement is smooth and (morally) one-to-one, hence it is a(n infinitesimal) diffeomorphism. This informal discussion will be made more rigorous along the following sections.

6.1 Integral Curves of a Vector Field Let M be a manifold, and let X be a vector field on M. In this chapter, we will always denote by t the canonical coordinate on R and on any open interval I ⊆ R. Definition 6.1.1 — Integral Curve. An integral curve of X is a curve γ : I → M on M, defined d on some open interval I ⊆ R, such that the canonical coordinate vector field dt on I and the vector field X are γ-related, i.e. d ◦ γ∗ = γ∗ ◦ X, dt or, equivalently,

d γ˙(t ) = d γ | = X , for every t ∈ I. 0 t0 dt t0 γ(t0) 0

If t0 ∈ I and γ(t0) = p0 ∈ M, we say that γ passes through p0 at time t0. In this case, if t0 = 0 104 Chapter 6. Flows and Symmetries

(so γ(0) = p0), we also say that γ starts from p0. Now, we show that, morally, an integral curve is the solution of system of ODEs. So, let γ : I → M be a curve, and let t0 ∈ I. Denote p0 = γ(t0). As γ is smooth we can choose an open 1 n subinterval J ⊆ I containing t0, and a chart (U,ϕ = (x ,...,x )) on M around p0, such that γ(J) ⊆U. i ∗ i Denote by γ = γ (x ) : J → R the components of γ in the chart (U,ϕ). Then, for every t ∈ J,

dγi ∂ γ˙(t) = (t) | dt ∂xi γ(t) (see (3.13)). On the other hand

∂ X| = Xi , U ∂xi for some Xi ∈ C∞(U), and

∂ X = Xi(γ(t)) | . γ(t) ∂xi γ(t) We conclude that γ : J → M is an integral curve of X if and only if

dγi (t) = Xi(γ(t)) = Xi(γ1(t),...,γn(t)), for all t ∈ J, dt b which is an ODE for the coordinate representation

1 n γb: J → Ub, t 7→ γb(t) = (γ (t),...,γ (t)) of γ. Most of the properties of integral curves are corollaries of the following standard result in Calculus, which we state without a proof.

Theorem 6.1.1 — Existence, Uniqueness, and Smooth Dependence on Initial Data of n Solutions of Systems of Explicit, Autonomous ODEs. Let U ⊆ R be an open subset, and let

1 n n Xb = Xb ,...,Xb : U → R

be a smooth (vector valued) map. Consider the following system of ODEs

dγi (t) = Xi(γ1(t),...,γn(t)), i ∈ {1,...,n}, (6.1) dt b

1 n imposed on a curve γb= (γ ,...,γ ) : I → U on U. We have the following. (1) UNIQUENESS. For every t0 ∈ R, every two solutions γ1 : I1 → U, and γ2 : I2 → U of (6.1) such that t0 ∈ I1 ∩ I2, and γ1(t0) = γ2(t0), agree on the (non-empty) intersection I1 ∩ I2 of

their domains, i.e. γ1|I1∩I2 = γ2|I1∩I2 . (2) EXISTENCE. For every t0 ∈ R, and every P0 ∈ U, there is an open interval I containing t0, and an open neighborhood V ⊆ U of P0, such that, for every P ∈ V, there exists a (necessarily unique) solution γP : I → U of (6.1) such that γP(t0) = P. (3) SMOOTH DEPENDENCEON INITIAL DATA. For every t0 ∈ R, and every P0 ∈ U, let I, V, P and γP be as in Point (2) EXISTENCE. Then the map Φ : I ×V → U, (t,P) 7→ Φ(t,P) := γP(t) depends smoothly on (both t and) P.

Proof. Omitted. 6.1 Integral Curves of a Vector Field 105

A first corollary of Theorem 6.1.1 is the existence and uniqueness of maximal integral curves of a vector field (starting from a prescribed point). Before stating this result, we have to define maximal integral curves. So let M be a manifold, let X be a vector field on M, and let p ∈ M be a point. Denote by Ip the set of integral curves of X starting from p. It is easy to see that Ip is non-empty, i.e., for every p ∈ M, there exists an integral curve starting from p. Indeed, choose a chart (U,ϕ) around p and look for an integral curve γ : I → M taking values in U. From Theorem 6.1.1, and the discussion preceding it, such γ exists. Next, any two integral curves γ1 : I1 → M, γ2 : I2 → M in Ip, agree on the intersection I1 ∩ I2 of their domains. This follows from Theorem 6.1.1 again and a little bit of Topology that we omit. Finally, Ip is equipped with a partial order ≤ defined as follows: let γ1 : I1 → M and γ2 : I2 → M be in Ip, i.e. γ1,γ2 are integral curves starting from p, then

γ1 ≤ γ2 if, by deﬁnition, I1 ⊆ I2 (and, in this case, necessarily γ1 = γ2|I1 ).

A maximal integral curve starting from p is, by deﬁnition, a maximal element of Ip with respect to this partial order, i.e. an integral curve γ of X starting from p such that, whenever γ0 is another integral curve starting from p such that γ ≤ γ0, in fact we have γ = γ0. Yet in other words, γ cannot be extended to an integral curve starting from p and deﬁned on a larger time interval.

Proposition 6.1.2 — Existence and Uniqueness of Maximal Integral Curves. Let M be a manifold, and let X be a vector ﬁeld on M. Then, for every point p ∈ M, there exists a unique maximal integral curve γp : Ip → M of X starting from p, and, for every other integral curve γ : I → M that starts from p, we have γ ≤ γp, i.e. I ⊆ Ip, and γ = γp|I.

Proof (a sketch). Deﬁne Ip ⊆ R as the union of the domains of all integral curves starting from p: [ Ip := I.

(γ:I→M)∈Ip

Being the union of open intervals containing 0, Ip is itself an open interval containing 0. Finally, we deﬁne a curve

γp : Ip → M as follows. For every t ∈ Ip, let γ : I → M be any integral curve starting from p such that t ∈ I, and put

γp(t) := γ(t).

From the above discussion, γp(t) is independent of the choice of γ. Additionally, γp agrees with γ in a whole open neighborhood of t in Ip. Hence, from the Gluing Lemma for Smooth Maps, γp is a well-deﬁned curve. As γp agrees locally, around every time in Ip, with an integral curve, it is an integral curve itself, and, by construction, its domain is the largest possible, so γp is maximal. 2 Example 6.1 On R with standard coordinates (x,y) consider the coordinate vector ﬁeld ∂ X = . ∂x

A curve γ = (x(t),y(t)) : I → R2 is an integral curve of X, if it is a solution of the system of ODEs:

dx dt (t) = 1 dy . dt (t) = 0 106 Chapter 6. Flows and Symmetries

This means that x(t) = x(0) +t . y(t) = y(0)

Hence, the maximal integral curve of X starting from P = (x0,y0) is

2 γP : R → R , t 7→ γP(t) = (x0 +t,y0).

∂ Figure 6.1: The coordinate vector ﬁeld ∂x and its integral curves.

2 Example 6.2 On R with standard coordinates (x,y) consider the vector ﬁeld ∂ ∂ X = x + y . ∂x ∂y

A curve γ = (x(t),y(t)) : I → R2 is an integral curve of X, if it is a solution of the system of ODEs:

dx dt (t) = x(t) dy . dt (t) = y(t) This means that x(t) = x(0)et . y(t) = y(0)et

Hence, the maximal integral curve of X starting from P = (x0,y0) is

2 t t γP : R → R , t 7→ γP(t) = (x0e ,y0e ).

6.1 Integral Curves of a Vector Field 107

∂ ∂ Figure 6.2: The vector ﬁeld x ∂x + y ∂y and its integral curves.

2 Example 6.3 On R with standard coordinates (x,y) consider the vector ﬁeld ∂ ∂ X = x − y . ∂y ∂x A curve γ = (x(t),y(t)) : I → R2 is an integral curve of X, if it is a solution of the system of ODEs: dx dt (t) = −y(t) dy . dt (t) = x(t) This means that x(t) = x(0)cost − y(0)sint . y(t) = x(0)sint + y(0)cost

Hence, the maximal integral curve of X starting from P = (x0,y0) is 2 γP : R → R , t 7→ γP(t) = (x0 cost − y0 sint,x0 sint + y0 cost).

Exercise 6.1 On R2 with standard coordinates (x,y), ﬁnd the maximal integral curves of the vector ﬁeld ∂ ∂ X = x − y . ∂x ∂y

Notice that maximal integral curves need not be deﬁned on the whole R as the following counter-example shows.

Example 6.4 Consider the standard line R with standard coordinate x and the open submanifold d (0,+∞) ⊆ R in it. Additionally, consider the coordinate vector ﬁeld X = dx |(0,+∞) on (0,+∞). Let x0 ∈ R. Any integral curve γ : I → R of X starting from x0 is of the type

γ : I → (0,+∞), t 7→ γ(t) = x0 +t. 108 Chapter 6. Flows and Symmetries "

∂ ∂ Figure 6.3: The vector ﬁeld x ∂y − y ∂x and its integral curves.

In particular, −x0 cannot belong to I, not even when I = Ix0 and γ is the maximal integral curve starting from x0.

We can change the “initial point” of an integral curve, by composing with a translation. Namely, let t¯ ∈ R. Then the translation

τt¯ : R → R, t 7→ τt¯(t) := t +t¯.

is a diffeomorphism whose inverse is τ−t¯. For any open interval I ⊆ R, we denote by I +t¯ := τt¯(I) its image under the translation τt¯.

Lemma 6.1.3 — Translation Lemma. Let M be a manifold, let X be a vector ﬁeld on M, and let γ : I → M be an integral curve of X. Then, for every t¯ ∈ R, the composition

γ ◦ τt¯ : I −t¯ → M

is an integral curve. If, additionally, I = Ip, for some point p ∈ M, γ = γp : Ip → M is the maximal ¯ ¯ ¯ integral curve of X that starts from p, and t ∈ Ip, then Ip −t = Iγp(t¯), and γ ◦ τt¯ : Ip −t → M is the maximal integral curve of X that starts from γp(t¯).

Proof. For the ﬁrst part of the statement, we compute the velocity

d(γ ◦ τ ) t¯ (t) dt 6.1 Integral Curves of a Vector Field 109

∂ ∂ Figure 6.4: The vector ﬁeld x ∂x − y ∂y .

∞ of the curve γ ◦ τt¯ at time t ∈ I −t¯. So, let f ∈ C (M) and compute

d(γ ◦ τ¯) d t (t)( f ) = f (γ(τ (t))) dt dt t¯ d = | f (γ(s +t¯)) ds s=t d 0 d = | 0 f (γ(s )) · | (s +t¯) ds0 s =t+t¯ ds s=t dγ = (t +t¯)( f ) dt = Xγ(t+t¯)( f )

= X(γ◦τt¯)(t)( f ).

This shows that γ ◦ τt¯ is an integral curve. For the second part of the statement, let p ∈ M be a point and consider the maximal integral curve γp : Ip → M that starts from p. For every t¯ ∈ Ip, γ ◦ τt¯ : Ip − t¯ → M is an integral curve that starts from γp(t¯). Indeed, ﬁrst of all, as t¯ ∈ Ip, then 0 = t¯−t¯ ∈ Ip −t¯. Now,

(γp ◦ τt¯)(0) = γp(0 +t¯) = γp(t¯). ¯ It remains to prove that γ ◦ τt¯ is a maximal integral curve (it will then follow that Ip −t = Iγp(t¯)). So, 0 0 let γ : I → M be another integral curve starting from γp(t¯) and such that

0 0 γ ◦ τt¯ ≤ γ , i.e. Ip −t¯ ⊆ I . In particular,

0 γ |Ip−t¯ = γp ◦ τt¯. 0 0 We also have that γ ◦ τ−t¯ : I +t¯ → M is an integral curve. Additionally,

0 I +t¯ ⊃ Ip −t¯+t¯ = Ip 3 0, 110 Chapter 6. Flows and Symmetries

so that 0 ∈ I0 +t¯, and

0 0 0 (γ ◦ τ−t¯)(0) = γ (0 −t¯) = γ (−t¯) = γp(τt¯(−t¯)) = γp(t¯−t¯) = γp(0) = p,

0 0 where we used that −t¯ = 0 −t¯ ∈ Ip −t¯. Summarizing, γ ◦ τ−t¯ : I +t¯ → M is an integral curve that 0 starts from p. As γp is the maximal integral curve that starts from p, then γ ◦ τ−t¯ ≤ γp, i.e. 0 0 I +t¯ ⊆ I ⇒ I ⊆ Ip −t¯.

0 0 We conclude that I = Ip −t¯, and γ = γp ◦ τt¯. This shows that γp ◦ τt¯ is a maximal integral curve and concludes the proof. We conclude this section discussing the relationship between integral curves and smooth maps.

Proposition 6.1.4 Let M,N be smooth manifolds, let F : M → N be a smooth map, and let X,Y be vector ﬁelds on M,N respectively. Then X and Y are F-related if and only if F maps integral curves of X to integral curves of Y, i.e., for every integral curve γ : I → M of X, the curve F ◦ γ : I → N is an integral curve of Y.

Proof. Assume that X and Y are F-related, and let γ : I → M be an integral curve of X. Then, d d ◦ (F ◦ γ)∗ = ◦ γ∗ ◦ F∗ (the pull-back construction inverts the compositions) dt dt = γ∗ ◦ X ◦ F∗ (γ is an integral curve of X) = γ∗ ◦ F∗ ◦Y (X is F-related to Y) = (F ◦ γ)∗ ◦Y (the pull-back construction inverts the compositions),

showing that F ◦ γ is an integral curve of Y. Conversely, assume that F maps integral curves of X to integral curves of Y, and let p ∈ M be a point. There exists an integral curve of X, say γ : I → M, starting from p (e.g. the maximal one), and by hypothesis, F ◦ γ : I → N is an integral curve of Y, hence, dγ d(F ◦ γ) d F(X ) = d F (0) = (0) = Y = Y , p p γ(0) dt dt F(γ(0)) F(p)

where, in the second step, we used Identity (3.14). This concludes the proof.

6.2 Flow of a Vector Field Let M be a manifold, and let X be a vector field on M. A point p ∈ M “moves” along the maximal integral curve γp : Ip → M. After a time t, it has moved to γp(t). Fix t ∈ R and move all points p of M in this way. We get a map Φt : M → M, p 7→ Φt (p) := γp(t) (possibly well-defined only on a suitable subset of M). Roughly, the flow of X is, by definition, the 1-parameter family of maps {Φt }t (see Figure 6.5 for a pictorial representation). The flow of X has several noteworthy properties that we discuss in this section. But we need a precise definition of the flow first. Notice, preliminarily, that a 1-parameter family of maps Φt : M → M, t ∈ R, can be encoded in a single map Φ : R × M → M by putting Φ(t, p) = Φt (p). Given a vector field X on a manifold M, we will define the flow of X as a map Φ : D → M from a suitable subset D ⊆ R × M that we now identify. In practice, D will be the subset of R × M consisting of pairs (t, p) such that the expression γp(t) is well-defined. We put

D := {(t, p) ∈ R × M : t ∈ Ip} ⊆ R × M. (6.2)

6.2 Flow of a Vector Field 111

M "#(%) !

Figure 6.5: The ﬂow of a vector ﬁeld X on a manifold M.

Deﬁnition 6.2.1 — Flow of a Vector Field. The ﬂow of X is the map

Φ : D → M, (t, p) 7→ Φ(t, p) := γp(t).

The domain D ⊆ R × M of Φ will be referred to as the flow domain of X. We have some remarks on Definition 6.2.1. First of all, (6.2) guarantees that Φ is well-defined. As, 0 ∈ Ip for all p ∈ M, it is clear that D contains {0} × M ⊆ R × M. More precisely, for all p ∈ M,

D ∩ (R × {p}) = Ip × {p}.

In particular, if Ip = R for all p ∈ M, then D = R × M. However, D is usually a proper subset in R × M (see Example 6.5, and Exercise 6.2).

Example 6.5 — A Non-Complete Vector Field. Consider the standard line R, with standard d coordinate x, the open submanifold (0,+∞) ⊆ R, and the coordinate vector ﬁeld dx |(0,+∞). For any d point x0 ∈ (0,+∞), the maximal integral curve γx0 of dx |(0,+∞) starting from x0 is

γx0 : Ix0 = (−x0,+∞) → (0,+∞), t 7→ γx0 (t) := x0 +t d (see also Example 6.4). It follows that the ﬂow domain of dx |(0,+∞) is

D = {(t,x0) ∈ R × (0,+∞) : −x0 < t}, which is a proper subset in R × (0,+∞).

Exercise 6.2 Consider the standard line R, with standard coordinate x, and the open submani- d fold (−1,+1) ⊆ R. Compute the ﬂow domain of the vector ﬁeld dx |(−1,+1).

The ﬂow Φ can be encoded in a 1-parameter family of maps Φt , as follows. For any t ∈ R, deﬁne

Mt := {p ∈ M : (t, p) ∈ D} = {p ∈ M : t ∈ Ip} ⊆ M.

Notice that Mt might be the empty subset (can you provide an example?). In any case, there is a well-deﬁned map

Φt : Mt → M, p 7→ Φt (p) := Φ(t, p) = γp(t), and Φ is equivalently encoded in the family of the Φt , that we denote {Φt }t and we also call the ﬂow of X. The main properties of the ﬂow are stated in the next 112 Chapter 6. Flows and Symmetries (0, +∞)

()

' −()

d Figure 6.6: The ﬂow domain of dx |(0,+∞).

Theorem 6.2.1 — Fundamental Theorem of Flows. Let M be a manifold, let X be a vector field on M, and let Φ : D → M be the flow of X. Then (1) the flow domain D is an open neighborhood of {0} × M and the flow Φ : D → M is a smooth map; (2) the flow Φ enjoys the following group properties: M0 = M, and Φ0 = idM, and, additionally,

(Φs ◦ Φt )(p) = Φt+s(p) (6.3)

whenever both sides are well-deﬁned (s,t ∈ R, p ∈ M); (3) for every t ∈ R, Mt ⊆ M is a (possibly empty) open subset, Φt (Mt ) ⊆ M−t , and, if Mt 6= ∅, Φt : Mt → M−t is a diffeomorphism with inverse Φ−t : M−t → Mt .

Proof (a sketch). Point (1) is a consequence of Theorem 6.1.1 and we omit the (technical) proof. Point (2) is a consequence of the Translation Lemma (Lemma 6.1.3), as we now show. First notice that the left hand side of Equation (6.3) is well-deﬁned when

t ∈ Ip and s ∈ IΦt (p) = Iγp(t), (6.4) while the right hand side is well-deﬁned when

t + s ∈ Ip. (6.5)

Actually, from the second part of the Translation Lemma, we have that (6.4) implies (6.5). So, let s,t, p be such that (6.4), and notice that

(Φs ◦ Φt )(p) = Φs(Φt (p)) = γΦt (p)(s) = γγp(t)(s) = γp(τt (s)) = γp(s +t) = Φs+t (p), where, in the fourth step, we used the Translation Lemma again (the maximal integral curve starting from γp(t) is obtained from the maximal integral curve starting from p by composition with the translation τt ). For Point (3), ﬁrst notice that Mt is the preimage of the ﬂow domain D under the map

Γt : M → R × M, p 7→ Γt (p) := (t, p). 6.2 Flow of a Vector Field 113

From Exercise 2.7, Γt is a smooth, hence continuous, map. As D ⊆ R × M is an open subset, it follows that Mt ⊆ M is an open subset. Second,

p ∈ Mt ⇒ t ∈ Ip ⇒ IΦt (p) = Iγp(t) = Ip −t, where we used the Translation Lemma again. As 0 ∈ Ip, we get

−t ∈ IΦt (p) ⇒ Φt (p) ∈ M−t , as claimed. Finally, assume that Mt 6= ∅. The map Φt : Mt → M can be seen as the composition of Γt : Mt → D followed by the ﬂow Φ. As both Φ and Γt are smooth, so is Φt : Mt → M, and its restriction to M−t in the codomain. From the arbitrariness of t, Φ−t : M−t → M is also smooth, and, from Point (2),

Φt ◦ Φ−t = Φ−t ◦ Φt = idM, i.e. Φt : Mt → M−t and Φ−t : M−t → Mt are mutually inverse diffeomorphisms.

R The flow {Φt }t of a vector field X is sometimes called the local 1-parameter group of local diffeomorphisms generated by X, and X is called the infinitesimal generator of {Φt }t . This terminology is motivated by the Fundamental Theorem of Flows.

As a vector field X is completely determined by its maximal integral curves γp, via Xp = γ˙p(0), and the flow of X determine the maximal integral curves via Φt (p) = γp(t), it follows that a vector field is completely determined by its flow, in the sense that two vector fields with the same flow necessarily coincide. Definition 6.2.2 A vector field X on a manifold M is complete if every maximal integral curve of X is defined on the whole R, i.e. Ip = R for all p ∈ M. In other words, the flow of X is defined on the whole R × M. 2 Example 6.6 — The Infinitesimal Generator of Translations. On R with standard coordinates (x,y) consider the coordinate vector field

∂ X = . ∂x It follows from Example 6.1 that the ﬂow of X is

2 2 Φ : R × R → R , (t,(x,y)) 7→ Φ(t,(x,y)) = (x +t,y).

As, for every t, Φt is a translation, we also say that X generates the translations along the x-axis. 2 Example 6.7 — The Inﬁnitesimal Generator of Dilations. On R with standard coordinates (x,y) consider the vector ﬁeld

∂ ∂ X = x + y . ∂x ∂y It follows from Example 6.2 that the ﬂow of X is

2 2 t t Φ : R × R → R , (t,(x,y)) 7→ Φ(t,(x,y)) = (xe ,ye ).

