Technische Universitat¨ Berlin Institut fur¨ Mathematik

Complex Analysis II

Riemann surfaces

Prof. Dr. Ulrich Pinkall

Lecture notes Dr.Felix Knoppel¨ and Oliver Gross

February 12, 2019

Table of Contents

1 Introduction1 1.1 Basic Concepts of Complex Analysis...... 1 1.2 A Historical Interlude...... 3

2 Complex Manifolds8 2.1 Crash Course in Topology...... 8 2.2 Smooth Structures...... 10 2.3 Introduction to Several Complex Variables...... 17 2.4 Complex Structures and Manifolds...... 22 2.5 Complex Linear Subspaces...... 24

3 The Tangent Bundle 27 3.1 Tangent Spaces...... 27 3.2 The tangent bundle as a smooth vector bundle...... 32 3.3 Vector Bundles...... 33 3.4 The Lie-bracket of Vector Fields...... 38

4 Almost Complex Structures 40 4.1 Tensors...... 40 4.2 The Newlander-Nirenberg Theorem...... 42 4.3 Connections on Vector Bundles...... 43 4.4 Partition of unity...... 51 4.5 Conformal Equivalence...... 53

5 Integration on manifolds 56 5.1 Volume Forms...... 56 5.2 Integration of Forms...... 59 5.3 Stokes Theorem...... 61 5.4 Fundamental theorem for flat vector bundles...... 63

I TABLE OF CONTENTS

6 Riemann surfaces 68 6.1 Holomorphic line bundles over a Riemann surface...... 71 6.2 Poincare-Hopf´ index theorem...... 73

7 Classification of Line Bundles 85 7.1 Tensor products of vector spaces and bundles...... 85 7.2 Line Bundles on Surfaces...... 87 7.3 Combinatorial Topology...... 91 7.4 Discrete Forms...... 96 7.5 Poincare´ Duality...... 98 7.6 ”Baby Riemann Roch Theorem”...... 101 7.7 A Natural Complex Structure on Dual Spaces...... 105 7.8 Holomorphic Structures on Vector Bundles...... 106 7.9 Elliptic Differential Operators...... 109

8 Appendix 122

II

1. Introduction

The first formal Definition of a manifold and a Riemann surface is found in the paper ”Die Idee der Riemann’schen Fl¨ache” by H. Weyl in 1910.

1.1 Basic Concepts of Complex Analysis

The central objects of discussion in this course will be Riemann surfaces, but before we start we will revise some basic concepts and vivid ideas of what we will be dealing with. Later we will then (hopefully) understand that these, sometimes vaguely stated, claims are in fact true.

Within this course we will see that the easiest examples of Riemann surfaces are open sets U ⊂ C = R2, e.g.

D2 = {(x, y) ∈ R2 | x2 + y2 < 1}, an annulus or the whole plane.

Definition 1.1 (homeomorphic). Let U, U˜ ⊂ Rn are called homeomorphic, if there is a bijection ϕ : U → U˜ such that ϕ and ϕ−1 both are continuous.

Example 1.2.

(i) The unit disc D2 and R2 are homeomorphic. A homeomorphism is given by πr ϕ : (r cos φ, r sin φ) 7→ tan 2 (cos φ, sin φ)

1 1.1 Basic Concepts of Complex Analysis

with inverse −1 2 ϕ : (r cos φ, r sin φ) 7→ π arctan(r)(cos φ, sin φ). It is easy to see that bot, ϕ and ϕ−1 are continuous.

(ii) U1 and U2 are homeomorphic.

(iii) U3 and U4 are homeomorphic.

Definition 1.3 (diffeomorphic). Let U, U˜ ⊂ Rn open are called diffeomor- phic if there is a bijective map ϕ : U → U˜ such that ϕ and ϕ−1 are both smooth.

2 The same statements as in example 1.2 about U1,..., U5, R are true replac- ing homeomorphic by diffeomorphic.

Definition 1.4 (biholomorphic). Let U, U˜ ⊂ Rn open are called biholomor- phic if there is a bijective map ϕ : U → U˜ such that ϕ and ϕ−1 are both holomorphic.

Theorem 1.5. The open unit disc D2 is not biholomorphic to C.

Proof. Let ϕ : C → D2 be holomorphic. Then ϕ is bounded and hence by Liouville’s theorem constant. Thus ϕ is no bijection.

Theorem 1.6 (Riemann mapping theorem). Let U, U˜ ⊂ C be open, U 6= C, U˜ 6= C and both homeomorphic to D2. Then U is biholomorphic to U.˜

Remark 1.7. Considering theorem 1.6 one may get the idea that also the annuli U3 and U4 are biholomorphic. Surprisingly this does not hold in general, but one can proof that indeed two annuli are biholomorphic if and only if the quotient of their respective radii is equal. Example 1.8 (Riemann sphere). An informal idea about Riemann surfaces is that they are ”surfaces” (2-dimensional manifolds) that ”locally” (meaning near every point) look like an open set U ⊂ C. Another example of such a surface is given by the Riemann sphere M = S2 = C¯ . This can be seen by considering the stereographic projection from the north pole 2 σn : S \ (0, 0, 1) → C as indicated in figure 1.1. Using the projection from the south pole is changing the chart by z 7→ 1/z. Thus we have completely covered S2 by charts.

2 Introduction

Figure 1.1: An image of how the stereographic projection works.

1.2 A Historical Interlude

Riemann surfaces naturally came up in mathematics when trying to solve so called elliptic integrals. To get an idea about what an elliptic integral is and what makes them special, we will consider two examples that math- ematicians in early times tried to conquer. This will then lead to an idea how elliptic integrals and Riemann surfaces fit together.

Ellipse: An ellipse is given by

2 = {( ) ∈ R2 | x2 + y = } E x, y a2 b2 1 .

Z a » The length of E is then given by L = 4 1 + f 0(x)2dx, where 0 … [ ] → R 7→ ( ) = − x2 f : 0, a , x f x b 1 a2 .

3 1.2 A Historical Interlude

Then 0 −xb f (x) = … 2 − x2 a 1 a2 and thus, with z = x/a and e2 = 1 − a2/b2,

Z 1 2 2 L = 4a √ 1−e z dz 0 (1−z2)(1−e2z2)

which is an elliptic integral and has no solution by elementary func- tions.

Pendulum: Considering the (mathematical) pendulum two related prob- lems arise naturally:

(i) What is the period? (ii) What is θ as a function of time?

We assume that m = g = L = 1. The pendulum equation is

θ00 = − sin θ.

By adding − sin θ to both sides and multiplying the equation with θ0 we get θ0θ00 + θ0 sin θ = 0. We see that Å ã0 Ä 1 0ä2 0 00 0 2 θ − cos θ = θ θ + θ sin θ = 0.

Ä 1 0ä2 Ä 1 0ä2 Thus 2 θ − cos θ, the sum of the kinetic energy 2 θ and the po- tential energy 1 − cos θ, must be constant. At maximum elongation α we have θ0 = 0, as the potential energy is

4 Introduction

maximal, what leads to const = − cos α. So we know the total energy of the system leading to

Ä 1 0ä2 d 0 » 2 θ = cos θ − cos α ⇔ dt θ = θ = 2 (cos θ − cos α)

Thus, when θ0(t) ≥ 0, by seperation of variables

dθ dt = » . 2(cos θ − cos α)

So that a quarter of a period, meaning from angle 0 to θ, is

Z θ t = √ dϕ 0 2(cos ϕ−cos α)

With 2 ϕ 2 ϕ 2 ϕ 2 ϕ cos ϕ = cos 2 − sin 2 = 1 − sin 2 − sin 2 we get 2 ϕ 1 − cos ϕ = 2 sin 2 , thus as the same holds for cos α

Z θ Z θ Z θ t = √ dϕ = 1 dϕ = 1 1 1 dϕ 2 q α ϕ 2 α Ã ϕ 0 2(cos ϕ−cos α) 0 sin2 −sin2 0 sin 2 2 2 2 sin 2 1− 2 α sin 2