As, for every t, Φt is a dilation, we also say that X generates the dilations. 114 Chapter 6. Flows and Symmetries

2 Example 6.8 — The Inﬁnitesimal Generator of Rotations. On R with standard coordinates (x,y) consider the coordinate vector ﬁeld

∂ ∂ X = x − y . ∂y ∂x It follows from Example 6.3 that the ﬂow of X is

2 2 Φ : R × R → R , (t,(x,y)) 7→ Φ(t,(x,y)) = (xcost − ysint,xsint + ycost).

As, for every t, Φt is a rotation, we also say that X generates the rotations around the origin. d The vector ﬁelds from Examples 6.6, 6.7, and 6.8 are all complete. The vector ﬁeld dx |(0,+∞) from Example 6.5 is not complete.

Exercise 6.3 On R2 with standard coordinates (x,y), find the flow of the vector field ∂ ∂ X = x − y ∂x ∂y

(see also Exercise 6.1). Is X complete?

We conclude this section describing a new construction involving the flow: the Lie derivative along a vector field. We begin with some informal remarks on vector fields and their flows. The flow {Φt }t of a vector field X moves the points of a manifold. At the time 0 this displacement is trivial: it is just the identity map that does not move any point. In any case, at an arbitrary time, this displacement is a diffeomorphism. The vector field is the velocity of this displacement at the initial time: informally

2 Φt = idM +t · X + O(t ).

In this sense X should be interpreted as an infinitesimal diffeomorphism, and the Fundamental Theorem of Flows then says that an infinitesimal diffeomorphism generates a whole 1-parameter family of diffeomorphisms. Notice that, moving all points of M, the flow {Φt }t of a vector field can often move a structure on M. By a structure on M we mean an additional datum, e.g. a submanifold, ∗ a smooth function, a vector field, etc.. In particular, a function f moves via the pull-back Φt ( f ) (and similarly for vector fields). The Lie derivative is then the initial velocity of this displacement. To be precise, let M be a manifold, let X be a vector field on M, with flow {Φt }t , and let f ∈ C∞(M) be a smooth function. The Lie derivative of f along X is the function

d ∗ LX f : M → , p 7→ (LX f )(p) := |t= Φ ( f )(p). (6.6) R dt 0 t ∗ This deﬁnition requires some explanations. First of all, notice that the expression Φt ( f )(p) under the t-derivative, does only make sense when p ∈ Mt , or, equivalently, t ∈ Ip. Indeed,

∗ Φt ( f )(p) = f (Φt (p)) = f (γp(t)). (6.7)

∗ Additionally, (6.7) shows that Φt ( f )(p) depends smoothly on t in Ip which is an open neighborhood of 0. Hence, we can take the t-derivative of (6.7) at t = 0. This discussion shows that the Lie derivative LX f is well-deﬁned. 6.2 Flow of a Vector Field 115

Proposition 6.2.2 — Lie Derivative of a Function. Let M be a manifold, let X ∈ X(M) be a vector ﬁeld, and let f ∈ C∞(M) be a smooth function. Then the Lie derivative of f along X is simply given by

LX f = X( f ). (6.8)

In particular, it is a smooth function.

Proof. We have

d ∗ d (LX f )(p) = |t= Φ ( f )(p) = |t= f (γp(t)) = γ˙p(0)( f ) = Xp( f ) = X( f )(p). dt 0 t dt 0

This concludes the proof.

Sometimes Deﬁnition (6.6) and Identity (6.8) are also written

d ∗ X( f ) = LX f = |t= Φ ( f ). dt 0 t

It is also useful to compute the following derivative at an arbitrary time t:

d d Φ∗( f )(p) = | Φ∗( f )(p). (6.9) dt t ds s=t s

This is slightly more complicated. First of all, (6.9) does only make sense if the expression ∗ Φs ( f )(p) = f (γp(s)) depends smoothly on s in an open neighborhood of t. This is the case when t ∈ Ip or, equivalently, p ∈ Mt (do you see it?). In other words, the correspondence

d d d Φ∗( f ) : M → , p 7→ Φ∗( f )(p) = | Φ∗( f )(p) dt t t R dt t ds s=t s is a well-deﬁned function.

Proposition 6.2.3 Let M be a manifold, let X ∈ X(M) be a vector ﬁeld with ﬂow {Φt }t , and let f ∈ C∞(M) be a smooth function. Then

d Φ∗( f ) = Φ∗(X( f )), (6.10) dt t t

d ∗ for all t such that Mt 6= ∅. In particular, dt Φt ( f ) is a smooth function (on Mt ).

Proof. Let p ∈ Mt , and compute

d ∗ d d d | Φ ( f )(p) = | 0 f (Φ 0 (p)) · | (s −t) = | 0 f (Φ 0 (p)). ds s=t s ds0 s =0 s +t ds s=t ds0 s =0 s +t

Now, we want to use the group property of the ﬂow:

Φs0+t (p) = Φs0 (Φt (p)). (6.11)

0 To do this, we have to make sure that t ∈ Ip (exactly what we are already assuming) and s ∈ IΦt (p). The latter condition can be safely assumed as we are computing the s0-derivative at 0. So we can go 116 Chapter 6. Flows and Symmetries on with the computation:

d ∗ d | Φ ( f )(p) = | 0 f (Φ 0 (p)) ds s=t s ds0 s =0 s +t d = | 0 f (Φ 0 (Φ (p))) ds0 s =0 s t d ∗ = | 0 (Φ 0 f )(Φ (p)) ds0 s =0 s t = (LX f )(Φt (p))

= X( f )(Φt (p)) ∗ = Φt (X( f ))(p). which, as claimed, depends smoothly on p ∈ Mt .

d ∗ R We can also compute dt Φt ( f ) following a different path. Namely we can use the group property of the ﬂow in the form:

Φs0+t (p) = Φt (Φs0 (p)). 0 To do this, we have to make sure that s ∈ Ip, which can be safely assumed as we are 0 0 computing the s -derivative at 0, and t ∈ I , i.e. Φ 0 (p) = γ (s ) ∈ M . The latter condition Φs0 (p) s p t 0 −1 can be guaranteed by taking s sufﬁciently close to 0, e.g. in Ip ∩ γp (Mt ) (which is an open neighborhood of 0). So we can go on with the computation:

d ∗ d | Φ ( f )(p) = | 0 f (Φ 0 (p)) ds s=t s ds0 s =0 s +t d = | 0 f (Φ (Φ 0 (p))) ds0 s =0 t s d ∗ ∗ = | 0 Φ 0 (Φ ( f ))(p). ds0 s =0 s t Exactly the same computation as for the Lie derivative (of a function) shows that

d ∗ ∗ ∗ | 0 Φ 0 (Φ ( f ))(p) = X| (Φ ( f ))(p), ds0 s =0 s t Mt t ∗ which, adopting our customary notation, we also denote simply by X(Φt ( f ))(p). Summariz- ing, we proved that, for every t ∈ R such that Mt 6= ∅, d Φ∗( f ) = X(Φ∗( f )). (6.12) dt t t

Finally, we discuss the Lie derivative of a vector field along a(n other) vector field. So, let M be a manifold, and let X,Y be vector fields on M. The Lie derivative of Y along X is a new vector field, denoted LXY and defined, through its values, as follows:

d ∗ (LXY)p := |t= Φ (Y)p. (6.13) dt 0 t ∗ This definition requires some explanations. First of all, what we actually mean by Φt (Y) is, more ∗ precisely, Φt (Y|M−t ): the pull-back of the vector field Y|M−t ∈ X(M−t ) along the diffeomorphism Φt : Mt → M−t (if X is complete, then Mt = M and we don’t need this level of precision). In ∗ ∗ particular, Φt (Y) is a vector field on Mt . Hence, the expression Φt (Y)p under the t-derivative, does ∗ only make sense when p ∈ Mt , or, equivalently, t ∈ Ip, and, in this case, Φt (Y)p is a t-dependent ∗ tangent vector in TpM. We can only take the t-derivative if Φt (Y)p depends smoothly on t (at least in an open neighborhood of 0). If this is the case, according to our interpretation of the velocity of a curve in a finite dimensional real vector space (see the end of Section 3.2), the derivative in (6.13) is again a vector in TpM. 6.2 Flow of a Vector Field 117

Proposition 6.2.4 — Lie Derivative of a Vector Field. Let M be a manifold, and let X,Y ∈ X(M) be vector fields. Then the Lie derivative of Y along X is a well-defined vector field simply given by

LXY = [X,Y]. (6.14)

∗ Proof. First we need to show that the tangent vector Φt (Y)p depends smoothly on t in an open neighborhood of 0 (so that we can take the t-derivative (6.13)). To do this, we work in local 1 n coordinates as follows. Fix a point p0 ∈ M, and let (U,ϕ = (x ,...,x )) be a chart around p0. Then ∂ ∂ X| = Xi and Y| = Y i U ∂xi U ∂xi for some smooth functions Xi,Y i ∈ C∞(U). The preimage Φ−1(U) of U under the ﬂow Φ : D → M, is an open subset of R × M containing (0, p0). Hence, there is an open interval J containing 0, −1 and a subchart (U0,ϕ) of (U,ϕ) around p0, such that J ×U0 ⊆ Φ (U), i.e. Φ(J ×U0) ⊆ U. In particular, we can consider the restriction

Φ : J ×U0 → U (6.15)

i ∗ i (yet in other words, for every t ∈ J, U0 ∈ Mt and Φt (U0) ⊆ U). As usual, we denote by Φ = Φ (x ) the components of (6.15). We will need the following identities: ∂Φi (0, p) = Xi(p), (6.16) ∂t and ∂Φi (0, p) = δ i (6.17) ∂xk k for all p ∈ U0. To prove them, notice ﬁrst that

i i Φ (t, p) = γp(t) for all t ∈ J, and all p ∈ U0 (do you see it?). In particular,

i i i i i i Φ (0, p) = γp(0) = x (γp(0)) = x (p), i.e. Φ (0,−) = x . Hence ∂Φi d d (0, p) = | Φi(t, p) = | γi (t) = Xi(p), ∂t dt t=0 dt t=0 p and ∂Φi ∂ ∂ (0, p) = | Φi(0,−) = | xi = δ i. ∂xk ∂xk p ∂xk p k

Now let (t, p) ∈ J ×U0, and, using (5.5), compute ∂Φi ∂ Φ∗(Y) = d Φ−1(Y ) = d Φ (Y ) = (−t,Φ(t, p))Y j(Φ(t, p)) | . t p Φt (p) t Φt (p) Φt (p) −t Φt (p) ∂x j ∂xi p The coefﬁcients in the last expression depend smoothly on t. This shows that we can take the derivative in (6.13). It remains to compute i d ∗ d ∂Φ k ∂ (LXY)p = |t= Φ (Y)p = |t= (−t,Φ(t, p))Y (Φ(t, p)) |p. dt 0 t dt 0 ∂tk ∂xi 118 Chapter 6. Flows and Symmetries

The i-th coefﬁcient is d ∂Φi | (−t,Φ(t, p))Y j(Φ(t, p)) dt t=0 ∂x j ∂ 2Φi ∂ 2Φi ∂Φk = − (0,Φ(0, p)) + (0,Φ(0, p)) (0, p) Y j(Φ(0, p)) ∂t∂x j ∂xk∂x j ∂t ∂Φi ∂Y j ∂Φk + (0,Φ(0, p)) (Φ(0, p)) (0, p) ∂x j ∂xk ∂t ∂Xi ∂Y i = − (p)Y j(p) + (p)X j(p) ∂x j ∂x j i = [X,Y]p, for all p ∈ U0. This concludes the proof. Sometimes Deﬁnition (6.13) (and Identity (6.14)) are also written

d ∗ [X,Y] = LXY = |t= Φ (Y). dt 0 t It is also useful to compute the following derivative at an arbitrary time t: d d Φ∗(Y) = | Φ∗(Y) . (6.18) dt t p ds s=t s p

Similarly as for functions, (6.18) does only make sense when t ∈ Ip or, equivalently, p ∈ Mt , and the correspondence d d d Φ∗(Y) : M → TM , p 7→ Φ∗(Y) = | Φ∗(Y) dt t t t dt t p ds s=t s p is a well-deﬁned (a priori non-necessarily smooth) section of TMt . One can show that it is actually a smooth section, hence it corresponds to a vector ﬁeld on Mt . To see this, let p ∈ Mt , and compute

d ∗ d ∗ d | Φ (Y) = | 0 Φ 0 (Y) · | (s −t) ds s=t s p ds0 s =0 s +t p ds s=t d ∗ ∗ = | 0 Φ (Φ 0 (Y)) ds0 s =0 t s p d −1 ∗ = | 0 d Φ (Φ 0 (Y) ). (6.19) ds0 s =0 Φt (p) t s Φt (p) At this point we need a

Lemma 6.2.5 Let V,W be ﬁnite dimensional real vector spaces, let γ : I → V be a curve in V, and let A : V → W be a linear map. Then A ◦ γ : I → W is a smooth curve and

d d | (A ◦ γ)(s) = A | γ(s) ds s=s0 ds s=s0

for all s0 ∈ I.

i Proof. Let R = (e1,...,en) be a frame of V, n = dimV. Then γ(s) = γ (s)ei for all s ∈ I, where i i the γ : I → R, s 7→ γ (s) are smooth functions. If

j= ,...,n j 1 Ai i=1,...,n 6.3 Symmetries and Inﬁnitesimal Symmetries 119

where, in the third step, we used Exercise 3.7. −1 Applying Lemma 6.2.5 to (6.19) (with A = dΦt (p)Φt ), we ﬁnd

d ∗ d −1 ∗ | Φ (Y) = | 0 d Φ (Φ 0 (Y) ) ds s=t s p ds0 s =0 Φt (p) t s Φt (p) −1 d ∗ = d Φ | 0 Φ 0 (Y) Φt (p) t ds0 s =0 s Φt (p) −1 = dΦt (p)Φt (LXY)Φt (p) −1 = dΦt (p)Φt [X,Y]Φt (p) ∗ = Φt ([X,Y])p .

Summarizing, we proved that, for every t ∈ R (such that Mt 6= ∅), d Φ∗(Y) = Φ∗ ([X,Y]). (6.20) dt t t

d ∗ R As for functions, we can also compute dt Φt (Y) following a different path. Namely,

d ∗ d ∗ d ∗ ∗ ∗ ∗ |s=t Φ (Y)p = | 0 Φ 0 (Y)p = | 0 Φ 0 (Φ (Y)) = (LX Φ (Y)) = [X,Φ (Y)]p, ds s ds0 s =0 s +t ds0 s =0 s t t p t

for all p ∈ Mt . This shows that d Φ∗(Y) = [X,Φ∗(Y)]. dt t t We leave to the reader to check that every single step in the above computation makes sense.

6.3 Symmetries and Infinitesimal Symmetries In Mathematics, a symmetry of an object O is always a transformation preserving O. The precise meaning of the words transformation and preserving depends on the nature of the object O. For instance, when O is a group, then a symmetry is an automorphism of O. Similarly, when O is a topological space, then a symmetry is a homeomorphism Φ : O → O, and when O is a manifold, a symmetry is a diffeomorphism Φ : O → O. The smoothness of objects in Differential Geometry allows us to talk not only about symmetries, but also about infinitesimal symmetries. As discussed in the previous section, it is natural to interpret vector fields on a manifold M as infinitesimal diffeomorphisms, hence they are also infinitesimal symmetris of M. We can also talk about symmetries and infinitesimal symmetries of a structure on a manifold M (see previous section). In this section we discuss in some details symmetries and infinitesimal symmetries of submanifolds, smooth functions, and vector fields on a manifold. In Chapter 8 we will quickly consider one more example: symmetries and infinitesimal symmetries of differential forms. We begin with submanifolds. Let M be a manifold, and let S ⊆ M be a submanifold. Recall that the image Φ(S) of S under a diffeomorphism Φ : M → M is a submanifold as well (Exercise 2.13). 120 Chapter 6. Flows and Symmetries

Deﬁnition 6.3.1 A symmetry of a submanifold S ⊆ M in a manifold M is a diffeomorphism Φ : M → M preserving S in the sense that Φ(S) = S.A local symmetry of S is a diffeomorphism Φ : U → Φ(U ) between open submanifolds U ,Φ(U ) ⊆ M, such that Φ(S ∩ U ) = S ∩ Φ(U ).

2 2 Example 6.9 In R with standard coordinates (x,y), consider a 1-dimensional subspace ` ⊆ R . Let λ ∈ R be a non-zero real number, and consider the dilation 2 2 Φ : R → R , (x,y) 7→ Φ(x,y) = (λx,λy).

Then Φ is clearly a symmetry of `. 2 1 2 Example 6.10 In R with standard coordinates (x,y), consider the circle S ⊆ R . Let θ ∈ R, and consider the rotation around the origin by an angle θ:

2 2 Φ : R → R , (x,y) 7→ Φ(x,y) = (xcosθ − ysinθ,xsinθ + ycosθ).

1 Then Φ is clearly a symmetry of S .

Proposition 6.3.1 Let M be a manifold, and let S ⊆ M be a submanifold. The subset

Diffeo(M,S) := {symmetries of S} ⊆ Diffeo(M)

of the group Diffeo(M) of diffeomorphisms of M is a subgroup.

Proof. Left as Exercise 6.4.

Exercise 6.4 Prove Proposition 6.3.1.

Definition 6.3.2 An infinitesimal symmetry of a submanifold S ⊆ M in a manifold M is a vector field X on M generating a flow {Φt }t by local symmetries of S, i.e. Φt (S ∩Mt ) = S ∩M−t for all t (such that Mt 6= ∅). 2 2 Example 6.11 In R with standard coordinates (x,y), consider a 1-dimensional subspace ` ⊆ R and the circle S1 ⊆ R2. Consider also the vector fields ∂ ∂ ∂ ∂ X = x + y and X = x − y . 1 ∂x ∂y 2 ∂y ∂x

It follows from Examples 6.7 and 6.9 that X1 is an infinitesimal symmetry of `. Similarly, it follows 1 from Examples 6.8 and 6.10 that X2 is an infinitesimal symmetry of S . Checking whether or not a vector field X is an infinitesimal symmetry of a submanifold S using the definition might be problematic, because it requires computing the flow of X, hence solving a system of ODEs for every possible initial data (which is far from being an easy task in general). However, infinitesimal symmetries of a submanifold can be (almost) characterized in a way which is (almost) independent of the flow.

Proposition 6.3.2 Let M be a manifold, let S ⊆ M be a submanifold, and let X be a vector field on M. X If X is an infinitesimal symmetry of S, then X is tangent to S. X Conversely, if X is tangent to S and, additionally, the restriction X|S is a complete vector field, then X is an infinitesimal symmetry of S.

Proof. Left as Exercise 6.5. 6.3 Symmetries and Inﬁnitesimal Symmetries 121

Exercise 6.5 Prove Proposition 6.3.2. (Hint: use Proposition 6.1.4).

We now pass to symmetries of smooth functions, noticing that the group Diffeo(M) of diffeomorphisms of a manifold M acts on the algebra of smooth functions C∞(M) (by algebra isomorphisms) via the pull-back, i.e., for any diffeomorphism Φ : M → M, we have an algebra isomorphism Φ∗ : C∞(M) → C∞(M). This suggests the following Deﬁnition 6.3.3 A symmetry of a smooth function f ∈ C∞(M) on a manifold M is a diffeomorphism Φ : M → M preserving f in the sense that Φ∗( f ) = f .A local symmetry of f is a diffeomorphism Φ : U → Φ(U ) between open submanifolds U ,Φ(U ) ⊆ M such that ∗ Φ ( f |Φ(U )) = f |U .

2 ∞ 2 Example 6.12 On R with standard coordinates (x,y), consider a smooth function f ∈ C (R ) depending on the sole y, i.e. there is a functions g ∈C∞(R) such that f (x,y) = g(y) for all (x,y) ∈ R2. Let x0 ∈ R, and consider the translation along the x-axis

2 2 Φ : R → R , (x,y) 7→ Φ(x,y) = (x + x0,y).

Then Φ is a symmetry of f , indeed, for every (x,y) ∈ R2:

∗ Φ ( f )(x,y) = f (Φ(x,y)) = f (x + x0,y) = g(y) = f (x,y).

Exercise 6.6 On R2 with standard coordinates (x,y), consider a smooth function f ∈ C∞(R2) depending on the sole x2 + y2, i.e. there is a functions g ∈ C∞(R) such that f (x,y) = g(x2 + y2) for all (x,y) ∈ R2. Let θ ∈ R, and consider the rotation around the origin by an angle θ:

2 2 Φ : R → R , (x,y) 7→ Φ(x,y) = (xcosθ − ysinθ,xsinθ + ycosθ).

Show that Φ is a symmetry of f .

Proposition 6.3.3 Let M be a manifold, and let f ∈ C∞(M) be a smooth function on M. The subset

Diffeo(M, f ) := {symmetries of f } ⊆ Diffeo(M)

of the group of diffeomorphisms of M is a subgoup.

Proof. Left as Exercise 6.7.

Exercise 6.7 Prove Proposition 6.3.3.

Proposition 6.3.4 Let M be a manifold, let f ,g ∈ C∞(M) and let Φ : M → M be a diffeomorphism. If Φ is a symmetry of both f and g, then it is also a symmetry of f + g, and f g.

Proof. Left as Exercise 6.8.

Exercise 6.8 Prove Proposition 6.3.4.

We now come to inﬁnitesimal symmetries of functions. 122 Chapter 6. Flows and Symmetries

Definition 6.3.4 An infinitesimal symmetry of a smooth function f ∈ C∞(M) on a manifold M ∗ is a vector field X on M generating a flow {Φt }t by local symmetries of f , i.e. Φt ( f |M−t ) = f |Mt for all t (such that Mt 6= ∅).

2 Example 6.13 On R with standard coordinates (x,y), consider a smooth function f1 depending 2 2 on the sole y, and a smooth function f2 depending on the sole x + y . Consider also the vector ﬁelds ∂ ∂ ∂ X = and X = x − y . 1 ∂x 2 ∂y ∂x

It follows from Examples 6.6 and 6.12 that X1 is an infinitesimal symmetry of f1. Similarly, it follows from Example 6.8 and Exercise 6.6 that X2 is an infinitesimal symmetry of f2. Similarly as for submanifolds, checking whether or not a vector field X is an infinitesimal symmetry of a smooth function f using the definition might be problematic. Luckily, infinitesimal symmetries of a smooth function can be characterized in a way which is totally independent of the flow. Proposition 6.3.5 Let M be a smooth manifold, let f ∈ C∞(M) be a smooth function, and let X be a vector field on M. Then X is an infinitesimal symmetry of f if and only if X( f ) = 0.

We postpone a little bit the proof of Proposition 6.3.5. First we propose an exercise and discuss a consequence. Both should convince the reader of the relevance of characterizing inﬁnitesimal symmetries of smooth functions in a “ﬂow-independent” way.

Exercise 6.9 On R2 with standard coordinates (x,y) consider a smooth function f ∈ C∞(R2) and the vector ﬁeld ∂ f ∂ ∂ f ∂ X = − . ∂x ∂y ∂y ∂x

Show that X is an inﬁnitesimal symmetry of f .

Corollary 6.3.6 Let M be a manifold, and let f ∈ C∞(M) be a smooth function of M. The subset

X(M, f ) := {inﬁnitesimal symmetries of f } ⊆ X(M)

of the Lie algebra of vector ﬁelds on M is a Lie subalgebra.

Proof. Left as Exercise 6.10.

Exercise 6.10 Prove Corollary 6.3.6.

Proof of Proposition 6.3.5. Let M, f ,X be as in the statement, and assume ﬁrst that X is an in- ﬁnitesimal symmetry of f . This means that

∗ Φt ( f ) = f |Mt for all t ∈ R (such that Mt 6= ∅). Then, for every p ∈ M,

d ∗ d X( f )(p) = LX f (p) = |t= Φ ( f )(p) = |t= f (p) = 0. dt 0 t dt 0 6.3 Symmetries and Inﬁnitesimal Symmetries 123

Conversely, let X( f ) = 0, and let t ∈ R be such that Mt 6= ∅. Then, for every p ∈ Mt , we have d Φ∗( f )(p) = Φ∗(X( f ))(p) = 0, (6.21) dt t t where we used (6.10). Identity (6.21) now shows that ∗ ∗ Φt ( f )(p) = const = Φ0( f )(p) = f (p), ∗ i.e. Φt ( f ) = f |Mt . This concludes the proof.

Corollary 6.3.7 Let M be a manifold, let X ∈ X(M), and let f ,g ∈ C∞(M). If X is an in- ﬁnitesimal symmetry of both f and g, then it is also an inﬁnitesimal symmetry of f + g, and f g.

Proof. Left as Exercise 6.11.

Exercise 6.11 Prove Corollary 6.3.7.

We now pass to symmetries (and infinitesimal symmetries) of vector fields. Definition 6.3.5 A symmetry of a vector field Y ∈ X(M) on a manifold M is a diffeomorphism Φ : M → M preserving Y in the sense that Φ∗(Y) =Y.A local symmetry of Y is a diffeomorphism ∗ Φ : U → Φ(U ) between open submanifolds U ,Φ(U ) ⊆ M such that Φ (Y|Φ(U )) = Y|U .

2 ∂ Example 6.14 On R with standard coordinates (x,y), consider the vector ﬁeld Y = ∂y . Let x0 ∈ R, and consider the translation along the x-axis 2 2 Φ : R → R , (x,y) 7→ Φ(x,y) := (x + x0,y). Then Φ is a symmetry of Y. To see this, let’s compute Φ∗ (Y). We have ∂ ∂ Φ∗ (Y) = A + B , ∂x ∂y for some functions A,B ∈ C∞(R2) given by A = Φ∗ (Y)(x) = Φ∗(Y((Φ−1)∗(x))), and B = Φ∗ (Y)(y) = Φ∗(Y((Φ−1)∗(y)))

− Now Φ 1 : R2 → R2 is given by −1 2 2 −1 Φ : R → R , (x,y) 7→ Φ (x,y) := (x − x0,y), −1 ∗ −1 and (Φ ) (x) is the ﬁrst component of Φ , i.e. x − x0. Hence ∂(x − x ) A = Φ∗(Y((Φ−1)∗(x))) = Φ∗(Y(x − x )) = Φ∗ 0 = Φ∗(0) = 0. 0 ∂y Similarly, (Φ−1)∗(y) is the second component of Φ−1(y), and ∂y B = Φ∗ (Y)(y) = Φ∗(Y((Φ−1)∗(y))) = Φ∗(Y(y)) = Φ∗ = Φ∗(1) = 1. ∂y We conclude that ∂ ∂ ∂ Φ∗(Y) = 0 · + 1 · = = Y, ∂x ∂y ∂y i.e. Φ is a symmetry of Y as claimed. 124 Chapter 6. Flows and Symmetries

Exercise 6.12 On R2 with standard coordinates (x,y), consider the vector ﬁeld ∂ ∂ Y = x + y . ∂x ∂y

Additionally, let θ ∈ R, and consider the rotation around the origin of an angle θ:

2 2 Φ : R → R , (x,y) 7→ Φ(x,y) = (xcosθ − ysinθ,xsinθ + ycosθ).

Show that Φ is a symmetry of Y.

Proposition 6.3.8 Let M be a manifold, and let Y ∈ X(M) be a vector ﬁeld on M. The subset

Diffeo(M,Y) := {symmetries of Y} ⊆ Diffeo(M)

of the group of diffeomorphisms of M is a subgoup.

Proof. Left as Exercise 6.13.

Exercise 6.13 Prove Proposition 6.3.8.

The following is a Corollary of Proposition 5.3.4.

Corollary 6.3.9 Let M be a manifold, let Y,Z ∈ X(M), let f ∈ C∞(M) and let Φ : M → M be a diffeomorphism. If Φ is a symmetry of Y, Z and f , then it is also a symmetry of Y + Z, Y( f ), fY and [Y,Z].

Proof. Left as Exercise 6.14.

Exercise 6.14 Prove Corollary 6.3.9.

We now come to infinitesimal symmetries of vector fields. Definition 6.3.6 An infinitesimal symmetry of a vector field Y ∈ X(M) on a manifold M is a(n ∗ other) vector field X on M generating a flow {Φt }t by local symmetries of Y, i.e. Φt (Y|M−t ) =

Y|Mt for all t (such that Mt 6= ∅). Beware that the two vector fields X,Y in Definition 6.3.6 play completely different roles. 2 Example 6.15 On R with standard coordinates (x,y), consider the vector fields ∂ ∂ ∂ Y = and Y = x + y , 1 ∂y 2 ∂x ∂y and the vector fields ∂ ∂ ∂ X = and X = x − y . 1 ∂x 2 ∂y ∂x

It follows from Examples 6.6 and 6.14 that X1 is an infinitesimal symmetry of Y1. Similarly, it follows from Example 6.8 and Exercise 6.12 that X2 is an infinitesimal symmetry of Y2. Similarly as for functions, checking whether or not a vector field X is an infinitesimal symmetry of an other vector field Y using the definition might be problematic, but as for functions, infinitesimal symmetries of a vector field can be characterized in a rather efficient way. 6.3 Symmetries and Infinitesimal Symmetries 125

Proposition 6.3.10 Let M be a smooth manifold, and let Y,X ∈ X(M) be vector ﬁelds. Then X is an inﬁnitesimal symmetry of Y if and only if [Y,X] = 0.

Proof. The proof is very similar to that of Proposition 6.3.5 and we leave it as Exercise 6.15.

Exercise 6.15 Prove Proposition 6.3.10.

Corollary 6.3.11 Let M be a smooth manifold, let Y,X ∈ X(M) be vector fields. Then X is an infinitesimal symmetry of Y if and only if Y is an infinitesimal symmetry of X.

Proof. Obvious.

Corollary 6.3.12 Let M be a manifold, and let Y ∈ X(M) be a vector ﬁeld on M. The subset

X(M,Y) := {inﬁnitesimal symmetries of Y} ⊆ X(M)

of the Lie algebra of vector ﬁelds on M is a Lie subalgebra.

Proof. Left as Exercise 6.16.

Exercise 6.16 Prove Corollary 6.3.12.

The following infinitesimal version of Corollary 6.3.9 is a corollary of either Corollary 6.3.9 itself, or (in the part about the commutator) Proposition 6.3.10 and the Jacobi identity for the commutator of vector fields. Corollary 6.3.13 Let M be a manifold, let X,Y,Z ∈ X(M), and let f ∈ C∞(M). If X is an infinitesimal symmetry of both Y, Z and f , then it is also an infinitesimal symmetry of Y + Z, Y( f ), fY, and [Y,Z].

Proof. Left as Exercise 6.17.

Exercise 6.17 Prove Corollary 6.3.13.

We conclude this chapter discussing the flows of two commuting vector fields, i.e. two vector fields X,Y such that their commutator vanishes: [X,Y] = 0. Let M be a manifold, and let X,Y ∈ X(M) be two vector fields on M. Denote by {Φt }t and {Ψs}s the flows of X and Y, respectively. We want to show that X and Y commute if and only if their flows commute. First we have to explain what does it mean that {Φt }t and {Ψs}s commute. We will only discuss in details the case when both Φ and Ψ are defined on the whole R × M, however, we report a definition in the general case for completeness.

Deﬁnition 6.3.7 The ﬂows {Φt }t and {Ψs}s commute if, for every p ∈ M, we have that, whenever I,J ⊆ R are open intervals containing 0 such that one of the expressions

Φt (Ψs(p)) or Ψs(Φt (p))

makes sense for all (t,s) ∈ I × J, then the other one also makes sense and they are equal. 126 Chapter 6. Flows and Symmetries

Proposition 6.3.14 — Commuting Vector Fields. The vector ﬁelds X and Y commute if and only if so do their ﬂows {Φt }t and {Ψs}s.

Proof. We provide a proof only in the case when both X and Y are complete. We begin remarking that, in this case, Deﬁnition 6.3.7 boils down to the condition

Φt ◦ Ψs = Ψs ◦ Φt for all (s,t) ∈ R × R.

Assume first that X and Y commute. Then, from Proposition 6.3.10, X is an infinitesimal symmetry of Y, i.e. Φt is a symmetry of Y for all t ∈ R. It now follows from Proposition 6.1.4 that Φt maps integral curves of Y to integral curves. In particular, for all p ∈ M, the maximal integral curve of Y starting from p, say γp : R → M, is mapped to an integral curve Φt ◦γp : R → M. Notice that, being defined on the whole R, not only Φt ◦ γp is an integral curve, but it is also maximal. As it starts from Φt (γp(0)) = Φt (p), it must coincide with the maximal integral curve

γΦt (p), i.e. Φt ◦ γp = γΦt (p) : R → M. In other words

Φt (γp(s)) = γΦt (p)(s), for all p ∈ M and all (t,s) ∈ R × R. The left hand side is

Φt (γp(s)) = Φt (Ψs(t)), while the right hand side is

γΦt (p)(s) = Ψs(Φt (p)).

This conclude the proof of the “only if” part of the statement. For the if part, suppose that

Φt ◦ Ψs = Ψs ◦ Φt for all (s,t) ∈ R × R.

Then, similarly as in the proof of Proposition 6.1.4, for all p ∈ M, and all t ∈ R, d d Φ (Y ) = d Φ | Ψ (p) (s 7→ Ψ (p) is an integral curve of Y) p t p p t ds s=0 s s d = | Φ (Ψ (p)) (tangent map applied to the velocity of a curve) ds s=0 t s d = | Ψ (Φ (p)) ({Φ } and {Ψ } commute) ds s=0 s t t t s s

= YΦt (p) (s 7→ Ψs(Φt (p)) is an integral curve of Y).

This shows that Φt is a symmetry of Y for all t, hence X is an inﬁnitesimal symmetry of Y and, from Proposition 6.3.10, X and Y commute. 7. Covectors and Differential 1-Forms

In this chapter, we introduce (tangent) covectors and differential 1-forms. Covectors and tangent vectors are dual objects. Similarly, differential 1-forms and vector fields are dual objects. Covectors are the points of a new manifold, the cotangent bundle, and differential 1-forms are equivalent to sections of the cotangent bundle, in a very similar way as for tangent vectors and vector fields. However, there are important differences between vector fields and differential forms. Among the most relevant, unlike vector fields, differential forms on a manifold can be transformed into differential forms on another manifold along any smooth map, via the pull-back construction.

7.1 Covectors and the Cotangent Bundle Let M be an n-dimensional manifold, let p ∈ M be a point, and let (U,ϕ = (x1,...,xn)) be a chart on M around p. Deﬁnition 7.1.1 — Cotangent Space. The cotangent space to M at the point p is the dual space

∗ ∗ Tp M := (TpM)

∗ of the tangent space. Vectors in Tp M are called tangent covectors, or cotangent vectors, or, simply, covectors. In other words, covectors are R-linear maps θ : TpM → R. The frame of ∗ Tp M dual to the coordinate frame

∂ ∂ | ,..., | ∂x1 p ∂xn p

is denoted

1 n dpx ,...,dpx

and it is called the coordinate coframe. 128 Chapter 7. Covectors and Differential 1-Forms

Let U ⊆ M be an open neighborhood of p. As TpU can be identiﬁed with TpM, the cotangent R ∗ ∗ space Tp U can be identiﬁed with Tp M, and, in what follows, we will always take this point of view.

Similarly as for the coordinate frame, if we change coordinates around p, the coordinate coframe changes. So, let (U˜ ,ϕ˜ = (x˜1,...,x˜n)) be another chart on M around p. Denote by t1,...,tn the standard coordinates on ϕ(U), and by t˜1,...,t˜n the standard coordinates on ϕ˜(U˜ ). Finally let P = ϕ(p). The transition matrix

MR∗,R˜∗ ∈ GL(n,R) between the coordinate coframes

∗ 1 n ∗ 1 n R˜ = dpx˜ ,...,dpx˜ and R = dpx ,...,dpx is the transpose of the transition matrix MR˜,R between the coordinate frames

∂ ∂ ∂ ∂ R = ,..., and R˜ = ,..., . ∂x1 ∂xn ∂x˜1 ∂x˜n

Recall that the latter is

j= ,...,n j=1,...,n ∂x˜j 1 ∂(ϕ˜ ◦ ϕ−1) j MR˜,R = i (p) = i (P) . ∂x i=1,...,n ∂t i=1,...,n Hence, the former is

j= ,...,n j=1,...,n ∂x˜i 1 ∂(ϕ˜ ◦ ϕ−1)i MR∗,R˜∗ = j (p) = j (P) . ∂x i=1,...,n ∂t i=1,...,n In other words ∂x˜i ∂(ϕ˜ ◦ ϕ−1)i d x˜i = (p)d x j = (P)d x j, p ∂x j p ∂t j p for all i = 1,...,n. The following notion of differential of a smooth function f at a point p encodes partial derivatives of f at p in a coordinate-free manner. Deﬁnition 7.1.2 — Differential at a Point. The differential at the point p is the map:

∞ ∗ dp : C (M) → Tp M, f 7→ dp f

deﬁned by

dp f (v) = v( f ), for all v ∈ TpM.

∞ The differential dp is well-deﬁned in the sense that dp f is a covector for every f ∈ C (M). Indeed, let v,w ∈ TpM, and let a,b ∈ R. Then

dp f (av + bw) = (av + bw)( f ) = a · v( f ) + b · w( f ) = a · dp f (v) + b · dp f (w), showing that dp f is indeed R-linear. 7.1 Covectors and the Cotangent Bundle 129

∞ Proposition 7.1.1 — Properties of the Differential at a Point. The differential dp : C (M) → ∗ Tp M enjoys the following properties: X dp is R-linear; X dp satisﬁes the following Leibniz rule at the point p:

∞ dp( f g) = f (p)dpg + g(p)dp f , for all f ,g ∈ C (M);

X dpc = 0 for every constant function c; X dp is local, in the sense that dp f does only depend on the values of f around p, i.e.

dp f = dpg

whenever f and g agree on an open neighborhood of p.

Proof. Left as Exercise 7.1

Exercise 7.1 Prove Proposition 7.1.1.

Let f ∈ C∞(M), let p ∈ M, and let (U,ϕ = (x1,...,xn)) be a chart on M around p. We now i compute the components of dp f in the coordinate coframe. If dp f = θidpx , then

∂ ∂ ∂ f θi = dp f i |p = |p f = i (p), ∂x ∂xi ∂x hence ∂ f d f = (p)d xi, (7.1) p ∂xi p showing that dp f encodes, in a coordinate-free way, partial derivatives of f , as announced. It follows from Equation (7.1) that the elements of the coordinate coframe are exactly the differentials i at the point p of the coordinate functions, motivating the notation dpx .

∗ Proposition 7.1.2 Let M be a manifold and let p ∈ M be a point. For any covector θ ∈ Tp M, there exists a (non-unique) smooth function f such that θ = dp f .

1 n ∗ Proof. Choose a chart (U,ϕ = (x ,...,x )) around p, and let θ ∈ Tp M. There exist real numbers θi i i ∞ such that θ = θidpx . Consider the smooth function g = θix ∈ C (U). In view of the Local Exten- sion Lemma, there exists a smooth function f ∈ C∞(M) agreeing with g in an open neighborhood of p, and we have

∂g d f = d g = (p)d xi = θ d xi = θ. p p ∂xi p i p

Similarly as tangent vectors, covectors on a manifold M can be organized into a manifold called

the cotangent bundle to M, and denoted T ∗M. Let’s construct T ∗M. First of all, we put ∗ ∏ ∗ ∗ T M := Tp M = (p,θ) : p ∈ M and θ ∈ Tp M . p∈M

A point (p,θ) in T ∗M will be sometimes simply denoted by θ. The set T ∗M comes with a natural surjection π : T ∗M → M, (p,θ) 7→ p, and the pair (T ∗M,π) is the cotangent bundle to M 130 Chapter 7. Covectors and Differential 1-Forms

(sometimes T ∗M itself is referred to as the cotangent bundle). The smooth structure on M induces an atlas on T ∗M as follows. Begin with a chart (U,ϕ = (x1,...,xn)) on M, and deﬁne a chart

(T ∗U,T ∗ϕ) on T ∗M as follows. First of all, put ∗ −1 ∏ ∗ ∗ T U := π (U) = Tp M = {(p,θ) ∈ T M : p ∈ U}. p∈U

So, T ∗U is exactly the cotangent bundle to U. Next, deﬁne a map: ∗ ∗ n ∗ ∂ ∂ T ϕ : T U → Ub × R , (p,θ) 7→ T ϕ(p,θ) := ϕ(p);θ |p ,...,θ |p . ∂x1 ∂xn

Notice that the last n entries of T ∗ϕ(p,θ) are the components of θ in the coordinate coframe

1 n dpx ,...,dpx .

Clearly, (T ∗U,T ∗ϕ) is a 2n-dimensional chart on T ∗M. Every such chart is called a standard chart. The ﬁrst n components of the coordinate map T ∗ϕ will be again denoted by (x1,...,xn), and the last n components will be denoted by (p1,..., pn). We now show that any two standard charts on T ∗M are compatible. So let (U,ϕ = (x1,...,xn)) and (U˜ ,ϕ˜ = (x˜1,...,x˜n)) be charts on M, and let (T ∗U,T ∗ϕ), (T ∗U˜ ,T ∗ϕ˜) be the associated ∗ ∗ ∗ standard charts on T M. If U ∩U˜ = ∅, then T U ∩ T U˜ = ∅ and there is nothing else to prove. If ∗ ∗ ∗ U ∩U˜ 6= ∅, then T U ∩ T U˜ = T (U ∩U˜ ), and we can consider the transition map

T ∗ϕ ◦ (T ∗ϕ˜)−1 : T ∗ϕ˜(T ∗U ∩ T ∗U˜ ) → Tϕ(T ∗U ∩ T ∗U˜ ).