ϕ sin 2 α Substitution with z = α and defining e := sin 2 leads to sin 2

ϕ sin = 2 = 2e = 2e = √ 2e z e dϕ ϕ dz q ϕ dz 2 2 dz cos 2 − 2 1−e z ; 1 sin 2

what finally yields

Z ρ t = √ dz . 0 (1−z2)(1−e2z2)

2 θ sin 2 where ρ := 2 α . Thus, with α = θ it is ρ = 1, therefore the full sin 2 period is given by Z 1 T = 4 √ dz . 0 (1−z2)(1−e2z2)

5 1.2 A Historical Interlude

But how does this now lead to Riemann surfaces? In two examples we have found integrals of the form

Z 1 Z 1 ( − 2 2) √ dz , √ 1 e z dz 0 (1−z2)(1−e2z2) 0 (1−z2)(1−e2z2)

2 2 2 1 1 1 The map z 7→ (1 − z )(1 − e z ) has zeros at − e , −1, 1, e . Define a := e .

√ √ What shall we do with some expression like z − a, say z? We have two choices here. √ For each z ∈ C \{negative real axis} consider two points representing ± z, visualize a z, but in two ’sheets’ (copies of the complex plane).

The same procedure, this time we glue two copies of C together along the » real intervalls [−a, −1] and [1, a], turns (1 − z2)(1 − e2z2) into an honest function on a Riemann surface – the two-sheeted cover of C branched over −a, −1, 1, a.

Remark 1.9. Note that the blue curve γ on the Riemann surface is closed, whereas the green η curve is not.

6 Introduction

Actually this can be regarded as a torus with two points removed – the missing points correspond to z = ∞.

7 2. Complex Manifolds

2.1 Crash Course in Topology

As a Riemann surface will be also a two-dimensional real manifold, which itself is nothing more than a special kind of topological space it makes sense to shortly revise the most important definitions and ideas of Topology.

Definition 2.1 (Topological space). A topological space is a set M together with O ⊂ P(M) (’collection of open sets’) such that

(i) ∅, M ∈ O

(ii)U α ∈ O for α ∈ A ⇒ ∪α∈AUα ∈ O

(iii)U 1,..., Un ∈ O ⇒ U1 ∩ · · · ∩ Un ∈ O

Example 2.2 (subspace topology). If M ⊂ M˜ , M˜ topological space. Then O = {U˜ ∩ M | U ∈ O˜ } defines a topology for M — the subset or relative topology.

Example 2.3 (Quotient topology). If π : M˜ → M, M˜ topological space. Then O = {U ∩ M | π−1U ∈ O˜ } defines a topology for M — the quo- tient topology. The map π yields an equivalence relation on M˜ given by x ∼ y :⇔ π(x) = π(y). Conversely, the natural projection of ∼ is such a map π. We denote the quotient space by M = M˜ /∼.

Example 2.4 (Torus). M˜ = R2, p ∼ q :⇔ q − p ∈ Z2.

Definition 2.5 (Continuity). Let M, M˜ be topological spaces. Then f : M → M˜ is called continuous, if f −1(U˜ ) ∈ O for all U˜ ∈ O˜ .

8 Complex Manifolds

Definition 2.6 (Homeomorphism). Let M, M˜ be topological spaces. Then f : M → M˜ is called a homoeomorphism, if f is bijective and f and f −1 both are continuous. To assure that the kind of topological spaces that we deal with have nice properties and we do not have to deal with any horrific examples of topo- logical spaces that topologists can think of we have certain demands on them.

Definition 2.7 (Hausdorff). A topological space M is called Hausdorff, if for all p, q ∈ M with p 6= q there are U, U˜ ∈ O such that p ∈ U, q ∈ U˜ and U ∩ U˜ = ∅.

Theorem 2.8.

(i) Rn is Hausdorff

(ii) M˜ Hausdorff, M ⊂ M.˜ Then M is Hausdorff (with respect to the relative topology).