Now,

∗ ∗ ∗ n T ϕ˜(T U ∩ T U˜ ) = (U ∩U˜ ) × R

∗ ∗ ∗ is an open subset and similarly for T ϕ(T U ∩ T U˜ ). Additionally, take a point (P˜;a1,...,an) ∈ n (U ∩U˜ ) × R , and compute

∗ ∗ −1 (T ϕ ◦ (T ϕ˜) )(P˜;a1,...,an) ∗ −1 ˜ i = T ϕ ϕ˜ (P),aidϕ−1(P˜)x˜ −1 i ∂ i ∂ = (ϕ ◦ ϕ˜ )(P˜);a d −1 ˜ x˜ | −1 ˜ ,...,a d −1 ˜ x˜ | −1 ˜ i ϕ (P) ∂x1 ϕ˜ (P) i ϕ (P) ∂xn ϕ˜ (P) ∂x˜i ∂x˜i = (ϕ ◦ ϕ˜ −1)(P˜);a (ϕ−1(P˜)),...,a (ϕ−1(P˜)) . i ∂x1 i ∂xn

As (U,ϕ),(U˜ ,ϕ˜) are compatible, the ﬁrst n entries depend smoothly on P˜. The last n entries ∗ ∗ −1 depend smoothly on P˜, and linearly, hence smoothly, on a1,...,an. This shows that T ϕ ◦ (T ϕ˜) is smooth. By the same argument, its inverse is also smooth, so they are both diffeomorphisms and the charts (T ∗U,T ∗ϕ),(T ∗U˜ ,T ∗ϕ˜) are compatible. Finally, standard charts cover T ∗M, so they form an atlas called the standard atlas. The standard atlas induces a Hausdorff and second countable topology on T ∗M, and this can be proved exactly as in the case of the tangent bundle. So T ∗M is a manifold, and one can also show that π : T ∗M → M is a smooth map. We leave details to the readers. Summarizing, we have proved the following 7.2 Differential 1-Forms and Fields of Covectors 131

Proposition 7.1.3 — Smooth Structure on the Cotangent Bundle. Standard charts on T ∗M form an atlas. With the associated smooth structure, T ∗M is a 2n-dimensional manifold, and π : T ∗M → M is a smooth map.

Deﬁnition 7.1.3 — Section of the Cotangent Bundle. A section of T ∗M is a map s : M → ∗ T M such that π ◦ s = idM. In other words, a section s is the assignment of a covector s(p) at p, for every point p ∈ M.

There is a natural C∞(M)-module structure on sections of T ∗M deﬁned exactly by the same formulas as for the tangent bundle (see Example 3.4), and smooth sections form a submodule denoted Γ(T ∗M). Now, let (U,ϕ = (x1,...,xn)) be a chart on M. Every section s : M → T ∗M maps the coordinate domain U into the standard coordinate domain T ∗U, and s is smooth if and only if ∗ the pull-backs si := s (pi) are smooth functions on U for every chart (U,ϕ) (in an atlas). All the claims in this paragraph can be proved exactly as for sections of the tangent bundle, and we leave details to the reader.

7.2 Differential 1-Forms and Fields of Covectors Let M be a manifold. Deﬁnition 7.2.1 — Differential 1-Form. A differential 1-form on M, or, simply, a 1-form, is a C∞(M)-linear map

θ : X(M) → C∞(M).

The space of 1-forms on M is denoted Ω1(M).

The space Ω1(M) of 1-forms on M is a C∞(M)-module. More precisely, it is the C∞(M)-module dual to the C∞(M)-module of vector fields. 1 n Example 7.1 — Coordinate Coframe. Let (U,ϕ = (x ,...,x )) be a chart on M. Recall that the C∞(U)-module X(U) of vector fields on U possesses a finite frame

∂ ∂ ,..., ∂x1 ∂xn

formed by coordinate vector ﬁelds. Accordingly, the dual module Ω1(U) does also possess a ﬁnite frame, the dual frame, denoted by

dx1,...,dxn (7.2)

and uniquely determined by the conditions

∂ dxi = δ i. ∂x j j

The frame (7.2) of Ω1(U) is (also) called the coordinate coframe. If

∂ X = Xi ∂xi is a vector ﬁeld on U, we have

dxi(X) = Xi, for all i = 1,...,n. 132 Chapter 7. Covectors and Differential 1-Forms

Every 1-form θ on U can be uniquely written as

i θ = θidx ,

∞ for some smooth functions θi ∈ C (U) given by

∂ θ = θ . i ∂xi

If (U,ϕ˜ = (x˜1,...,x˜n)) is another chart with the same coordinate domain, then θ can be also written as

i θ = θ˜idx˜ and

∂ ∂x˜j ∂ ∂x˜j ∂ ∂x˜j θ = θ = θ = θ = θ˜ . i ∂xi ∂xi ∂x˜j ∂xi ∂x˜i ∂xi j

Deﬁnition 7.2.2 — Differential. The differential is the map:

d : C∞(M) → Ω1(M), f 7→ d f

deﬁned by

d f (X) = X( f ), for all X ∈ X(M).

The differential d is well-deﬁned in the sense that d f is a 1-form for every f ∈ C∞(M). Check it as an exercise!

Proposition 7.2.1 — Properties of the Differential. The differential d : C∞(M) → Ω1(M) enjoys the following properties: X d is R-linear; X d satisﬁes the following Leibniz rule: d( f g) = f dg + gd f , for all f ,g ∈ C∞(M);

X dc = 0 for every constant function c.

Proof. Left as Exercise 7.2

Exercise 7.2 Prove Proposition 7.2.1.

Let (U,ϕ = (x1,...,xn)) be a chart on M, and let f ∈ C∞(U). We want to compute the 1 i components of d f ∈ Ω (U) in the coordinate coframe. If d f = θidx , then

∂ ∂ f θ = d f = , i ∂xi ∂xi hence ∂ f d f = dxi. ∂xi 7.2 Differential 1-Forms and Fields of Covectors 133

So d f encodes the gradient of f in a coordinate free manner. In particular the differential of the i-th coordinate function is exactly the i-th coordinate 1-form, motivating the notation dxi. In the second part of this section we show that differential 1-forms are equivalent to sections of the cotangent bundle. More precisely, there is a canonical C∞(M)-module isomorphism Ω1(M) =∼ ∗ 1 ∗ Γ(T M). We begin proving that a differential 1-form θ ∈ Ω (M) determines a covector θp ∈ Tp M for every point p ∈ M. In order to deﬁne θp we need two lemmas.

1 Lemma 7.2.2 Let θ ∈ Ω (M), let p0 ∈ M, and let X ∈ X(M). Then the value

θ(X)(p0)

∞ of the function θ(X) ∈ C (M) at p0, does only depend on (θ and) the value Xp0 of the vector ﬁeld X at p0.

Proof. The proof consists of various steps.

Step I: Differential 1-Forms are Local Operators For every open subset U ⊆ M, the func- 0 tion θ(X)|U does only depend on (θ and) the restriction X|U . To see this, let X be another vector 0 ∞ ﬁeld on M such that X |U = X|U , let p ∈ U , and let β ∈ C (M) be a bump function relative to the data (U , p). In other words, β = 1 in an open neighborhood of p, and suppβ ⊆ U , in particular, β = 0 outside U . Put η = 1 − β. Then

X − X0 = η(X − X0), hence

θ(X − X0)(p) = θ(η(X − X0))(p) = ηθ(X − X0)(p) = η(p)θ(X − X0)(p) = 0, so that

θ(X)(p) = θ(X − X0)(p) + θ(X0)(p) = θ(X0)(p).

It follows from the arbitrariness of p ∈ U , that

0 θ(X)|U = θ(X )|U .

Step II: Local Extension Lemma for Vector Fields Let S ⊆ M be a submanifold, and let X be a vector field on S. For every point p0 in S, there exists an open neighborhood U ⊆ M of p0 and a vector field Xe on M such that Xe|U is tangent to S ∩U, and Xe|U |S∩U = X|S∩U . We prove this claim in the case when S = U is an open submanifold, and leave the general case as an exercise for the reader (the proof is not very different from that of the Local Extension Lemma for functions). So, let U ⊆ M be an open submanifold, let p0 ∈ U , and let X ∈ X(U ). Choose a bump function ∞ β ∈ C (M) relative to the data (U , p0), and consider the vector field Xe defined (thorugh its values) as follows: β(p)Xp if p ∈ U Xep := . 0 if p ∈/ U

In view of the Gluing Lemma for Vector Fields, Xe is a well-defined vector field on M. Additionally, Xe agrees with X on an open neighborhood of p0 (specifically, the open neighborhood where β = 1). This concludes the proof of the Local Extension Lemma (for S = U an open submanifold). 134 Chapter 7. Covectors and Differential 1-Forms

Step III: Vector Fields Vanishing at a Point Let X be a vector ﬁeld vanishing at a point p, i.e. Xp = 0, then there exist vector ﬁelds X1,...,Xn, and smooth functions f1,..., fn such that: 1) the fi all vanish at p, i.e. fi(p) = 0, and 2)

X = f1X1 + ··· + fnXn in an open neighborhood of p. To see this, consider (U,ϕ = (x1,...,xn)) a chart around p. Then

∂ X| = gi U ∂xi

∞ for some smooth functions gi ∈ C (U). Then

∂ 0 = X = gi(p) | . p ∂xi p

i ∞ Hence, g (p) = 0 for all i = 1,...,n. Now, let f1,..., fn ∈ C (M) be smooth functions agreeing with 1 n g ,...,g in an open neighborhood of p. In particular, fi(p) = 0, for all i = 1,...,n. Finally, use X ,...,X ∈ X(M) ∂ ,..., ∂ Step II to ﬁnd vector ﬁelds 1 n agreeing with ∂x1 ∂xn in an open neighborhood of p, and notice that

X = f1X1 + ··· + fnXn in an open neighborhood of p as required. This concludes Step III.

0 Step IV: Conclusion Let p0 ∈ M be a point, and let X,X ∈ X(M) be vector ﬁelds such that X = X0 (X)(p ) = (X0)(p ) p0 p0 . We want to show that θ 0 θ 0 . To do this notice that

= X − X0 = (X − X0) . 0 p0 p0 p0

∞ Hence, it follows from Step III, that there are functions f1,..., fn ∈ C (M) and vector ﬁelds X1,...,Xn ∈ X(M) such that fi(p0) = 0, and

0 X − X = f1X1 + ··· + fnXn in an open neighborhood of p0. It now follows from Step I that

0 θ(X − X ) = θ( f1X1 + ··· + fnXn) = f1θ(X1) + ··· + fnθ(Xn) in an open neighborhood of p0. In particular

0 θ(X −X )(p0) = ( f1θ(X1) + ··· + fnθ(Xn))(p0) = f1(p0)θ(X1)(p0)+···+ fn(p0)θ(Xn)(p0) = 0.

We conclude noticing (again) that

0 0 0 θ(X)(p0) = θ(X − X )(p0) + θ(X )(p0) = θ(X )(p0).

Lemma 7.2.3 Let p ∈ M, and let v ∈ TpM. Then there exists a vector ﬁeld X ∈ X(M) such that Xp = v. 7.2 Differential 1-Forms and Fields of Covectors 135

Proof. Let (U,ϕ = (x1,...,xn)) be a chart around p. Then ∂ v = vi | ∂xi p for some real numbers vi. Interpreting the vi as constant functions on U we can deﬁne a vector ﬁeld on U: ∂ Y = vi ∈ X(U). ∂xi

Clearly, Yp = v. From the Local Extension Lemma for Vector Fields (Step II in the proof of Lemma 7.2.2) there exists a vector ﬁeld X ∈ X(M) on M such that X and Y agree in an open neighborhood of p. In particular

Xp = Yp = v.

This concludes the proof. We are now ready to present the main result of this section. Begin with a differential 1-form θ ∗ on a manifold M. For any point p ∈ M we want to define a covector θp ∈ Tp M. So, let v ∈ TpM, and let X ∈ X(M) be any vector field such that Xp = v. The vector field X exists in view of Lemma 7.2.3. We put

θp(v) := θ(X)(p).

From Lemma 7.2.2, θp(v) does only depend on (θ and) v. Hence, θp is a well-deﬁned map

θp : TpM → R.

It is easy to see that θp is a covector. Indeed, let v,w ∈ TpM, let a,b ∈ R, and let X,Y ∈ X(M) be vector ﬁelds such that Xp = v and Yp = w. Then, from Exercise 5.5,

(aX + bY)p = aXp + bYp = av + bw, and

θp(av + bw) = θ(aX + bY)(p) = (aθ(X) + bθ(Y))(p)

= aθ(X)(p) + bθ(Y)(p) = aθp(v) + bθp(w).

∗ The covector θp ∈ Tp M is called the value at p of the 1-form θ.

Exercise 7.3 Prove that, for every θ,κ ∈ Ω1(M), every f ∈ C∞(M), and every p ∈ M,

(θ + κ)p = θp + κp,

( f θ)p = f (p)θp,

(d f )p = dp f . (7.3)

Additionally, let (U,ϕ = (x1,...,xn)) be a chart around p. Prove that the value of the coordinate i i 1-form dx at p is exactly the coordinate covector dpx .

1 Let θ ∈ Ω (M). The assignment p 7→ θp is a(n a priori non-necessarily smooth) section sθ of the cotangent bundle T ∗M:

∗ sθ : M → T M, p 7→ sθ (p) := θp. 136 Chapter 7. Covectors and Differential 1-Forms

Proposition 7.2.4 The map

1 ∗ Ω (M) → Γ(T M), θ 7→ sθ (7.4)

is a well-deﬁned C∞(M)-module isomorphism.

1 Proof. First we prove that for every θ ∈ Ω (M), the section sθ is actually smooth. As for vector 1 n ﬁelds, it is enough to check that, for every chart (U,ϕ = (x ,...,x )) on M, the components (sθ )i of sθ in the chart (U,ϕ) (see the end of Section 7.1) are smooth functions on U. To do this, we ﬁx p0 ∈ U and show that (sθ )i is smooth around p0. So, let p ∈ U and compute

∂ (s ) (p) = p (s (p)) = p (θ ) = i-th component of θ in the coordinate coframe = θ | . θ i i θ i p p p ∂xi p

∂ To continue the computation, we need to choose a vector ﬁeld Xi ∈ X(M) such that (Xi)p = ∂xi |p. It is convenient to choose Xi in a more precise way. Namely, we choose Xi such that it agrees with ∂ ∂xi in an open neighborhood V ⊆ U of p0. Such vector ﬁeld exists in view of the Local Extension Lemma for Vector Fields, and we have

∂ (s ) (p) = θ | = θ(X )(p) θ i p ∂xi p i for all p ∈ V. Clearly, θ(Xi) depends smoothly on p, so that (sθ )i|V is smooth. We leave it as Exercise 7.4 (Point (1)) to check that the map (7.4) is C∞(M)-linear. It remains to check that (7.4) is invertible. To do this we deﬁne its inverse. Given a section s ∈ Γ(T ∗M) of ∗ 1 T M, we construct a 1-form θs ∈ Ω (M) in such a way that

sθs = s and θsθ = θ, (7.5) for all s ∈ Γ(T ∗M), and all θ ∈ Ω1(M). For every X ∈ X(M) we deﬁne

θs(X) : M → R, p 7→ θs(X)(p) := s(p)(Xp).

We have to show that ∞ ∗ X θs(X) ∈ C (M) for all s ∈ Γ(T M) and all X ∈ X(M), ∞ ∞ X θs : X(M) → C (M) is a C (M)-linear map, hence a differential 1-form, and ∗ 1 X the identities (7.5) hold for all s ∈ Γ(T M) and all θ ∈ Ω (M). We only discuss the ﬁrst point and leave it to the reader to prove the remaining two points as ∗ Exercise 7.4 (Points (2) and (3)). Let s ∈ Γ(T M) and X ∈ X(M). In order to prove that θs(X) is 1 n smooth, we prove that the restriction θs(X)|U is smooth for every chart (U,ϕ = (x ,...,x )). Now

∂ X| = Xi U ∂xi for some smooth functions Xi on U, and, for p ∈ U, we have

∂ θ (X)(p) = s(p)(X ) = s(p) Xi(p) | = Xi(p)s (p) = (Xis )(p). s p ∂xi p i i

As the si are smooth, the last expression depends smoothly on p. This concludes the proof. 7.2 Differential 1-Forms and Fields of Covectors 137

Exercise 7.4 Complete the proof of Proposition 7.2.4 showing that ∞ (1) The map θ 7→ sθ is C (M)-linear, ∗ ∞ ∞ (2) for every s ∈ Γ(T M), the map θs : X(M) → C (M) is C (M)-linear, and (3) the map s 7→ θs inverts θ 7→ sθ .

As for vector fields, Proposition 7.2.4 has some important consequences. First of all, a 1-form θ on M can be “restricted” to an open submanifold U ⊆ M by restricting the associated section, 1 exactly as for vector fields (we leave details to the reader). We denote by θ|U ∈ Ω (U ) the 1 restricted 1-form. Clearly we have θ|U (XU ) = θ(X)|U for all θ ∈ Ω (M) and all X ∈ X(M). A second consequence of Proposition 7.2.4 is a Gluing Lemma for Differential 1-Forms. The statement and the proof are formally identical to that of the Gluing Lemma for Vector Fields. Finally, we can combine the Gluing Lemma for Differential 1-Forms with Example 7.1 to find that, for every 1-form θ on M, (1) θ is completely determined by its restrictions θ|U to some coordinate domains U covering M, 1 n (2) for every chart (U,ϕ = (x ,...,x )), the restriction θ|U can be uniquely written in the form

i θ|U = θidx

∞ for some smooth functions θi ∈ C (U), called the components of θ in the chart (U,ϕ). (3) given a vector ﬁeld X ∈ X(M) locally given by ∂ X| = Xi , U ∂xi we have

i θ(X)|U = θ|U (X|U ) = θiX .

The proof of Proposition 7.2.4 shows that the components θi of a 1-form in the chart (U,ϕ = (x1,...,xn)) coincide with the components of the corresponding section. When the chart (U,ϕ) changes, the components of θ change. The transformation rule has been already discussed in Example 7.1.

Exercise 7.5 Consider R2 with standard coordinates x,y, and the polar chart (U,ψ = (r,φ)) on it. Recall that U = R2 r {y = 0,x ≥ 0}, Ub = (0,+∞) × (0,2π) and p ψ : U → Ub, (x,y) 7→ ψ(x,y) = (r,φ) := x2 + y2,φ(x,y) ,

with  arccos √ x if y ≥ 0  2 2 φ(x,y) := x +y . −arccos √ x if y < 0  x2+y2

Recall also that the inverse map is

−1 −1 ψ : Ub → U, (r,φ) 7→ ψ (r,φ) = (r cosφ,r sinφ).

Consider the following differential 1-forms on R2:

θ = xdy − ydx and κ = xdx + ydy. 138 Chapter 7. Covectors and Differential 1-Forms

Prove that

2 θ|U = r dφ and κ|U = rdr.

Let (U,ϕ = (x1,...,xn)) be a chart on M. Example 7.1 shows not only that the C∞(U)-module Ω1(U) possesses a ﬁnite frame, but also that it is generated by the differentials dx1,...,dxn of the coordinate functions x1,...,xn. In particular, it is generated by 1-forms of the type d f for some f ∈ C∞(U). The last fact is true even globally according to the following

Theorem 7.2.5 Let M be a manifold. The C∞(M)-module Ω1(M) of differential 1-forms on M is generated by differentials d f of smooth functions f ∈ C∞(M), i.e. every differential 1-form θ ∈ Ω1(M) can be written (in a possibly non-unique way) in the form

θ = f1dg1 + ··· + fkdgk

∞ for some smooth functions f1,g1,..., fk,gk ∈ C (M).

Proof. The proof is non-trivial, in the sense that it is based on more involved results on smooth manifolds, and we omit it.

7.3 Differential 1-Forms and Smooth Maps Let M,N be manifolds and let F : M → N be a smooth map. While we can move tangent vectors from M to N via the tangent maps to F, there is no way to move vector ﬁelds from M to N, nor from N to M, unless F is a diffeomorphism. On the other hand, there is a natural way to move differential forms from N to M via a pull-back construction in many ways similar to the pull-back of functions. Before deﬁning the pull-back of differential forms along F, consider a point p in M and the tangent map

dpF : TpM → TF(p)N. The dual map is denoted by:

∗ ∗ ∗ ∗ dpF : TF(p)N → Tp M, θ 7→ dpF(θ) := θ ◦ dpF.

Example 7.2 Let S ⊆ M be a submanifold, and let p ∈ S. The dual map

∗ ∗ ∗ dpiS : Tp M → Tp S

to the tangent map to the inclusion iS : S ,→ M is a surjection consisting in restricting a covector θ : TpM → R on M at p ∈ S to tangent vectors to S.

Proposition 7.3.1 — Naturality of the Differential at a Point. For every smooth function f ∈ C∞(N) we have

∗ ∗ dp (F ( f )) = dpF dF(p) f .

Proof. Let v ∈ TpM, then ∗ ∗ ∗ dp (F ( f ))(v) = v(F ( f )) = dpF(v)( f ) = dF(p) f (dpF(v)) = dpF dF(p) f (v).

7.3 Differential 1-Forms and Smooth Maps 139

Now let (U,ϕ = (x1,...,xm)) be a chart on M around p, and let (V,ψ = (y1,...,yn)) be a chart on N around F(p) such that F(U) ⊆ V. Put P = ϕ(p). The representative matrix of the dual map ∗ ∗ ∗ dpF : TF(p)N → Tp M in the coordinate coframes

1 n 1 m dF(p)y ,...,dF(p)y and dpx ,...,dpx is the transpose of the representative matrix of the tangent map dpF : TpM → TF(p)N in the coordinate frames

∂ ∂ ∂ ∂ | ,..., | and | ,... | . ∂x1 p ∂xm p ∂y1 F(p) ∂yn F(p)

Hence it is

!i=1,...,m ∂Fa i=1,...,m ∂Fa (p) = b (P) ∂xi ∂ti a=1,...,m a=1,...,m

Where, as usual, Fa = F∗(ya) = ya ◦ F, a = 1,...,m. In other words,

∂Fa d∗F(d ya) = (p)d xi,d Fa = d F∗(ya), p F(p) ∂xi p p p for all a ∈ {1,...,m}, which is duly consistent with Proposition 7.3.1. We now come to the main result of this section. Proposition 7.3.2 — Pull-Back of 1-Forms. Let M,N be smooth manifolds and let F : M → N be a smooth map. Then there exists a unique map

F∗ : Ω1(N) → Ω1(M),

called the pull-back of differential 1-forms along F such that, for all θ,κ ∈ Ω1(N), and all f ∈ C∞(N), (1) F∗(θ + κ) = F∗(θ) + F∗(κ), (2) F∗( f θ) = F∗( f )F∗(θ), and (3) F∗(d f ) = dF∗( f ).