Example 2.9. An example of a non-Hausdorff space is M = R together with p ∼ q :⇔ q − p ∈ Q. Then M/∼ is not Hausdorff. You can check that as an exercise.

In particular, quotient spaces tend not to be Hausdorff. So when we are dealing with quotient spaces it is always necessary to, at least, shortly think about it.

Definition 2.10 (2nd-countable space). A topological space M is said to satisfy the second axiom of countability (or, in short, is called 2nd countable) if there are U1, U2, U3,... ⊂ O such that for each U ∈ O there is I ⊂ N such that U = ∪i∈IUi.

Theorem 2.11.

(i) Rn is 2nd-countable.

(ii) If M˜ is 2nd-countable, M ⊂ M.˜ Then M is 2nd-countable (with respect to the subspace topology).

n Proof. (only for a)) Let U1, U2, . . . be an enumeration of all open balls in R n with center Q and radius r ∈ Q. Then this is a basis of ORn .

9 2.2 Smooth Structures

Definition 2.12 (topological manifold). A topological space M is called an n-dimensional topological manifold, if M is a 2nd-countable Haussdorf space and for every p ∈ M there is an open set U 3 p, an open set V ⊂ Rn and a homeomorphism ϕ : U → V.

Definition 2.13 (coordinate chart, atlas). A pair (U, ϕ) as in definition 2.12 is called coordinate chart on a topological manifold M. An collection (Uα, ϕα)α∈I of corrdinate charts on a topological manifold M is called an atlas if M = ∪α∈IUα.

2.2 Smooth Structures

Definition 2.14 (coordinate change). Given two charts ϕ : U → Rn and ψ : V → Rn of a topological manifold M, then the map

−1 f : ϕ(U ∩ V) → ψ(U ∩ V) with f = ψ ◦ ( ϕ|U ∩ V) is a homeomorphism, called the coordinate change or transition map.

10 Complex Manifolds

Definition 2.15 (smoothly compatible). Two charts (U, ϕ), (V, ψ) on a topological manifold M are called smoothly compatible if the corresponding −1 coordinate change f = ψ ◦ ( ϕ|U ∩ V) is a diffeomorphism.

Example 2.16. Consider M = Sn ⊂ Rn+1, and define charts as follows: For i = 0, . . . , n, ± 2 Ui = {x ∈ S | ±xi > 0} with ± ± ± ϕi ,: Ui → B, ϕi (x0,..., xn) = (x0,..., x“i,..., xn), where the hat means omission. To check that ϕi are homeomorphisms we make ourselves aware of the fact that this, as a projection, is indeed continuous and that inverse is given by the continuous map

Ä ±ä−1  q P 2  ϕi (x0,..., x“i,..., xn) = x0,..., 1 − j6=i xj ,..., xn .

Since Sn, as a subset of Rn+1, is Hausdorff and second countable, Sn is an ± n-dimensional topological manifold. All ϕi are compatible, so this atlas with 2n + 2 charts turns Sn into a smooth manifold.

Figure 2.1: This illustration for the case n = 2 is taken from the title page of the book ”Riemannian Geometry” by Manfredo do Carmo (Birkenhauser¨ 1979).

As compatibility defines an equivalence relation on charts, the following definition makes sense.

Definition 2.17 (maximal atlas). An atlas (Uα, ϕα)α∈I of mutually com- patible charts on M is called maximal if every chart (U, ϕ) on M which is compatible with all charts in (Uα, ϕα)α∈I is already contained in the atlas. Remark 2.18. A maximal atlas of mutually compatible charts is also often called smooth structure on M.

11 2.2 Smooth Structures

Definition 2.19 (smooth manifold). A smooth structure defined on a topo- logical manifold M turns M into a smooth manifold.