Proof. Let θ ∈ Ω1(N). We deﬁne F∗(θ) ∈ Ω1(M) through its values. So, for every p ∈ M, put

∗ ∗ F (θ)p = dpF(θF(p)).

In other words, for every v ∈ TpM,

∗ F (θ)p(v) = θF(p) (dpF(v)).

∗ ∗ This deﬁnes a section s : p 7→ F (θ)p of T M, and we have to show that it is a smooth section. To do this, we choose charts (U,ϕ = (x1,...,xm)) and (V,ψ = (y1,...,yn)) on M and N respectively, ∗ such that F(U) ⊆ V, and compute the components si = s (pi) of s in the chart (U,ϕ). We have

a θ|V = θady , 140 Chapter 7. Covectors and Differential 1-Forms

∞ for some smooth functions θa ∈ C (V), a = 1,...,n. Then, for every p ∈ U, ∂ s (p) = s(p) | i ∂xi p ∂ = F∗(θ) | p ∂xi p ∂ = θ d F | F(p) p ∂xi p ∂Fa ∂ = θ (p) | F(p) ∂xi ∂ya F(p) ∂Fa ∂ = (p)θ | ∂xi F(p) ∂ya F(p) ∂Fa = (p)θ (F(p)). ∂xi a a ∗ a a ∂Fa for all i ∈ {1,...,m}, where, as usual, F = F (y ) = y ◦ F, a = 1,...,n. Both ∂xi and θa ◦ F = ∗ F (θa) depend smoothly on p ∈ U, hence si is smooth. It follows from the arbitrariness of (U,ϕ) and (V,ψ) that s is smooth, and F∗(θ) is a well-deﬁned 1-form on M. We leave it as Exercise 7.6 to show that the map F∗ : Ω1(N) → Ω1(M) enjoys Properties (1) and (2) in the statement, and we prove here Property (3). So, let f ∈ C∞(N), and p ∈ M, and compute

∗ ∗ F (d f )p = dpF((d f )F(p)) ∗ = dpF(dF(p) f ) (Equation (7.3) in Exercise 7.3) ∗ = dp(F ( f )) (Proposition 7.3.1) ∗ = (dF ( f ))p. (Equation (7.3) in Exercise 7.3) It follows from the arbitrariness p, that F∗(d f ) = dF∗( f ) as claimed. It remains to show that Properties (1)–(3) uniquely deﬁne F∗ : Ω1(N) → Ω1(M). To do this we use Theorem 7.2.5. So, let 0F∗ : Ω1(N) → Ω1(M) be another map satisfying (1)–(3). Take an 1 ∞ arbitrary θ ∈ Ω (N). From Theorem 7.2.5 there exist f1,g1,..., fk,gk ∈ C (N) such that

θ = f1dg1 + ··· + fkgk. Hence 0 ∗ 0 ∗ F (θ) = F ( f1dg1 + ··· + fkgk) ∗ 0 ∗ ∗ 0 ∗ = F ( f1) F (dg1) + ··· + F ( fk) F (dgk) ∗ ∗ ∗ ∗ = F ( f1)dF (g1) + ··· + F ( fk)dF (gk) = F∗(θ).

This concludes the proof.

Exercise 7.6 Complete the proof of Proposition 7.3.2 showing that the map F∗ : Ω1(N) → 1 Ω (M) enjoys Properties (1) and (2) in the statement.

Let F : M → N be a smooth map, and let (U,ϕ) and (V,ψ) be charts as in the proof of Proposition 7.3.2. It immediately follows from that proof that ∂Fa F∗(θ)| = F∗(θ| ) = F∗(θ ) dxi. U V a ∂xi 7.3 Differential 1-Forms and Smooth Maps 141

2 3 Example 7.3 Consider R with standard coordinates (u,v), and R with standard coordinates (x,y,z). Consider also the smooth map

2 3 2 2 F : R → R , (u,v) 7→ F(u,v) = (u ,uv,v ).

We want to compute the pull-back along F of the 1-form θ = ydx − xdy ∈ Ω1(R3): F∗(θ) = F∗(ydx − xdy) = F∗(y)dF∗(x) − F∗(x)dF∗(y) = uvdu2 − u2d(uv) = 2u2vdu − u2vdu − u3dv = u2vdu − u3dv.

Example 7.4 — Restriction of a 1-Form to a Submanifold. If iU : U → M is the inclusion of 1 ∗ an open submanifold, and θ ∈ Ω (M) is a 1-form, then iU (θ) = θ|U is the (usual) restriction of θ to U . More generally, we can “restrict” a 1-form θ ∈ Ω1(M) to any submanifold S ⊆ M by pulling it ∗ back along the inclusion iS : S ,→ M. The pull-back iS(θ) is sometimes denoted by θ|S and called the restriction of θ to S. In practice, it is obtained by restricting the values of θ at points of S to tangent vectors to S. For instance, let M be R3 with standard coordinates (x,y,z), let S = S2 ⊆ R3 be the 2- dimensional sphere, and let

1 3 θ = xdx + ydy + zdz ∈ Ω (R ).

2 We want to show that θ|S2 = 0. We do this in two different ways. First of all let P = (a,b,c) ∈ S and let ∂ ∂ ∂ v = A | + B | +C | ∈ T S2. ∂x P ∂y P ∂z P P This means that

aA + bB + cC = 0.

Hence ∂ ∂ ∂ θ (v) = (ad x + bd y + xd z) A | + B | +C | = aA + bB + cC = 0. P P P P ∂x P ∂y P ∂z P

It follows from the arbitrariness of P and v that θ|S2 = 0. Alternatively, notice that 1 θ = d x2 + y2 + z2. 2 Hence ∗ ∗ 1 2 2 2 1 ∗ 2 2 2 1 θ| 2 = i 2 (θ) = i 2 d x + y + z = di 2 x + y + z = d1 = 0. S S S 2 2 S 2

2 Exercise 7.7 Consider R with standard coordinates (x,y), and let (U+,ϕ+) be the stereographic chart (from the north) on the circle S1 ⊆ R2. Consider the 1-form

θ = ydx − xdy. 142 Chapter 7. Covectors and Differential 1-Forms

Write θ|U+ in terms of the stereographic coordinate X+. (Hint: notice that ∂x|U+ ∂y|U+ θ|U+ = y|U+ d (x|U+ ) − x|U+ d (y|U+ ) = y|U+ − x|U+ dX+. ∂X+ ∂X+

Finally use the explicit expressions of x|U+ ,y|U+ in terms of X+.) 8. Differential Forms and Cartan Calculus

In this chapter we define higher degree differential forms. Differential forms are of great importance in modern Differential Geometry. Numerous interesting geometric structures (and physical systems) can be formalized via differential forms, and they play a key role in Algebraic Topology: the branch of Mathematics that uses algebraic constructions to study topological properties of spaces (in our case, smooth manifolds and related geometric objects). Additionally, differential forms serve as integrands in a coordinate free integration theory (not discussed in these notes). The basic rules of the calculus with vector fields and differential forms are collectively called Cartan calculus, after the French mathematician Èlie Cartan (in the head picture of the chapter), who first defined higher degree differential forms as we know them in modern Differential Geometry. In this chapter we will skip, or just sketch, most of the proofs.

8.1 Algebraic Preliminaries: Alternating Forms Let A be a real (associative, commutative, unital) algebra, let M be an A-module, and let k be a non-negative integer. Deﬁnition 8.1.1 — Alternating Forms. An alternating (or skewsymmetric) A-multilinear k- form (or, simply, a k-form) on M is a map

ω : M × ··· × M → A | {z } k times such that X ω is A-multilinear, X ω is alternating, i.e. ω(µ1,..., µk) = 0 whenever two arguments among µ1,..., µk ∈ M coincide. When k = 0 we interpret this deﬁnition requiring ω to be simply an element in A.

R An alternating k-form ω : M × ··· × M → A is automatically skewsymmetric i.e., for all 144 Chapter 8. Differential Forms and Cartan Calculus

µ1,..., µk ∈ M , and for all permutations σ ∈ Sk,

σ ω(µσ(1),..., µσ(k)) = (−) ω(µ1,..., µk),

where we denoted by (−)σ the sign of the permutation σ. As we are working over a ﬁeld of 0 characteristic, the converse is also true, i.e. skewsymmetric forms are alternating.

n Example 8.1 Let A = R, and M = R . The determinant

n n det : R × ··· × R → R, (A1,...,An) 7→ det(A1 ···An) | {z } n times

n is an n-form on R .

We denote by Altk(M ,A) the space of alternating k-forms on M . In particular, Alt0(M ,A) = A, and Alt1(M ,A) = Hom(M ,A) (the dual module of M ). If ω ∈ Altk(M ,A), we also denote |ω| = k, and call it the degree of ω (so negative degree alternating forms vanish). It is easy to see that Altk(M ,A) is an A-module for every k. The module operations are the following: for every ω,ω1,ω2 ∈ Altk(M ,A), and all a ∈ A, the sum ω1 + ω2 is given by

(ω1 + ω2)(µ1,..., µk) = ω1(µ1,..., µk) + ω2(µ1,..., µk), and the product aω is given by

(aω)(µ1,..., µk) = aω(µ1,..., µk), for all µ1,..., µk ∈ M . When k is a negative integer, we put Altk(M ,A) = 0: the zero module. In other words, by definition, negative degree alternating forms vanish. There are more operations on alternating forms. In order to define them, we need one more Definition 8.1.2 For any two non-negative integers k,l, a (k,l)-unshuffle, or simply an unshuffle, is a permutation σ ∈ Sk+l such that

σ(1) < ··· < σ(k) and σ(k + 1) < ··· < σ(k + l).

The subset in Sk+l consisting of (k,l)-unshufﬂes is denoted by Sk,l.

Now, for every k,l there is a wedge product (also called exterior product):

∧ : Altk(M ,A) × Altl(M ,A) → Altk+l(M ,A), (ω,ρ) 7→ ω ∧ ρ. (8.1)

The (k + l)-form ω ∧ ρ is deﬁned by

σ (ω ∧ ρ)(µ1,..., µk+l) = ∑ (−) ω(µσ(1),..., µσ(k))ρ(µσ(k+1),..., µσ(k+l)), (8.2) σ∈Sk,l where, as indicated, the sum runs over the (k,l)-unshuffles. When k = 0, ω = a is just an element in A, and we interpret Formula (8.2) by putting ω ∧ ρ = aρ. Similarly, when l = 0, ρ = b ∈ A, and we put ω ∧ ρ = bω. Finally, when either k or l is negative, then either ω or ρ vanish and we put ω ∧ ρ = 0. It is not too hard to see that ω ∧ ρ is indeed a well-defined (k + l)-form: the A-multilinearity of ω ∧ ρ immediately follows from the A-multilinearity of ω and ρ, and the skew-symmetry follows from both the skew-symmetry of ω and ρ, and the sign (−)σ in the defining formula (8.2). 8.1 Algebraic Preliminaries: Alternating Forms 145

1 2 3 4 5 6 7 8 9 1 3 4 8 9 2 5 6 7

Figure 8.1: A (5,4)-unshufﬂe σ.

R The wedge product ω ∧ ρ can be equivalently defined by the following formula 1 (ω ∧ρ)(µ ,..., µ ) = (−)σ ω(µ ,..., µ )ρ(µ ,..., µ ). (8.3) 1 k+l k!l! ∑ σ(1) σ(k) σ(k+1) σ(k+l) σ∈Sk+l We stress that, in (8.3), the sum runs over all permutations, not just the unshuffles. However, as ω and ρ are alternating, hence skew-symmetric, several terms in (8.3) are counted several times, and we have to normalize by the factor 1/(k!l!). Using the unshuffles, instead, is a way to avoid such repetitions. Additionally, it makes it easier to prove several properties of the wedge product.

Proposition 8.1.1 The wedge product enjoys the following properties: X (bilinearity) the wedge product is A-bilinear; X (associativity) for any three alternating forms ω,ρ,τ,

(ω ∧ ρ) ∧ τ = ω ∧ (ρ ∧ τ);

X (graded commutativity) for any two alternating forms ω,ρ,

ρ ∧ ω = (−)|ω||ρ|ω ∧ ρ.

Proof. The bilinearity should be clear. The associativity and the graded commutativity can be easily proved by playing with unshufﬂes. We omit the technical details. It immediately follows from the graded commutativity of the wedge product that, if ω is any alternating form of odd degree, then

ω ∧ ω = 0

(do you see it?). 146 Chapter 8. Differential Forms and Cartan Calculus

The wedge products (8.1) can be combined into a single interior operation in the direct sum M Alt•(M ,A) := Altk(M ,A). k∈Z Before discussing this, we brieﬂy recall what is the direct sum of a sequence of modules. So let ( ) be a sequence of A-modules. By deﬁnition, the direct sum Nk k∈Z M N• = Nk (8.4) k∈Z consists of sequences

(νk)k∈Z, νk ∈ Nk for all k ∈ Z, with the additional property that only ﬁnitely many of the νk are non-zero. The direct sum N• is an A-module with the following two term-wise operations:

0 0 (νk)k∈Z + (νk)k∈Z := (νk + νk)k∈Z, and

a(νk)k∈Z := (aνk)k∈Z,

0 for all (νk)k∈Z,(νk)k∈Z ∈ N•, and all a ∈ A. Notice that the zero vector in N• is the zero sequence (0)k∈Z, and the opposite of an element (νk)k∈Z is the opposite sequence (−νk)k∈Z. A module of the form (8.4) is also called a graded module. If A = R we rather speak about a graded vector space. 1 n Example 8.2 The real vector space R[t ,...,t ] of real polynomials in the real indeterminates 1 n 1 n (t ,...,t ) is a graded vector space. Indeed, denote by R[t ,...,t ]k the space of homogeneous 1 n 1 n polynomials of degree k. We also put R[t ,...,t ]k = 0 for k < 0. Then R[t ,...,t ] is (canonically 1 n isomorphic to) the direct sum of the R[t ,...,t ]k:

1 n M 1 n R[t ,...,t ] = R[t ,...,t ]k k∈Z

(do you see it?). L The graded module N• = k∈Z Nk comes with canonical monomorphisms of A-modules:

ιk0 : Nk0 → N•, ν 7→ ιk0 (ν) := (νk)k∈Z where ν if k = k0 νk := . 0 if k 6= k0

The image of ιk0 consists of sequences whose terms all vanish except, possibly, for the k0-th one. If

(νk)k∈Z ∈ N• is a sequence whose only non-zero entries are νk1 ,...,νk` , then, clearly,

(νk)k∈Z = ιk1 (νk1 ) + ··· + ιk` (νk` ). (8.5)

In the following, we will always interpret the Nk as subspaces of N , identifying them with their images under the ιk (if we do so, then N• is exactly the direct sum of its subspaces Nk, in the usual sense of the direct sum of subspaces). For instance, rather than (8.5), we will simply write

(νk)k∈Z = νk1 + ··· + νk` . 8.1 Algebraic Preliminaries: Alternating Forms 147

In this way, N• will equivalently consist of ﬁnite sums of elements in the Nk. An element ν of N• in (the image of) Nk (under ιk) is called homogeneous, the integer k is called the degree of ν, and we write |ν| = k. So N• is generated by homogeneous elements. Now, we go back to alternating forms. So, let M be an A-module. The Altk(M ,A) will play the role of the Nk. Elements in the graded module M Alt•(M ,A) = Altk(M ,A) k∈Z will be also referred to as alternating forms. Homogeneous forms are then those with a “deﬁnite number of entries”.

Proposition 8.1.2 — Wedge Product. There exists a unique R-bilinear map

∧ : Alt•(M ,A) × Alt•(M ,A) → Alt•(M ,A), (ω,ρ) 7→ ω ∧ ρ (8.6)

extending the wedge products (8.1), and also called the wedge product. The wedge product (8.6) enjoys the following properties: X (bilinearity) the wedge product is A-bilinear; X (associativity) for any three alternating forms ω,ρ,τ,

(ω ∧ ρ) ∧ τ = ω ∧ (ρ ∧ τ);

X (graded commutativity) for any two homogeneous alternating forms ω,ρ,

ρ ∧ ω = (−)|ω||ρ|ω ∧ ρ;

X (unitality) for any alternating form ω

1 ∧ ω = ω ∧ 1 = ω.

Proof. Let ω = (ωk)k∈Z and ρ = (ρl)l∈Z be two (non-necessarily homogeneous) alternating forms. By deﬁnition, their wedge product is

ω ∧ ρ = (τm)m∈Z, where

τm := ∑ ωk ∧ ρl, k+l=m where, in the last formula, we used the wedge products (8.1). It is easy to see that the operation in Alt•(M ,A) so deﬁned enjoys all the required properties. We leave the simple details to the reader.

The vector space Alt•(M ,A) of alternating forms, together with the wedge product (8.6), is an important instance of a graded algebra. Deﬁnition 8.1.3 — Graded Algebra. A real, associative, graded commutative algebra with unit, or simply a graded R-algebra, or just a graded algebra, is a graded vector space L (A• = k∈ZAk,+,·)

equipped with an additional interior operation ∧ : A• × A• → A•, such that (1) ∧ is R-bilinear, (2) ∧ is associative, 148 Chapter 8. Differential Forms and Cartan Calculus

equivalently, (A•,+,∧) is a ring and, additionally, (3) ∧ preserves the degree, i.e. for every two homogeneous elements α,β ∈ A•, the product α ∧ β is also homogeneous and

|α ∧ β| = |α| + |β|,

(4) ∧ is graded commutative, i.e. for every two homogeneous elements α,β ∈ A•

β ∧ α = (−)|α||β|α ∧ β,

(1) ∧ possesses a unit 1 ∈ A0.

If A• and B• are graded algebras, a graded algebra homomorphism between A• and B• is an R-linear map H : A• → B• such that (I) H preserves the degree, i.e. for every homogeneous element α ∈ A•, the image H(α) ∈ B• is also homogeneous of the same degree:

|H(α)| = |α|;

(II) H preserves the product, i.e. for every two elements α,β ∈ A•,

H(α ∧ β) = H(α) ∧ H(β);

(III) H preserves the unit, i.e.

H(1) = 1.

As usual, the identity is a graded algebra homomorphism, and the composition of graded algebra homomorphisms is a graded algebra homomorphism.

Theorem 8.1.3 Let A be an algebra, let M be an A-module possessing a ﬁnite frame

(ε1,...,εn),

and let

ε1,...,εn

be the dual frame of Hom(M ,A) = Alt1(M ,A). Then, for every 0 ≤ k ≤ n, the A-module Altk(M ,A) of alternating k-forms on M is generated by

i1 ik ε ∧ ··· ∧ ε , i1,...,ik ∈ {1,...,n}.

More precisely, the k-forms

εi1 ∧ ··· ∧ εik i1<···

ordered (for instance) lexicographically, form a frame of Altk(M ,A) of cardinality

n . k

On the other hand, for k > n, Altk(M ,A) = 0. 8.1 Algebraic Preliminaries: Alternating Forms 149

In the hypothesis of Theorem 8.1.3, let i1,...,ik ∈ {1,...,n} be any k-tuple of indexes, and consider the multiple wedge product

εi1 ∧ ··· ∧ εik .

We notice preliminarily that, from the graded commutativity of the wedge product, when at least two i i i i indexes among i1,...,ik coincide, then ε 1 ∧ ··· ∧ ε k = 0. So ε 1 ∧ ··· ∧ ε k can only be non-trivial when the i j are all different, j = 1,...,k, for instance when i1 < ··· < ik.

Proof of Theorem 8.1.3. The proof generalizes the proof of the existence of the dual frame and we only sketch it. It is enough to prove the “more precise version” of the statement. Let k ≥ 0. First we prove that the system

εi1 ∧ ··· ∧ εik (8.7) i1<···

bi1···ik := ω(εi1 ,...,εik ) ∈ A.

We want to show that

i1 ik ω = ∑ bi1···ik ε ∧ ··· ∧ ε . (8.8) i1<···

Denote by ω0 the right hand side of (8.8). By multilinearity it is enough to show that ω0 acts as ω on elements in the frame (ε1,...,εn). By skew-symmetry it is enough to check that

0 ω (ε j1 ,...,ε jk ) = ω(ε j1 ,...,ε jk ) = b j1··· jk for all 1 ≤ j1 < ··· < jk ≤ n. So let 1 ≤ j1 < ··· < jk ≤ n and compute

0 ω (ε j1 ,...,ε jk ) ! i1 ik = ∑ bi1···ik ε ∧ ··· ∧ ε (ε j1 ,...,ε jk ) i1<···

As both i1,...,ik and j1,..., jk are increasingly ordered, the only non-trivial contributions to

i1 ik ε ∧ ··· ∧ ε (ε j1 ,...,ε jk ) come from

εi1 (ε )···εik (ε ) = δ i1 ···δ ik . j1 jk j1 jk

Hence

0 ω (ε j1 ,...,ε jk ) i1 ik = ∑ bi1···ik ε ∧ ··· ∧ ε (ε j1 ,...,ε jk ) i1<···

It remains to show that the system (8.7) consists of linearly independent k-forms. So, let bi1···ik ∈ A be scalars such that

0 i1 ik ω = ∑ bi1···ik ε ∧ ··· ∧ ε = 0. i1<···

0 0 = ω (ε j1 ,...,ε jk ) = b j1··· jk .

This concludes the proof. Theorem 8.1.3 says that, when M possesses a ﬁnite frame (ε1,...,εn), then every k-form ω ∈ Altk(M ,A) can be written in the form

i1 ik ω = ai1···ik ε ∧ ··· ∧ ε (8.9) for some ai1···ik ∈ A, where the sum (omitted according to the Einstein convention) runs over all indexes i1,...,ik. On the other hand ω can be uniquely written in the form

i1 ik ω = ∑ bi1,...,ik ε ∧ ··· ∧ ε . (8.10) i1<···

Notice that the ai1···ik can be uniquely chosen in such a way that they enjoy the following skew- symmetry property

σ aiσ(1)···iσ(k) = (−) ai1···ik (8.11) for every permutation σ ∈ Sk (do you see it?). In this case, the relationship between the ai1···ik and the bi1···ik is easily obtained by appropriately reordering the factors in (8.9), and it is

bi1···ik = k!ai1···ik = ω(εi1 ,...,εik ), 1 ≤ i1 < ··· < ik ≤ n.

It is sometimes convenient to expand ω as in (8.9) (with skew-symmetric coefﬁcients ai1···ik ) rather than as in (8.10).

Corollary 8.1.4 — Alternating Forms on a Module with a Finite Frame. Let A be an algebra, let M be an A-module possessing a ﬁnite frame

(ε1,...,εn),

and let

ε1,...,εn,

be the dual frame of Hom(M ,A) = Alt1(M ,A). Then the graded R-algebra Alt•(M ,A) of alternating forms is generated by (1) 0-forms, i.e. elements in Alt0(M ,A) = A, and (2) the 1-forms ε1,...,εn. This means that every form ω ∈ Alt•(M ,A) can be written in the form

i1 ik ω = ∑ ai1···ik ε ∧ ··· ∧ ε , (8.12) k∈Z

for some ai1···ik ∈ A, where only ﬁnitely many of the ai1···ik are non-zero. 8.2 Higher Degree Differential Forms 151

Proof. Obvious.

Notice that, from the above discussion, the ai1···ik can be uniquely chosen so that they enjoy the skew-symmetry property (8.11). Additionally, any form ω ∈ Alt•(M ,A) can be uniquely written in the form

i1 ik ω = ∑ ∑ bi1···ik ε ∧ ··· ∧ ε , k∈Z i1<···

where only ﬁnitely many of the bi1···ik are non-zero. 3 1 2 3 Example 8.3 — Vector Product. Le A = R, and M = R . Let (ε ,ε ,ε ) be the canonical 3 ∗ ∼ 3 frame in (R ) = R , and consider two 1-forms

1 2 3 1 2 3 A = A1ε + A2ε + A3ε , and B = B1ε + B2ε + B3ε .

3 3 The space Alt2(R ,R) of 2-forms on R is 3-dimensional, and it is spanned by

ε2 ∧ ε3, ε3 ∧ ε1, ε1 ∧ ε2.

A direct computation shows that the wedge product A ∧ B is given by

A2 A3 2 3 A1 A3 3 1 A1 A2 1 2 A ∧ B = ε ∧ ε − ε ∧ ε + ε ∧ ε , B2 B3 B1 B3 B1 B2

whose coefﬁcients are the components of the vector product of the coordinate vectors (A1,A2,A3) 3 and (B1,B2,B3). This shows that the wedge product generalizes the standard vector product in R .

8.2 Higher Degree Differential Forms Let M be a manifold, let p ∈ M be a point, and let (U,ϕ = (x1,...,xn)) be a chart on M around p. We will apply the language developed in the previous section to two cases: (1) A = R and M = TpM, and (2) A = C∞(M) and M = X(M). We begin with the ﬁrst one. Let k be a non-negative integer. Deﬁnition 8.2.1 — k-Covectors. The space of k-covectors of M at the point p is the space

k ∗ ∧ Tp M := Altk(TpM,R)

k ∗ of alternating k-forms on the tangent space TpM. Forms in ∧ Tp M are k-covectors. In other words, a k-covector at the point p is an R-multilinear, alternating map:

ω : TpM × ··· × TpM → R. | {z } k times The frame

i1 ik dpx ∧ ··· ∧ dpx i1<···

k ∗ of ∧ Tp M, is called the coordinate frame. 152 Chapter 8. Differential Forms and Cartan Calculus

Notice that 0-covectors are just real numbers, while 1-covectors are just covectors. k ∗ Every k-covector ω ∈ ∧ Tp M can be uniquely written as

i1 ik ω = ai1···ik dpx ∧ ··· ∧ dpx , for some ai1···ik ∈ R with the skew-symmetry property (8.11), where the (omitted) sum runs over all multi-indexes. The k-covector ω can also be uniquely written as

i1 ik ω = ∑ bi1···ik dpx ∧ ··· ∧ dpx . i1<···

k ∗ k ∗ R Let U ⊆ M be an open neighborhood of p. Then ∧ Tp U can be identiﬁed with ∧ Tp M in the obvious way, and we will always do this.

As already for tangent vectors and covectors, k-covectors can be organized into a manifold

called the bundle of k-covectors of M, and denoted ∧kT ∗M, deﬁned as k ∗ ∏ k ∗ n k ∗ o ∧ T M := ∧ Tp M = (p,ω) : p ∈ M and ω ∈ ∧ Tp M . p∈M As usual, a point (p,ω) ∈ ∧kT ∗M will be often simply denoted by ω. The set ∧kT ∗M comes with an obvious surjection π : ∧kT ∗M → M, (p,ω) 7→ p, and the pair (∧kT ∗M,π) is the bundle of k- covectors of M. The smooth structure on M induces an atlas on ∧kT ∗M as follows. Begin with a chart (U,ϕ = (x1,...,xn)) on M and deﬁne a chart (∧kT ∗U,∧kT ∗ϕ) on ∧kT ∗M as follows. First

of all, put k ∗ −1 ∏ k ∗ n k ∗ o ∧ T U := π (U) = ∧ Tp M = (p,ω) ∈ ∧ T M : p ∈ U . p∈U Next put n N(k,n) = , k and deﬁne a map: ∧kT ∗ϕ : ∧kT ∗U → U × N(k,n), (p,ω) 7→ ∧kT ∗ϕ(p,ω) := ϕ(p);(b ) , b R i1···ik i1<···

Recall that the bi1···ik are the components of ω in the coordinate frame

i1 ik dpx ∧ ··· ∧ dpx . i1<···

Proposition 8.2.1 — Smooth Structure on the Bundle of k-Covectors. Standard charts on ∧kT ∗M form an atlas. With the associated smooth structure ∧kT ∗M is a manifold of dimension n n + , k

and π : ∧kT ∗M → M is a smooth map.

0 ∗ R Notice that a point in the bundle ∧ T M of 0-covectors is simply a pair (p,a) where p ∈ M, ∗ and a ∈ R. Accordingly, as a set, ∧0T M is simply the cross product M × R, and it is easy to see that the manifold structure on ∧0T ∗M agrees with the product manifold structure on M × R. Additionally, the discussion preceding Proposition 8.2.1, shows that the bundle of 1-covectors, with its manifold structure, is exactly the cotangent bundle.

Deﬁnition 8.2.2 — Fields of k-Covectors. A section of ∧kT ∗M is a map s : M → ∧kT ∗M such that π ◦s = idM. In other words, a section s is the assignment of a k-covector s(p) at p, for every point p ∈ M.

There is a natural C∞(M)-module structure on sections of ∧kT ∗M defined exactly by the same formulas as for the tangent and the cotangent bundle, and smooth sections form a submodule denoted Γ(∧kT ∗M). We now come to alternating forms on the C∞(M)-module X(M) of vector fields. Let k be a non-negative integer. Definition 8.2.3 — Differential k-Form. A differential k-form on M, or, simply, a k-form, is an alternating k-form

ω : X(M) × ···X(M) → C∞(M). | {z } k times

The C∞(M)-module of k-forms is denoted Ωk(M). For k < 0, we also put Ωk(M) = 0. The graded algebra of alternating forms on X(M) is denoted M Ω•(M) = Ωk(M). k∈Z Elements in Ω•(M) are called differential forms.

Differential 0-forms on M are just functions on M. By definition, negative degree differential forms vanish. 1 n Example 8.4 — Basis k-Forms on a coordinate domain. Let (U,ϕ = (x ,...,x )) be a chart on M. As the C∞(U)-module X(U) possesses a finite frame, the space Ωk(U) of k-forms on U does also possess a finite frame, the coordinate frame: dxi1 ∧ ··· ∧ dxik . i1<···

i1 ik ω = ai1···ik dx ∧ ··· ∧ dx , ∞ for some ai1···ik ∈ C (U) with the skew-symmetry property (8.11), where the (omitted) sum runs over all multi-indexes. The k-form ω can also be uniquely written as

i1 ik ω = ∑ bi1···ik dx ∧ ··· ∧ dx , i1<···

∞ for some bi1···ik ∈ C (U) related to the ai1···ik by the formula

∂ ∂ bi ···i = k!ai ···i = ω ,..., , i1 < ··· < ik. 1 k 1 k ∂xi1 ∂xik

Differential k-forms are equivalent to sections of the bundle of k-covectors. More precisely, there is a canonical C∞(M)-module isomorphism Ωk(M) =∼ Γ(∧kT ∗M). To see this, ﬁrst notice that k k ∗ a k-form ω ∈ Ω (M) determines a k-covector ωp ∈ ∧ Tp M, for every point p ∈ M. In order to deﬁne ωp, we need the degree k analogue of Lemma 7.2.2.

k Lemma 8.2.2 Let ω ∈ Ω (M), let p0 ∈ M, and let X1,...,Xk ∈ X(M). Then the value

ω(X1,...,Xk)(p0)

∞ of the function ω(X1,...,Xk) ∈ C (M) at p0 does only depend on (ω and) the values

(X1)p0 ,...,(Xk)p0

of the vector ﬁelds X1,...,Xk at p0.

Proof. The proof is essentially the same of that of Lemma 7.2.2 and we omit it. Now, consider a differential k-form ω on M. For any point p ∈ M we deﬁne a k-covector k ∗ ωp ∈ ∧ Tp M as follows. Let v1,...,vk ∈ TpM, and let X1,...,Xk ∈ X(M) be vector ﬁelds such that (Xi)p = vp. The Xi exist in view of Lemma 7.2.3. Put

ωp(v1,...,vk) := ω(X1,...,Xk)(p).

It is easy to see, exactly as in the case of 1-covectors, that ωp is a well-deﬁned k-covector

ωp : TpM × ··· × TpM → R. | {z } k times called the value of ω at p.

Exercise 8.1 Let k,l be non-negative integers. Prove that, for every ω,ω0 ∈ Ωk(M), every ρ ∈ Ωl(M), and every p ∈ M,

0 0 (ω + ω )p = ωp + ωp

(ω ∧ ρ)p = ωp ∧ ρp. (8.13)

R Consider Exercise 8.1. Notice that when l = 0, then ρ = f is a function. Then Equation (8.13) says that

( f ω)p = f (p)ωp.

Additionally, if (U,ϕ = (x1,...,xn)) is a chart around p, then the values at p of the k-forms in the coordinate frame are

i1 ik i1 ik (dx ∧ ··· ∧ dx )p = dpx ∧ ··· ∧ dpx , i1 < ··· < ik. 8.2 Higher Degree Differential Forms 155

k ∗ The assignment p 7→ ωp is a section sω of the bundle of k-covectors ∧ T M:

k ∗ sω : M → ∧ T M, p 7→ sω (p) := ωp.

Proposition 8.2.3 The map

k k ∗ Ω (M) → Γ(∧ T M), ω 7→ sω

is a well-deﬁned C∞(M)-module isomorphism.

Proof. The proof is essentially the same as that of Proposition 7.2.4 and we omit it. As for vector ﬁelds and 1-forms, Proposition 8.2.3 has important consequences. Differential forms can be restricted to open submanifolds, by restricting the associated sections, exactly as vector ﬁelds and 1-forms, and we use the same notation ω|U for the restriction of a differential form ω to an open submanifold U . Second, there is a(n obvious) Gluing Lemma for Differential Forms, and we leave it to the reader to state and prove it. Finally, given a differential k-form ω ∈ Ωk(M), we have that (1) ω is completely determined by its restrictions ω|U to some coordinate domains U covering M; 1 n (2) for every chart (U,ϕ = (x ,...,x )), the restriction ω|U (as already remarked) can be uniquely written as

i1 ik ω|U = ai1···ik dx ∧ ··· ∧ dx ,

∞ for some ai1···ik ∈ C (U) with the skew-symmetry property (8.11), where the (omitted) sum runs over all multi-indexes. The k-form ω|U can also be uniquely written as

i1 ik ω|U = ∑ bi1···ik dx ∧ ··· ∧ dx , i1<···

∞ for some bi1···ik ∈ C (U); (3) given vector ﬁelds X1,...,Xk ∈ X(M) locally given by

∂ X | = Xi , α = 1,...,k, α U α ∂xi we have

i1 ik i1 ik ω(X1,...,Xk)|U = ω|U (X1|U ,...Xk|U ) = k!ai1···ik X1 ···Xk = ∑ bi1···ik X1 ···Xk ; i1<···

(4) if k > n = dimM, then ω|U = 0 for all carts (U,ϕ), hence ω = 0, i.e. there are no non-trivial differential forms on M of degree greater than dimM.

Exercise 8.2 Investigate how does the local coordinate expression of a differential form change

under a change of coordinates.

n Now, let (U,ϕ = (x1,...,x )) be a chart on M. From Example 8.4, the graded R-algebra Ω•(U) is generated by functions C∞(U) and 1-forms dx1,...,dxn. In particular, it is generated by functions, and 1-forms of the type d f for some f ∈ C∞(U). This is true even globally according to the following analogue of Theorem 7.2.5. 156 Chapter 8. Differential Forms and Cartan Calculus

• Theorem 8.2.4 Let M be an n-dimensional manifold. The graded R-algebra Ω (M) is generated by functions C∞(M) and differentials d f of functions f ∈ C∞(M), i.e. every differential form ω ∈ Ωk(M) can be written (in a possibly non-unique way) as a ﬁnite sum of forms of the type

f dg1 ∧ ··· ∧ dgk, 0 ≤ k ≤ n,

∞ where f ,g1,...,gk ∈ C (M).

Proof. Omitted. We conclude this section showing that the pull-back of functions and differential 1-forms along a smooth map F : M → N extends uniquely to a homomorphism of graded algebras

F∗ : Ω•(N) → Ω•(M), also called the pull-back, according to the following

Proposition 8.2.5 — Pull-Back of Differential Forms. Let M,N be smooth manifolds and let F : M → N be a smooth map. Then there exists a unique homomorphism of graded algebras

F∗ : Ω•(N) → Ω•(M)

such that, on functions and 1-forms, F∗ agrees with the pull-backs already deﬁned.

Proof. First of all, let ω ∈ Ωk(N) be a homogeneous differential form (of degree k). When k = 0, then ω = f is a functions and we put F∗( f ) = f ◦ F. When k > 0, we deﬁne F∗(ω) ∈ Ωk(M) through its values as follows. For every p ∈ M, and every v1,...,vk ∈ TpM, we put

∗ F (ω)p(v1,...,vk) = ωF(p)(dpF(v1),...,dpF(vk)). (8.14) In particular, when k = 1, we recover the pull-back of 1-forms already defined. For generic k, ∗ k ∗ Formula (8.14) defines a section s : p 7→ F (ω)p of ∧ T M. One can prove that s is a smooth section in a very similar way as for 1-forms and we leave the details to the reader. In this way we defined the pull-back of homogeneous forms. We extend the definition to possibly non-homogeneous forms • by additivity in the obvious way, i.e. for (ωk)k∈Z ∈ Ω (M) a generic form, we put F∗ (( ) ) = (F∗( )) . (8.15) ωk k∈Z ωk k∈Z Next we have to prove that

F∗ : Ω•(N) → Ω•(M) is R-linear and preserves the wedge product of differential forms. We leave this easy check as Exercise 8.3. Finally, we have to prove uniqueness. This immediately follows from Theorem 8.2.4 (do you see it?).

Exercise 8.3 Complete the proof of Proposition 8.2.5 showing that the pull-back

F∗ : Ω•(N) → Ω•(M)

defined via (8.14) and (8.15) is R-linear and preserves the wedge product. (Hint: to show ∗ that F is R-linear, discuss first linear combinations of homogeneous differential forms of the same degree. To show that F∗ preserves the wedge product, discuss first the wedge product of 8.2 Higher Degree Differential Forms 157

homogeneous forms (of possibly different degrees). To do this use Exercise 8.1. Then discuss the

general case.)

Exercise 8.4 Let F : M → N and G : N → Q be smooth maps between manifolds. Show that, for every ω ∈ Ω•(Q),

∗ idQ(ω) = ω,

and

(G ◦ F)∗(ω) = F∗(G∗(ω)).

2 3 Example 8.5 Consider R with standard coordinates (u,v), and R with standard coordinates (x,y,z). Consider also the smooth map

2 3 2 2 F : R → R , (u,v) 7→ F(u,v) = u ,uv,v , and the differential 2-form

2 3 ω = dx ∧ dy + dy ∧ dz + dz ∧ dx ∈ Ω R . We want to compute the pull-back of ω along F. We have

F∗(ω) = F∗(dx ∧ dy + dy ∧ dz + dz ∧ dx) = F∗(dx) ∧ F∗(dy) + F∗(dy) ∧ F∗(dz) + F∗(dz) ∧ F∗(dx) = dF∗(x) ∧ dF∗(y) + dF∗(y) ∧ dF∗(z) + dF∗(z) ∧ dF∗(y) = du2 ∧ d(uv) + d(uv) ∧ dv2 + dv2 ∧ du2 = (2udu) ∧ (udv + vdu) + (udv + vdu) ∧ (2vdv) + (2vdv) ∧ (2udu) = (2u2 + 2v2 − 4uv)du ∧ dv = 2(u − v)2du ∧ dv, where we used, among other things, that du ∧ du = dv ∧ dv = 0, and dv ∧ du = −du ∧ dv (can you motivate any single step in the above computation?).

Example 8.6 — Restriction of a k-Form to a Submanifold. If iU : U → M is the inclusion of k ∗ an open submanifold, and ω ∈ Ω (M) is a k-form, then iU (ω) = ω|U is the restriction of ω to U . More generally, exactly as for 1-forms (and functions), we can “restrict” a generic k-form k ω ∈ Ω (M) to any submanifold S ⊆ M by pulling it back along the inclusion iS : S ,→ M. The ∗ pull-back iS(ω) is sometimes denoted ω|S and called the restriction of ω to S. In practice, it is obtained by restricting the values of ω at points of S to tangent vectors to S.

Exercise 8.5 Consider R3 with standard coordinates (x,y,z) and the differential 2-form

2 3 ω = zdx ∧ dy + xdy ∧ dz + ydz ∧ dx ∈ Ω R .

2 3 Let (U+,ϕ+ = (X+,Y+)) be the stereographic chart on the 2-dimensional sphere S ⊆ R . Compute

ω|U+ = ω|S2 |U+ 158 Chapter 8. Differential Forms and Cartan Calculus

in terms of the stereographic coordinates (X+,Y+).

8.3 Cartan Calculus We conclude this chapter and this lecture notes presenting some natural operations on differential forms, besides the wedge product. These operations and the associated computational rules are sometimes collectively called Cartan calculus in honor of Èlie Cartan, who first discovered (some of) them. L We begin with some algebraic preliminaries. Let A• = k∈Z Ak be a graded algebra, and let l ∈ Z be an integer. Definition 8.3.1 — Graded Derivation. A degree l, graded derivation of A•, or, simply, a graded derivation, is an R-linear map ∆ : A• → A• such that (1) ∆(Ak) ⊆ Ak+l, for all k ∈ Z, (2) ∆ satisfies the following graded Leibniz rule: for all homogeneous elements α,β ∈ A•

∆(α ∧ β) = ∆(α) ∧ β + (−)l|α|α ∧ ∆(β).

The (possibly negative) integer l is called the degree of ∆ and is also denoted by |∆|.

Proposition 8.3.1 Degree l graded derivations of A• form a real vector space under the sum and the product by a scalar deﬁned in the following (obvious) way:

(∆ + ∆0)(α) := ∆(α) + ∆0(α), (a∆)(α) := a(∆(α)),

0 for all degree l graded derivations ∆,∆ , all a ∈ R, and all α ∈ A•.

Proof. A straightforward computation that we omit.

Now, let ∆,∇ be graded derivations (of possibly different degrees), and let β ∈ A• be a homogeneous element. We deﬁne new operators:

β ∧ ∆ : A• → A• and [∆,∇] : A• → A•,

by putting

(β ∧ ∆)(α) := β ∧ ∆(α), [∆,∇](α) := ∆(∇(α)) − (−)|∆||∇|∇(∆(α)).

Proposition 8.3.2 Both β ∧ ∆ and [∆,∇] are graded derivations. Their degrees are

|β ∧ ∆| = |β| + |∆| and |[∆,∇]| = |∆| + |∇|.

Additionally, the bracket [−,−] (1) is R-bilinear; (2) is graded skew-symmetric, i.e.