Example 2.20. n 1. Let M ⊂ R open with the chart (U, ϕ) where U = M and ϕ = idRn , then this defines a smooth manifold. 2. (Product manifolds): Let M and N be topological manifolds of dimension m and n, respec- tively. Then their Cartesian product M × N together with the product topology is a topological manifold of dimension m + n. Further, if (Uα, ϕα)α∈A is a smooth atlas of M and (Vβ, ψβ)β∈B is a smooth atlas of N, then (Uα × Vβ, ϕα × ψβ)(α,β)∈A×B is a smooth atlas of M × N. Here ϕα × ψβ : Uα × Vβ → ϕα(Uα) × ψβ(Vβ) is defined by ϕα × ψβ(p, q) := (ϕα(p), ψβ(q)). 3. (Quotient spaces): Let M˜ be a topological space and π : M˜ → M surjective. Define ¶ © O := U ⊂ M | π−1(U) ∈ O˜ then this defines the quotient topology on M, i.e. M is also a topological space now. Define an equivalence relation ∼ on M˜ by p ∼ q :⇔ π(p) = π(q), then M is in bijective correspondence to the equivalence classes [π(p)] on M˜ , thus M˜ M = ∼. As mentioned before, in particular for quotient spaces, the Hausdorff prop- erty and the property of being second countable have to be carefully checked each time as quotient spaces tend to lack of these. Example 2.21. Examples of manifolds that are obtained by quotient spaces are the following: 1. (Real projective space): n+1 n M˜ Let n ∈ N and M˜ := R \{0}. The quotient space RP = ∼ with equivalence relation given by x ∼ y :⇐⇒ x = λy, λ ∈ R is called the n-dimensional real projective space.

12 Complex Manifolds

2. (Torus): Let M˜ = Rn and define

p ∼ q :⇔ p − q ∈ Zn.

M˜ n Then ∼ is the n-dimensional Torus T and is a smooth manifold. For n = 2 this manifold is actually diffeomorphic to the donut-shaped Torus embedded in R3 and S1 × S1.

Definition 2.22 (Smooth map). Let M and M˜ be smooth manifolds. Then a map f :M → M˜ is called smooth if for every chart (U, ϕ) of M and every chart (V, ψ) of M˜ the map

ψ ◦ f ◦ ϕ−1 : ϕ( f −1(V) ∩ U) → ψ(V), x 7→ ψ( f (ϕ−1(x)))

is smooth.

Definition 2.23 (Diffeomorphism). Let M and M˜ be smooth manifolds. Then a bijective map f :M → M˜ is called a diffeomorphism if both f and f −1 are smooth.

Note that the property of being diffeomorphic defines an equivalence real- tion on smooth manifolds. Diffeomorphic manifolds are indistinguishable from the viewpoint of differential topology. In fact, the following interest- ing theorems hold.

13 2.2 Smooth Structures

Theorem 2.24. Every connected 1-dimensional smooth manifold is diffeomor- phic to either (0, 1), or S1.

Theorem 2.25. Every connected and compact 2-dimensional smooth manifold is diffeomorphic to exactly one in the following list:

The poorly drawn manifolds in the second row of the theorem above are supposed to be Boy’s surface, then the same with a hole and so on. But this is quite hard to visualize. If we demand the manifolds to be orientable, then the theorem also holds, but only the first row of cases can appear.

Manifolds of dimension 3 or higher can sadly not be ordered by a list like in the lower dimensional cases. There are too many of them.

If we have a manifold of dimension k, it makes sense to think about whether the locally structure transfers to lower dimensional subsets. How this can be the case is formalized in the following definition.

Definition 2.26 (Submanifold). A subset M ⊂ M˜ in a k-dimensional smooth manifold M˜ is called an n-dimensional submanifold if for every point p ∈ M there is a chart ϕ : U → V of M˜ with p ∈ U such that

ϕ(U ∩ M) = V ∩(Rn × {0}) ⊂ Rk.

14 Complex Manifolds

If we restrict ourselves to considering manifolds M = Rk, we have the following equivalent definitions of a submanifold.