[∇,∆] = −(−)|∆||∇|[∆,∇]

for all graded derivations ∆,∇; 8.3 Cartan Calculus 159

(3) satisﬁes the following graded Jacobi identity:

|∆|(|∇|+| |) | |(|∆|+|∇|) [∆,[∇,]] + (−) [∇,[,∆]] + (−) [,[∆,∇]] = 0.

for all graded derivations ∆,∇,. Finally, (4) the wedge product of a homogeneous element of A• and a graded derivation is an R- bilinear operation; (5)

(β ∧ γ) ∧ ∆ = β ∧ (γ ∧ ∆);

(6)

[∆,β ∧ ∇] = ∆(β) ∧ ∇ + (−)|∆||β|β ∧ [∆,∇],

for all graded derivations ∆,∇, and all homogeneous elements β,γ ∈ A•.

Proof. A long but straightforward computation left as Exercise 8.6.

Exercise 8.6 Prove Proposition 8.3.2.

The derivation [∆,∇] is called the graded commutator of ∆ and ∇. Notice that, while the graded commutator of graded derivations is a graded derivation, the usual commutator ∆ ◦ ∇ − ∇ ◦ ∆ is not a graded derivation in general.

R The reader has probably already understood that the wedge product in a graded algebra and the graded commutator of graded derivations are graded analogues of the product in an algebra and the usual commutator of usual derivations. These operations enjoy similar properties as their standard cousins. However the “graded formulas” differ from the usual ones by signs depending on the degrees of the objects involved. There is a (mnemonic) meta-mathematical rule called the Koszul sign rule that might help the reader remembering these signs:

Koszul Sign Rule: A graded formula differs from its standard (non-graded) analogue by signs appearing everytime two (homogeneous) graded objects swap. Namely, everytime that 0 two graded objects O,O0 swap, there appears a sign (−)|O||O |.

The reader is invited to check that the Koszul Sign Rule applies to all cases discussed so far: commutativity rule of the product, Leibniz rule, deﬁnition of the commutator, skew-symmetry of the commutator, Jacobi identity, etc..

Proposition 8.3.3 A graded derivation ∆ of the graded algebra A• is completely determined by its action on homogeneous generators of A•, i.e. if {αi}i∈I is a family of homogeneous 0 generators of A• (indexed by some set I) and ∆ is another graded derivation of A• such that |∆0| = |∆| and

0 ∆ (αi) = ∆(αi), for all i ∈ I,

then ∆ = ∆0. 160 Chapter 8. Differential Forms and Cartan Calculus

Proof. First of all, we clarify/recall what we mean by a family of homogeneous generators {αi}i∈I of A•. This means that the αi are homogeneous elements of A•, and every other element α ∈ A• can be written as a ﬁnite sum of products of the form

αi1 ∧ ··· ∧ αik , i1,...,ik ∈ I, k ∈ Z.

0 Now, assume ∆,∆ are as in the statement, and let α ∈ A•. Then

α = ∑αi1 ∧ ··· ∧ αik i1,...,ik ∈ I, k ∈ Z, and

0 0 ∆ (α) = ∆ ∑αi1 ∧ ··· ∧ αik

k 0 |∆ |(|αi1 |+···|αi j−1 |) 0 = ∑ ∑(−) αi1 ∧ ··· ∧ αi j−1 ∧ ∆ (αi j ) ∧ αi j+1 ∧ ··· ∧ αik j=1 k |∆|(|αi1 |+···|αi j−1 |) = ∑ ∑(−) αi1 ∧ ··· ∧ αi j−1 ∧ ∆(αi j ) ∧ αi j+1 ∧ ··· ∧ αik j=1 = ∆(α), where we used the graded Leibniz rule. This concludes the proof.

Theorem 8.3.4 — Interior Product. Let M be a manifold, and let X ∈ X(M) be a vector ﬁeld • on M. There exists a unique degree −1 graded derivation iX of the graded algebra Ω (M) of differential forms on M such that X iX f = 0, X iX d f = X( f ), for all smooth functions f ∈ C∞(M) = Ω0(M).

∞ Proof. First, we deﬁne iX on homogeneous forms ω. If |ω| = 0, then ω = f ∈ C (M), and we put

iX f := 0.

If |ω| = k > 0, then

iX ω := ω(X,−,...,−), i.e. iX ω is the (k − 1)-form deﬁned by

iX ω(X1,...,Xk−1) := ω(X,X1,...,Xk−1), (8.16) for all X1,...,Xk−1 ∈ X(M). It is clear that iX ω is a (k − 1)-form, indeed the expression in the right ∞ hand side of (8.16) is obviously C (M)-multilinear and alternating in the arguments X1,...,Xk−1, ∞ and it is R-linear (actually even C (M) linear) in the argument ω. Now we extend iX to all forms • by additivity in the following obvious way: for all (ωk)k∈Z ∈ Ω (M) we put

i ( ) = (i ) . X ωk k∈Z X ωk k∈Z The map

• • iX : Ω (M) → Ω (M) 8.3 Cartan Calculus 161 deﬁned in this way is R-linear and maps k-forms to (k − 1)-form. We now prove that it is a graded derivation (of degree −1). So, let ω ∈ Ωk(M) and ρ ∈ Ωl(M). We assume k,l > 0 and we leave the simpler cases k = 0 or l = 0 to the reader. Let X1,...,Xk+l−1 ∈ X(M) and compute

iX (ω ∧ ρ)(X1,...,Xk+l−1)

= ω ∧ ρ(X,X1,...,Xk+l−1) σ = ∑ (−) ω(X,Xσ(1),...,Xσ(k−1))ρ(Xσ(k),...,Xσ(k+l−1)) σ∈Sk−1,l σ k + ∑ (−) (−) ω(Xσ(1),...,Xσ(k))ρ(X,Xσ(k),...,Xσ(k+l−1)) σ∈Sk,l−1 σ = ∑ (−) iX ω(Xσ(1),...,Xσ(k−1))ρ(Xσ(k),...,Xσ(k+l−1)) σ∈Sk−1,l |ω| σ + (−) ∑ (−) ω(Xσ(1),...,Xσ(k))iX ρ(Xσ(k),...,Xσ(k+l−1)) σ∈Sk,l−1 |ω| = (iX ω) ∧ ρ + (−) ω ∧ iX ρ (X1,...,Xk+l−1).

Finally, let f ∈ C∞(M) and compute

iX d f = d f (X) = X( f ).

This shows that iX satisﬁes all the required properties. Uniqueness immediately follows from Proposition 8.3.3 and Theorem 8.2.4.

The graded derivation iX is called the insertion of X, or contraction with X, or also interior product with X. We want to describe it in local coordinates. So, let (U,ϕ = (x1,...,xn)) be a chart on M, let ω ∈ Ωk(M) be locally given by

i1 ik ω|U = ai1···ik dx ∧ ··· ∧ dx ,

∞ for some ai1···ik ∈ C (U) enjoying the usual skew-symmetry property (8.11), and let X ∈ X(M) be locally given by

∂ X| = Xi . U ∂xi

We want to show that

i1 i2 ik (iX ω)|U = kai1i2···ik X dx ∧ ··· ∧ dx .

First of all, it is easy to see using the deﬁnition of iX , that (iX ω)|U = iX|U ω|U (do you see it?). Now 162 Chapter 8. Differential Forms and Cartan Calculus compute

iX|U ω|U i1 ik = iX|U ai1···ik dx ∧ ··· ∧ dx k j−1 i1 i j−1 i j i j+1 ik = ∑(−) ai1···ik dx ∧ dx ∧ iX dx ∧ dx ∧ ··· ∧ dx j=1 k j−1 i1 i j−1 i j i j+1 ik = ∑(−) ai1···ik dx ∧ dx ∧ X(x ) ∧ dx ∧ ··· ∧ dx j=1 k j−1 i j i1 i j ik = ∑(−) ai1···ik X dx ∧ ··· ∧ dxd ∧ ··· ∧ dx j=1 k = a Xi j dxi1 ∧ ··· ∧ dxdi j ∧ ··· ∧ dxik (reordering the indexes) ∑ i ji1···ibj···ik j=1 k i1 i2 ik = ∑ ai1i2···ik X dx ∧ ··· ∧ dx (renaming the indexes) j=1

i1 i2 ik = kai1i2···ik X dx ∧ ··· ∧ dx

3 Example 8.7 On R with standard coordinates (x,y,z), consider the 2-form

ω = xdy ∧ dz + ydz ∧ dx + zdx ∧ dy and the vector ﬁeld ∂ ∂ ∂ X = x + y + z . ∂x ∂y ∂z Compute the interior product of ω with X:

iX ω = iX (xdy ∧ dz + ydz ∧ dx + zdx ∧ dy)

= xiX (dy ∧ dz) + yiX (dz ∧ dx) + ziX (dx ∧ dy)

= x(iX dy) ∧ dz − xdy ∧ iX dz + y(iX dz) ∧ dx − ydz ∧ iX dx + z(iX dx) ∧ dy − zdx ∧ iX dy = xX(y)dz − xX(z)dy + yX(z)dx − yX(x)dz + zX(x)dy − zX(y)dx = xydz − xzdy + yzdx − yxdz + zxdy − zydx = 0.

Proposition 8.3.5 Let X,Y ∈ X(M), and let f ∈ C∞(M). Then

iX+Y = iX + iY (8.17) i f X = fiX .

Proof. Both the left hand sides and the right hand sides of (8.17) are degree −1 graded derivations of Ω•(M). In order to check that they agree, it is enough to check that they agree on generators, and this is an easy consequence of Theorem 8.3.4. We leave details to the reader. 8.3 Cartan Calculus 163

Theorem 8.3.6 — Lie Derivative. Let M be a manifold, and let X ∈ X(M) be a vector ﬁeld • on M. There exists a unique degree 0 graded derivation LX of the graded algebra Ω (M) of differential forms on M such that X LX f = X( f ), X LX d f = dX( f ), for all smooth functions f ∈ C∞(M) = Ω0(M).

Proof. First, we define LX on homogeneous forms ω. So let |ω| = k. Denote by {Φt }t the flow of X. We define LX ω through its values, as follows:

d ∗ (LX ω)p := |t= Φ (ω)p, (8.18) dt 0 t for all p ∈ M. As already for the Lie derivative of a vector field, we have to show that Formula (8.18) defines correctly a smooth section of the bundle ∧kT ∗M of k-covectors. We work in local 1 n coordinates. So, fix a point p0 ∈ M, and let (U,ϕ = (x ,...,x )) be a chart around p0. Then ∂ X| = Xi and ω| = a dxi1 ∧ ··· ∧ dxik , U ∂xi U i1···ik

i ∞ for some smooth functions X , and ai1···ik ∈ C (U) (enjoying the usual skew-symmetry property). Additionally, let J ⊆ R be an open interval containing 0, and let (U0,ϕ) be a subchart of (U,ϕ) around p0 exactly as in the proof of Proposition 6.2.4, i.e. such that Φ(J ×U0) ⊆ U. Now, let (t, p) ∈ J ×U0 and, using the same notation and formulas as in the proof of Proposition 6.2.4, compute

∗ ∗ i1 ik Φt (ω)p = Φt ai1···ik dx ∧ ··· ∧ dx p

∗ i1 ik = Φt (ai1···ik )(p)dpΦt ∧ ··· ∧ dpΦt i1 ik ∂Φ ∂Φ j1 jk = ai ···i (Φ(t, p)) (t, p)··· (t, p)dpx ∧ ··· ∧ dpx . (8.19) 1 k ∂x j1 ∂x jk The coefﬁcients in the last expression depend smoothly on t, and we can take the derivative in (8.18) (see Exercise 3.7):

d d i1 ik ∗ ∂Φ ∂Φ j1 jk (LX ω)p = |t=0Φt (ω)p = |t=0 ai1···ik (Φ(t, p)) (t, p)··· (t, p) dpx ∧···∧dpx . dt dt ∂x j1 ∂x jk The coefﬁcients are all smooth in the argument p, and this shows that the section

p 7→ (LX ω)p

k ∗ of ∧ Tp M is well-deﬁned and smooth. For future use, we write explicitly the coefﬁcient labelled by ( j1,..., jk). A direct computation exploiting (6.16) and (6.17) shows that

d ∂Φi1 ∂Φik |t=0 ai ···i (Φ(t, p)) (t, p)··· (t, p) dt 1 k ∂x j1 ∂x jk (8.20) i i i ∂a j1··· jk ∂X ∂X = X + ai j ··· j + ··· + a j ··· j i (p). ∂xi ∂x j1 2 k ∂x jk 1 k−1 We leave the computational details to the reader. • Next we extend LX to all forms by additivity as follows: for all (ωk)k∈Z ∈ Ω (M) we put ( ) = ( ) . LX ωk k∈Z LX ωk k∈Z 164 Chapter 8. Differential Forms and Cartan Calculus

The map

• • LX : Ω (M) → Ω (M) deﬁned in this way is R-linear and maps k-forms to k-forms. We now prove that it is a graded derivation (of degree 0). So, let ω,ρ be homogeneous forms, let p ∈ M, and compute

d ∗ d ∗ ∗ d ∗ ∗ (LX (ω ∧ ρ)) = |t= Φ (ω ∧ρ)p = |t= (Φ (ω) ∧ Φ (ρ)) = |t= Φ (ω)p ∧Φ (ρ)p. p dt 0 t dt 0 t t p dt 0 t t Now we apply Lemma 8.3.7 below to the case B = ∧ to ﬁnd

d ∗ ∗ (LX (ω ∧ ρ)) = |t= Φ (ω)p ∧ Φ (ρ)p p dt 0 t t d d = | Φ∗(ω) ∧ ρ + ω ∧ | Φ∗(ρ) dt t=0 t p p p dt t=0 t p

= (LX ω)p ∧ ρp + ωp ∧ (LX ρ)p

= ((LX ω) ∧ ρ + ω ∧ LX ρ)p . The graded Leibniz rule now follows from the arbitrariness of p. ∞ Next, let f ∈ C (M). Property LX f = X( f ) is provided by Proposition 6.2.2. In order to 1 n compute LX d f , ﬁx p0 ∈ M, and let (U,ϕ = (x ,...,x )) and U0 be as in the ﬁrst part of the proof. Then, for all p ∈ U0,

(can you reproduce all the above steps?) and, from the arbitrariness of p, we have LX d f = dX( f ). Finally, uniqueness immediately follows from Proposition 8.3.3 and Theorem 8.2.4.

Lemma 8.3.7 Let V1,V2,V be ﬁnite dimensional real vector spaces, and let

B : V1 ×V2 → V

be a bilinear map. Let I ⊆ R be an open interval. Then, for any two smooth curves γ1 : I → V1, γ2 : I → V2, (1) the map I → V, t 7→ B(γ1(t),γ2(t)) is a smooth curve, and (2) for all t0 ∈ I, the following Leibniz rule holds: d d d | B(γ (t),γ (t)) = B | γ (t),γ (t ) + B γ (t ), | γ (t) . dt t=t0 1 2 dt t=t0 1 2 0 1 0 dt t=t0 2

Proof. Left as Exercise 8.7. 8.3 Cartan Calculus 165

Exercise 8.7 Prove Lemma 8.3.7.

The graded derivation LX is called the Lie derivative along X. We want to describe it in local coordinates. So, let (U,ϕ = (x1,...,xn)) be a chart on M, let ω ∈ Ωk(M) be locally given by

i1 ik ω|U = ai1···ik dx ∧ ··· ∧ dx ,

with the ai1···ik skew-symmetric, and let X ∈ X(M) be locally given by

∂ X| = Xi . U ∂xi It immediately follows from (8.20) that

(LX ω)|U = LX|U ω|U ∂a Xi Xi (8.21) i j1··· jk ∂ ∂ j1 jk = X + ai j ··· j + ··· + a j ··· j i dx ∧ ··· ∧ dx . ∂xi ∂x j1 2 k ∂x jk 1 k−1

Example 8.8 On the 3-dimensional standard Euclidean space with standard coordinates (x,y,z) consider the 2-form

ω = (dx + 2zdy) ∧ dz, and the vector ﬁeld ∂ ∂ X = y − x . ∂x ∂y

Compute the Lie derivative of ω along X:

LX ω = LX ((dx + 2dy) ∧ dz)

= (LX (dx + 2zdy)) ∧ dz + (dx + 2zdy) ∧ LX (dz) = (dX(x) + 2X(z)dy + 2zdX(y)) ∧ dz + (dx + 2zdy) ∧ dX(z) = (dy − 2zdx) ∧ dz.

Proposition 8.3.8 Let X,Y ∈ X(M), and let f ∈ C∞(M). Then

LX+Y = LX + LY (8.22) L f X = f LX + d f ∧ iX .

Proof. Both the left hand sides and the right hand sides of (8.22) are degree 0 graded derivations of Ω•(M). In order to check that they agree it is enough to check that they agree on generators, and this is an easy consequence of Theorem 8.3.6.

Theorem 8.3.9 — Exterior Differential. Let M be a manifold. There exists a unique degree 1 graded derivation d of the graded algebra Ω•(M) of differential forms on M such that X d f is exactly the differential of f , X dd f = 0, for all smooth functions f ∈ C∞(M) = Ω0(M). 166 Chapter 8. Differential Forms and Cartan Calculus

Proof. First, we define d on homogeneous forms ω. If |ω| = 0, then ω = f ∈ C∞(M), and, by definition, d f is simply the usual differential of a function. If |ω| = k > 0, then dω is the (k + 1)-form defined by the following Chevalley-Eilenberg Formula:

i+1 dω(X1,...,Xk+1) = ∑(−) Xi ω(X1,...,Xbi,...,Xk+1) i (8.23) i+ j + ∑(−) ω([Xi,Xj],X1,...,Xbi,...,Xbj,...,Xk+1), i< j for all X1,...,Xk+1 ∈ X(M). It is clear that dω is R-multilinear. The fact that dω is alternating i i+ j easily follows from (the fact that ω is R-multilinear and) the signs (−) and (−) in (8.23) and we leave the details to the reader. Next we have to show that dω is C∞(M)-multilinear. It is enough to show that dω is C∞(M)-linear in the ﬁrst argument. The C∞(M)-linearity in the other arguments then follows from skew-symmetry. So we have to show that

dω( f X1,X2,...,Xk+1) = f dω(X1,X2,...,Xk+1),

∞ for all X1,X2,...,Xk+1 ∈ X(M), and all f ∈ C (M). To see this we compute

dω( f X1,X2,...,Xk+1) i+1 = f X1 (ω(X2,...,Xk+1)) + ∑(−) Xi ω( f X1,X2,...,Xbi,...,Xk+1) 1

= f dω(X1,X2,...,Xk+1).

We extend d to all forms by additivity in the usual way. The map

d : Ω•(M) → Ω•(M) deﬁned in this way is clearly R-linear and maps k-forms to (k + 1)-forms. We now prove that it is a graded derivation (of degree 1). So, let ω ∈ Ωk(M) and ρ ∈ Ωl(M). If k = 0, then ω = f ∈ C∞(M), 8.3 Cartan Calculus 167 and ω ∧ ρ = f ρ. So, for every X1,...,Xl+1 ∈ X(M),

d(ω ∧ ρ)(X1,...,Xl+1) = d( f ω)(X1,...,Xl+1) i+1 = ∑(−) Xi f ω(X1,...,Xbi,...,Xk+1) i i+ j + ∑(−) f ω([X1,Xj],X1,...,Xbi,...,Xbj,...,Xk+1) i< j i+1 = ∑(−) Xi( f )ω(X1,...,Xbi,...,Xk+1) i i+1 + ∑(−) f Xi ω(X1,...,Xbi,...,Xk+1) i i+ j + ∑(−) f ω([X1,Xj],X1,...,Xbi,...,Xbj,...,Xk+1) i< j i+1 = ∑(−) d f (Xi)ω(X1,...,Xbi,...,Xk+1) + f dω(X1,...,Xl+1) i

= (d f ∧ ω + f dω)(X1,...,Xl+1). This shows that the graded Leibniz rule for d(ω ∧ ρ) works when |ω| = 0. We leave it to the reader to show that it also works when |ω| = 1 as Exercise 8.8 (it’s a long computation but, if well organized, it is not conceptually complicated). This allows us to prove the graded Leibniz rule for d(ω ∧ ρ) by induction on |ω| (notice that the graded Leibniz rule could also be proved by a direct, but cumbersome, computation). So assume that the graded Leibniz rule for d(ω ∧ ρ) works when |ω| < r − 1, and let |ω| = r > 1. By Theorem 7.2.5, ω can be written as a sum of terms of the form

θ1 ∧ ··· ∧ θr,

1 where θ1,...,θr ∈ Ω (M). Then

d(ω ∧ ρ) = ∑d(θ1 ∧ θ2 ∧ ··· ∧ θr ∧ ρ) = ∑(dθ1) ∧ θ2 ∧ ··· ∧ θr ∧ ρ − ∑θ1 ∧ d(θ2 ∧ ··· ∧ θr ∧ ρ) (base of induction) = ∑(dθ1) ∧ θ2 ∧ ··· ∧ θr ∧ ρ − ∑θ1 ∧ d(θ2 ∧ ··· ∧ θr) ∧ ρ r−1 − ∑(−) θ1 ∧ θ2 ∧ ··· ∧ θr ∧ dρ (induction hypothesis) = (dω) ∧ ρ + (−)rω ∧ dρ (base of induction). Finally, specializing the Chevalley-Eilenberg Formula (8.23) to k = 1, we get

dd f (X,Y) = X(d f (Y)) −Y(d f (X)) − d f ([X,Y]) = X(Y( f )) −Y(X( f )) − [X,Y]( f ) = 0, for all X,Y ∈ X(M). This concludes the proof of the existence. The uniqueness again follows from Theorem 7.2.5.