Theorem 2.27. Let M ⊂ Rk be a subset. Then the following are equivalent:

a) M is an n-dimensional submanifold,

b) locally M looks like the graph of a map from Rn to Rk−n, which means: For every point p ∈ M there are open sets V ⊂ Rn and W ⊂ M,W 3 p, a smooth map f : V → Rk−n and a coordinate permutation π : Rk → Rk,

π(x1, ..., xk) = (xσ1 , ..., xσk ) such that

π(W) = {(x, f (x)) | x ∈ V},

c) locally M is the zero set of some smooth map into Rk−n, which means: For every p ∈ M there is an open set U ⊂ Rk,U 3 p and a smooth map g : U → Rk−n such that

M ∩ U = {x ∈ U | g(x) = 0}

and the Jacobian g0(x) has full rank for all x ∈ M,

d) locally M can be parametrized by open sets in Rn, which means: For every p ∈ M there are open sets W ⊂ M,W 3 p, V ⊂ Rn and a smooth map ψ : V → Rk such that ψ maps V bijectively onto W and ψ0(x) has full rank for all x ∈ V.

Proof.

(b) ⇒ (a): Let p ∈ M. By b) after reordering coordinates in Rk we find open sets V ∈ Rn, W ⊂ Rk−n such that p ∈ V × W and we find a

15 2.2 Smooth Structures

smooth map f : V → W such that V × W ∩ M = {(x, f (x)) | x ∈ V}. Then ϕ : V × W → Rk, (x, y) 7→ (x, y − f (x)) is a diffeomorphism and ϕ(M ∩(V × W)) ⊂ Rn × {0}.

(a) ⇒ (c): Let p ∈ M. By a) we find an open U ∈ Rk, U 3 p and a diffeo- morphism ϕ : U → Uˆ ⊂ Rk such that ϕ(U ∩ M) ⊂ Rn × {0}. Now define g : U → Rk−n to be the last k − n component functions of ϕ, −1 i.e. ϕ = (ϕ1,..., ϕn, g1,..., gk−n). Then M ∩(V × W) = g ({0}). For q ∈ V × W we have  ∗   .   .   .    0  ∗  ϕ (q) =   .  0 ( )   g1 q     .   .  0 gk−n(q) Hence g0 has rank k − n.

(c)⇒ (b): This is just the implicit function theorem.

(b)⇒ (d): Let p ∈ M. After reordering the coordinates by b) we have an open neighborhood of p of the form V × W and a smooth map f : V → W such that M ∩(V × W) = {(x, f (x)) | x ∈ V}. Now define ψ : V → Rk by ψ(x) = (x, f (x)), then ψ is smooth Ç å Id n ψ0(x) = R f 0(x)

So ψ0(x) has rank n for all x ∈ V. Moreover, ψ(V) = M × (V × W).

(d) ⇒ (b): Let p ∈ M. Then by d) there are open sets Vˆ ⊂ Rn, U ⊂ Rk, U 3 p and a smooth map ψ : Vˆ → Rk such that ψ(Vˆ ) = M ∩ U such that rank ψ0(x) is n for all x ∈ Vˆ . After reordering the coordinates on Rk t n 0 we can assume that ψ = (φ, fˆ) with φ : Vˆ → R with det φ (x0) 6= 0, where ψ(x0) = p. Passing to a smaller neighborhood V ⊂ Vˆ , V 3 p, we then achieve that φ : V → φ(V) is a diffeomorphism (by the inverse function theorem). Now for all y ∈ φ(V) we have ! ! φ(φ−1(y) y ψ(φ−1(y)) = =: fˆ(φ−1(y)) f (φ−1(y))

16 Complex Manifolds

Now we will address ourselves to the topic of Riemann surfaces. To do that we first need to translate the just defined machinery to holomorphic maps. Most of the definitions can be adopted word by word only changing smooth or diffeomorphic to holomorphic or biholomorphic. But before we do this we will first need to define holomorphicity for higher dimensions.

2.3 Introduction to Several Complex Variables

Every n-dimensional complex vector space V is automatically a 2n-dimensional real vector space: If v1,..., vn is a complex basis of V, then v1, iv1,..., vn, ivn is a real basis of V. On V as a real vector space we have a real linear map J : V → V, v 7→ Jv = iv (complex scalar multiplication) and J satisfies J2 = −Id.