Exercise 8.8 Complete the proof of Theorem 8.3.9 showing that

d(ω ∧ ρ) = (dω) ∧ ρ − ω ∧ dρ,

1 for all ω ∈ Ω (M), and all homogeneous forms ρ.

The graded derivation d is called the exterior differential, or De Rham differential, or just differential. We want to describe it in local coordinates. So, let (U,ϕ = (x1,...,xn)) be a chart on M, and let ω ∈ Ωk(M) be locally given by

i1 ik ω|U = ai1···ik dx ∧ ··· ∧ dx , 168 Chapter 8. Differential Forms and Cartan Calculus

for some skew-symmetric ai1···ik . We want to show that

∂ i1 i2 ik+1 (dω)|U = ai ···i dx ∧ dx ∧ ··· ∧ dx . (8.24) ∂xi1 2 k+1 Before proving this formula we stress that, while it is simple and useful, the coefficients in it are not skew-symmetric in the indexes i1 ···ik+1. As we know, they can be uniquely replaced by skew-symmetric coefficients di1···ik+1 given by k+1 1 j+1 ∂ di ···i = (−) a . 1 k+1 ∑ i j i1···ibj···ik+1 k + 1 j=1 ∂x We leave the details to the reader. We now prove (8.24). First of all, it is easy to see, using the definition of the exterior differential, that (dω)|U = d(ω|U ). Now, ∂ d(ω| ) = d a dxi1 ∧ ··· ∧ dxik = da ∧dxi1 ∧···∧dxik = a dxi ∧dxi1 ∧···∧dxik , U i1···ik i1···ik ∂xi i1···ik where we used that ∂ da = a dxi i1···ik ∂xi i1···ik and that ddxi = 0. After renaming the indexes this proves (8.24). 3 Example 8.9 On R with standard coordinates (x,y,z) consider the 2-form ω = xdy ∧ dz + ydz ∧ dx + zdx ∧ dy. Compute the exterior differential of ω: dω = dx ∧ dy ∧ dz + dy ∧ dz ∧ dx + dz ∧ dx ∧ dy = 3dx ∧ dy ∧ dz.

3 1 2 3 Example 8.10 — Gradient, Rotor, Divergence. On R with standard coordinates (x ,x ,x ), consider a generic function f = f (x1,x2,x3), a generic 1-form 1 2 3 A = A1dx + A2dx + A3dx , and a generic 2-form 2 3 3 1 1 2 B = B1dx ∧ dx + B2dx ∧ dx + B3dx ∧ dx . Then ∂ f ∂ f ∂ f d f = dx1 + dx2 + dx3 ∂x1 ∂x2 ∂x3 encodes, as we already know, the gradient grad f of f , ∂A ∂A ∂A ∂A ∂A ∂A dA = 3 − 2 dx2 ∧ dx3 + 1 − 3 dx3 ∧ dx1 − 2 − 1 dx1 ∧ dx2 ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 encodes the rotor rotA of A = (A1,A2,A3), and ∂B ∂B ∂B dB = 1 + 2 + 3 dx1 ∧ dx2 ∧ dx3 ∂x1 ∂x2 ∂x3 encodes the divergence divB of B = (B1,B2,B3), showing that the exterior differential generalizes the standard differential operators grad,rot,div. 8.3 Cartan Calculus 169

Now, we want to compute the graded commutators of interior products, Lie derivatives, and the exterior differential. These formulas are the main formulas in Cartan calculus. We collect them in the following table of graded commutators:

[−,−] iY LY d iX 0 i[X,Y] LX LX i[X,Y] L[X,Y] 0 d LY 0 0

Table 8.1: Distinguished graded commutators.

Theorem 8.3.10 — Cartan Calculus. Let M be a manifold and let X,Y ∈ X(M) be vector ﬁelds on M. The interior products iX ,iY , the Lie derivatives LX ,LY and the exterior differential d ﬁt in the Table 8.1 of graded commutators.

Proof. All the graded commutators in Table 8.1 can be computed in the same way. We discuss one example and leave all the rest to the reader as Exercise 8.9. The formula

[d,iX ] = LX (8.25) is known as Cartan Magic Formula and can be proved as follows. Both sides of (8.25) are graded derivations of degree 0. To show that they coincide, it is enough to show that they agree on generators. So let f ∈ C∞(M), and compute

[d,iX ] f = diX f + iX d f = iX d f = X( f ) = LX f , and

[d,iX ]d f = diX d f + iX dd f = dX( f ) = LX d f .

This concludes the proof of the Cartan Magic Formula.

Exercise 8.9 Complete the proof of Theorem 8.3.10 computing the remaining graded commu-

tators in Table 8.1. Notice that the formula [d,d] = 0, in the bottom-right corner of Table 8.1, is not at all trivial. Indeed, as d is a degree 1 derivation, we have [d,d] = d ◦ d + d ◦ d = 2d ◦ d. Hence [d,d] = 0 is the same as d ◦ d = 0, which is in turn equivalent to imd ⊆ kerd. A form in the kernel of d is called a closed form or a cocycle. A form in the image of d is called an exact form or a coboundary. So [d,d] = 0 means that exact forms are also closed. The converse might not be true. The quotient vector space kerd H (M) := dR imd is a graded vector space called the De Rham cohomology of M, and contains important topological information about M. 170 Chapter 8. Differential Forms and Cartan Calculus

R It should be clear how to define symmetries and infinitesimal symmetries of a differential form. A symmetry of a differential form ω on a manifold M is a diffeomorphism Φ : M → M preserving ω in the sense that Φ∗(ω) = ω.A local symmetry of ω is a diffeomorphism ∗ Φ : U → Φ(U ) between open submanifolds U ,Φ(U ) ⊆ M such that Φ (ω|Φ(U )) = ω|U . An infinitesimal symmetry of ω is a vector field X on M generating a flow {Φt }t by local ∗ symmetries of ω, i.e. Φt (ω|M−t ) = ω|Mt for all t. The subset Diffeo(M,ω) := {symmetries of ω} ⊆ Diffeo(M)

of the group of diffeomorphisms of M is a subgroup. The subset

X(M,ω) := {inﬁnitesimal symmetries of ω} ⊆ X(M)

of the Lie algebra of vector fields is a Lie subalgebra. To prove this last claim it is convenient to characterize infinitesimal symmetries of ω in a suitable way. Namely, a vector field X on M is an infinitesimal symmetry of ω if, and only if, LX ω = 0. This statement can be proved, exactly as Proposition 6.3.5, after proving the following formulas:

d ∗ ∗ ∗ Φ (ω) = Φ (LX ω) = LX (Φ (ω)), (8.26) dt t t t

where {Φt }t is the flow of X. By definition, given t such that Mt 6= ∅, the left hand side of (8.26) is the differential form on Mt defined through its values as follows: for all p ∈ Mt , the d ∗ value of dt Φt (ω) at p is d d Φ∗(ω) = | Φ∗(ω) . dt t p ds s=t t p One can show that this a well-defined form and that Formulas (8.26) indeed hold. We leave details to the reader. As already mentioned, from those formulas, one can easily prove that X is an infinitesimal symmetry of ω if and only if LX ω = 0. Finally, to see that X(M,ω) ⊆ X(M) is a Lie subalgebra, take X,Y ∈ X(M,ω), and a ∈ R. Then

LX+Y ω = LX ω + LY ω = 0. Additionally,

LaX ω = aLX ω = 0. The last two formulas show that X(M,ω) is a vector subspace of X(M). We conclude computing

L[X,Y]ω = [LX ,LY ]ω = LX LY ω − LY LX ω = 0,

where we used the commutator in the center of Table 8.1. Hence X(M,ω) is also a Lie subalgebra of X(M), as claimed.

Besides the deﬁnition provided in the proof of Theorem 8.3.6, and the one provided by the Cartan Magic Formula, there is yet another equivalent deﬁnition of the Lie derivative.

Proposition 8.3.11 Let M be a manifold, and let X ∈ X(M). The Lie derivative along X satisﬁes the following Leibniz rule with respect to insertions of vector ﬁelds: for any degree k differential form ω, and any X1,...,Xk ∈ X(M),

k (LX ω)(X1,...,Xk) = X (ω(X1,...,Xk)) − ∑ ω(X1,..., [X,Xi],...,Xk). (8.27) i=1 | {z } i-th place

Proof. The proof is by induction on k. The case k = 1 is easily obtained applying the formula

[iX1 ,LX ] = i[X1,X] 8.3 Cartan Calculus 171 to a 1-form (do you see it?). Now, rewrite the left hand side of (8.27) as follows:

(LX ω)(X1,...,Xk) = (iX1 LX ω)(X2,...,Xk)

= ([iX1 ,LX ]ω)(X2,...,Xk) + (LX iX1 ω)(X2,...,Xk)

= (i[X1,X]ω)(X2,...,Xk) + (LX iX1 ω)(X2,...,Xk)

= ω([X1,X],X2,...,Xk) + (LX iX1 ω)(X2,...,Xk). and use induction on the last summand. We conclude this section, this chapter, and these notes discussing how do the interior product, the Lie derivative, and the exterior differential interact with smooth maps. So let M,N be smooth manifolds and let F : M → N be a smooth map. We begin with the most important case: that of the exterior differential. Proposition 8.3.12 — Naturality of the Exterior Differential. The exterior differential “commutes” with pull-backs, i.e.

d ◦ F∗ = F∗ ◦ d. (8.28)

Before proposing a proof we stress that the reader should not be confused by the terminology “the exterior differential commutes with pull-backs”. Namely, the exterior differential in the left hand side of (8.28) is that in Ω•(M) while the one in the right hand side is that in Ω•(N).

Proof of Proposition 8.3.12 (a sketch). The statement could be proved by a direct computation. We propose an alternative proof exploiting Theorem 8.2.4. Consider the R-linear map D : Ω•(N) → Ω•(M), ω 7→ D(ω) := dF∗(ω) − F∗(dω).

We have to prove that D = 0. To do this, we ﬁrst list some properties of D. Besides the R-linearity, D enjoys the following properties: k k+ (1) D is a degree 1 map, i.e. D(Ω (N)) ⊆ Ω 1(M), for all k ∈ Z, (2) D satisﬁes the following (F-relative) version of the graded Leibniz rule: for all homogeneous forms ω,ρ ∈ Ω•(N),

D(ω ∧ ρ) = D(ω) ∧ F∗(ρ) + (−)|ω|F∗(ω) ∧ D(ρ).

Property (1) is obvious. Property (2) can be easily checked using that d is a graded derivation and that F∗ is a graded algebra homomorphism, and we leave the details to the reader. In a very similar way as in the proof of Proposition 8.3.3, it follows from Properties (1) and (2) that, if D vanishes on generators, then D = 0. So it remains to check that D( f ) = 0, D(d f ) = 0, i.e.

dF∗( f ) = F∗(d f ), (8.29) dF∗(d f ) = F∗(dd f ), (8.30) for all f ∈ C∞(M). Now, the ﬁrst identity has already been proved as Proposition 7.3.2 Point (3), while the second identity follows from Proposition 7.3.2 Point (3), and d ◦ d = 0. This concludes the proof. We now pass to interior products and Lie derivatives. 172 Chapter 8. Differential Forms and Cartan Calculus

Proposition 8.3.13 Let X ∈ X(M) and Y ∈ X(N) be F-related vector ﬁelds. Then

∗ ∗ iX ◦ F = F ◦ iY , ∗ ∗ LX ◦ F = F ◦ LY .

In particular, if Φ : M → N is a diffeomorphism, and Y ∈ X(N), then

∗ ∗ iΦ∗(Y) ◦ Φ = Φ ◦ iY , ∗ ∗ LΦ∗(Y) ◦ Φ = Φ ◦ LY .

Proof. Left as Exercise 8.10.

Exercise 8.10 Prove Proposition 8.3.13. (Hint: there is a proof very similar to the proof of

Proposition 8.3.12.)

• Corollary 8.3.14 Let M be a manifold, let Y ∈ X(M), let ω,ω1,ω2 ∈ Ω (M), and let Φ : M → M be a diffeomorphism. If Φ is a symmetry of ω,ω1,ω2, then it is also a symmetry of dω,ω1 + ω2,ω1 ∧ ω2. If, additionally, Φ is a symmetry of Y, then it is also a symmetry of iY ω and LY ω.

Proof. Left as Exercise 8.11.

Exercise 8.11 Prove Corollary 8.3.14.

The following inﬁnitesimal version of Corollary 8.3.14 is a corollary of either Corollary 8.3.14 itself, or Theorem 8.3.10.

• Corollary 8.3.15 Let M be a manifold, let X,Y ∈ X(M), and let ω,ω1,ω2 ∈ Ω (M). If X is an infinitesimal symmetry of ω,ω1,ω2, then it is also an infinitesimal symmetry of dω,ω1 + ω2,ω1 ∧ ω2. If, additionally, X is an infinitesimal symmetry of Y, then it is also an infinitesimal symmetry of iY ω and LY ω.

Proof. Left as Exercise 8.12.

Exercise 8.12 Prove Corollary 8.3.15. Index

Symbols Magic Formula ...... 169 Chain rule ...... 61 k-Covector ...... 151 Chart ...... 13 bundle of ...... 152 adapted to a submanifold ...... 48 on an open submanifold ...... 152 around a point ...... 13 centered at a point ...... 13 A compatible ...... 17 linear chart on a vector space...... 13 n Affine chart on RP ...... 16 standard n Algebra ...... 33 on R , 13 graded ...... 147 on the bundle of k-covectors, 152 of real valued functions ...... 34 on the cotangent bundle, 130 of smooth functions ...... 34 on the tangent bundle, 68 Alternating form ...... 143 stereographic ...... 14 are skewsymmetric ...... 144 submanifold ...... 52 on a module with a finite frame . 148, 150 Chevalley-Eilenberg Formula ...... 167 Atlas ...... 17 Closed subset ...... 22 affine atlas on Pn ...... 18 R Commutator ...... 91 compatible ...... 18 graded ...... 159 linear atlas on a vector space ...... 18 of vector fields ...... 91 maximal ...... 20 coordinate formula, 101 standard atlas on n ...... 18 R Coordinate stereographic ...... 18 coframe ...... 127, 131 submanifold ...... 52 domain ...... 13 C frame ...... 62, 94 map ...... 13 Cartan representation Èlie ...... 143, 158 of a function, 31 Calculus ...... 169 of a map, 35 174 INDEX

Cotangent bundle ...... 129 Embedding ...... 83 Cotangent space ...... 127 and submanifolds ...... 83 of an open submanifold ...... 128 Segre ...... 84 Curve ...... 55 integral ...... 103 F are solutions of an ODE, 104 of F-related vector fields, 110 Flow ...... 111 maximal integral...... 105 commuting ...... 125 existence and uniqueness, 105 domain ...... 111 velocity of ...... 56 group property ...... 112 coordinate expression, 63 Function in a vector space, 66 bump ...... 33 smooth...... 31 D G De Rham cohomology...... 169 Graded vector space...... 146 Derivation ...... 89 Gradient ...... 168 graded ...... 158 Group degree, 158 general linear ...... 30 Diffeomorphism ...... 40 of diffeomorphisms ...... 40 between P1 and S1 ...... 41 R of symmetries coordinate maps are ...... 41 of a differential form, 170 Differential of a function, 121 exterior...... 165, 167 of a submanifold, 120 coordinate expression, 168 of a vector field, 124 naturality, 171 orthogonal ...... 80 squares to zero, 169 special linear ...... 81 of a function ...... 132 special orthogonal...... 81 at a point, 128 coordinate expression, 132 H naturality, 139 properties, 132 Homeomorphism ...... 23 Differential 1-form ...... 131 Homomorphism ∗ and sections of T M ...... 133, 136 of algebras ...... 41 components in a chart ...... 137 of graded algebras ...... 148 restriction to a submanifold ...... 141 of modules ...... 70 value at a point ...... 135 Differential 1-forms I are local operators ...... 133 Differential form...... 153 Immersion ...... 73 components in a chart ...... 155 Infinitesimal diffeomorphism...... 114 generators...... 156 Infinitesimal generator on a coordinate domain...... 153 of dilations ...... 106, 113 restriction to a submanifold ...... 157 of rotations ...... 107, 114 value at a point ...... 154 of translations...... 105, 113 Divergence ...... 168 Infinitesimal symmetry ...... 172 of a differential form ...... 170 E and Lie derivative, 170 of a function ...... 122 Eight figure...... 82 and Lie derivative, 122 INDEX 175

of a submanifold...... 120 coordinate expression, 165 and tangent vector fields, 120 of a function ...... 114, 115 of a vector field ...... 124 at time t, 115, 116 and Lie derivative, 125 of a vector field ...... 116, 117 Interior product ...... 160, 161 at time t, 118, 119 coordinate expression ...... 161 Lie subalgebra ...... 91 Isomorphism Local diffeomorphism ...... 73 of algebras ...... 41 M J Manifold ...... 27 Jacobi identity ...... 91 diffeomorphic ...... 40 graded ...... 159 product ...... 29 Jacobian matrix ...... 43 projective space...... 28 sphere ...... 27 K standard Euclidean space ...... 27 Map Koszul Sign Rule ...... 159 constant rank, smooth properties, 77 L continuous ...... 23 dual of the tangent ...... 138 Leibniz rule ...... 90, 132 graph of a smooth ...... 47 at a point ...... 57, 129 regular value of ...... 43 graded ...... 158 smooth ...... 35 F -relative, 171 constant rank, 76 with respect to insertions of vector fields smooth ⇒ continuous ...... 38 170 tangent ...... 60 Lemma to the inclusion of a submanifold, 61 Gluing valued in a subspace ...... 23 for differential 1-forms, 137 Module ...... 69 for differential forms, 155 graded ...... 146 for smooth functions, 33 degree, 147 for smooth maps, 38 homogeneous elements, 147 for vector fields, 94 of differential 1-forms ...... 131 Hadamard ...... 58 of sections of ∧kT ∗M ...... 153 Local Extension of sections of T ∗M ...... 131 for smooth functions, 53 of sections of TM ...... 70 for vector fields, 133 of vector fields ...... 91 Translation ...... 108 Lie algebra ...... 91 O of endomorphisms ...... 91 of infinitesimal symmetries Open subset ...... 22 of a differential form, 170 with respect to an atlas ...... 21 of a function, 122 of a vector field, 125 P of vector fields ...... 91 Lie bracket ...... 91 Projective space...... 16 Lie derivative Pull-back of a differential form ...... 163, 165 of 1-forms ...... 139 at time t, 170 of differential forms...... 156 176 INDEX

of real valued functions ...... 41 to a submanifold ...... 71 of smooth functions ...... 42 Tangent covector ...... 127 of vector fields...... 101 Tangent space...... 57 to a vector space ...... 66 R to the sphere...... 79 Tangent vector ...... 57 Rank of a smooth map ...... 73 coordinate ...... 57 and Jacobian matrix ...... 74 elementary properties...... 57 Regular system ...... 43 is a local operator ...... 57 Rotor ...... 168 Theorem S Bump Function...... 33, 34 Existence, Uniqueness, and Smooth De- Section pendence on Initial Data of Solutions of a surjective submersion ...... 83 of ODEs ...... 104 of the bundle of k-covectors...... 153 Fundamental Theorem of Flows . . . . . 112 of the cotangent bundle ...... 131 Implicit Function ...... 45 components, 131 Inverse Function ...... 81 of the tangent bundle ...... 69 Rank ...... 76 components, 71 Submanifold Chart ...... 49 smooth, 70 Tangent Vector ...... 58, 62 n Smooth structure ...... 20 on R , 59 containing an atlas ...... 20 Topological space ...... 22 of the cotangent bundle ...... 131 Hausdorff ...... 24 on the bundle of k-covectors ...... 153 II-countable ...... 25 on the tangent bundle...... 68 product ...... 25 n standard smooth structure on R ...... 20 product of Hausdorff spaces...... 26 Stereographic projection ...... 14 product of II-countable spaces...... 26 Subalgebra ...... 33 Topological subspace ...... 22 Subchart ...... 18 of a Hausdorff space ...... 24 Submanifold ...... 47 of a II-countable space ...... 25 0-dimensional ...... 48 Topology...... 22 n-dimensional ...... 48 atlas...... 21, 22 affine subspace ...... 47 basis for ...... 24 characterizations ...... 48 concrete ...... 22 graph of a smooth map ...... 47 discrete ...... 22 in a submanifold ...... 53 generated by a basis ...... 25 sphere ...... 47 product ...... 25 Submersion ...... 73 properties of the atlas topology ...... 26 n Submodule ...... 69 standard topology on R ...... 22 Symmetry ...... 119 subspace ...... 22 of a differential form ...... 170 Torus ...... 30 of a function...... 121 of a submanifold...... 120 U of a vector field ...... 123 Unshuffle ...... 144 T V Tangent bundle ...... 68 is a manifold ...... 68 Vector field ...... 90 section of ...... 69 F-related ...... 97 INDEX 177

and algebraic structures, 99 a non-complete ...... 111 and sections of TM ...... 92 commuting ...... 125 components in a chart ...... 95 coordinate...... 90 computing with, 101 restriction to an open submanifold . . . . 93 tangent to a submanifold...... 98 and algebraic structures, 100 characterization, 98 value at a point ...... 92 vanishing at a point ...... 134 Vector product ...... 151

Wedge product ...... 144, 147 graded commutativity...... 145, 147

Zero locus...... 43