Conversely: If V is a real vector space with an R-linear map J : V → V such that J2 = −Id, then we can make V into a complex vector space by defining a complex scalar multiplication by (α + iβ)v := (α + βJ)v .

Corollary 2.28. No such J exists on a real vector space of odd dimension.

Definition 2.29 (multi-variable holomorphicity). A map f : U → Ck, for U ⊂ Cn open, is called holomorphic if is continuously differentiable in the real sense and for all p ∈ U the linear map f 0(p) : R2n → R2k is complex-linear.

Remark 2.30. An R-linear map is also complex-linear if

A(JR2n v) = JR2k (Av), i.e. if AJ = JA holds.

Theorem 2.31. Let (V, JV) and (W, JW ) be two complex vector spaces. Every real linear map A : V → W can be uniquely decomposed into A = A0 + A00 . where A0 is a complex-linear map

0 0 0 A : V → W such that A JV = JW A and A00 is a complex-anti-linear map

00 00 00 A : V → W such that A JV = −JW A

17 2.3 Introduction to Several Complex Variables

0 1 0 1 0 Proof. Define A = 2 (A − JW AJV) and A = 2 (A + JW AJV) and check A is complex-linear and A00 is complex-anti-linear. Uniqueness is clear.

Theorem 2.32. Let U ⊂ Cn and V ⊂ Ck be open, f : U → Ck, h : V → U. Then:

(i)f = ( f1,..., fk) is holomorphic if and only if fj : U → C are holomorphic for all j = 1, . . . , k.

(ii) If f , g : U → C holomorphic then f + g and f · g are holomorphic and if g(z) 6= 0 for all z ∈ U then f /g is holomorphic.

(iii)g , h holomorphic then g ◦ h is holomorphic.

Proof. The proof of this theorem will be a homework exercise.

Multiindices: Dealing with functions in several complex variables can of- ten lead to a mess when it comes to indexing. A neat way of dealing with this problem is to introduce so called multiindices that are de- fined as follows: n n Let z = (z1,..., zn) ∈ C , k = (k1,..., kn) ∈ N , then

n k k Y j k1 kn z := zj = z1 · ... · zn . j=1

n If all zj 6= 0, then this works also for k ∈ Z .

n 1 −1 Let further be k ∈ C . Later terms of the form w−z will appear. These will have to be understood as n Y 1 = (w − z)−1 = (w − z)(−1,...,−1) = 1 . w−z (wj−zj) j=1

In a similar manner we will deal with expressions of the form k + 1, we simply add 1 to each of the components of k, i.e.

k + 1 = (k1 + 1, . . . , kn + 1) .

As we translate theorems of complex analysis of one variable to the higher-dimensional case, we will also have to use integration, thus expressions of the form dw will appear. For w ∈ Cn these are defined as dw := dw1 . . . dwn.

18 Complex Manifolds

Definition 2.33 (Polydisc, n-Torus). For a ∈ Cn, r ∈ Rn, we define the n-dimensional polydisc with center a and radius r by

n Da,r := {(z1,..., zn) ∈ C | |zj − aj| < rj∀j} = Da1,r1 × · · · × Dan,rn

and the n-dimensional Torus with center a and radius r by

Tn(a r) = {(z z ) ∈ Cn | |z − a | = r ∀j} = S1 × · · · × S1 , : 1,..., n j j j a1,r1 an,rn n Remark 2.34. Note that T is not the actual topological boundary of Da,r in Cn. Anyways, it will act like it in the sense that it will be the right choice to generalize one-dimensional theorems into higher dimensions.

n Theorem 2.35 (Cauchy formula). Let U ⊂ C be open, Da,r ⊂ U and f : U → C holomorphic. Then for all z ∈ D˚ a,r Z 1 f (w) f (z) = ( )n w−z dw. 2πi Tn(a,r)

Proof. Proof by induction on n ∈ N: For n = 1 that’s just the well-known Cauchy formula. For the induction step we just consider the function g = f (·, z2,..., zn) defined on Da1,r1 . Then, by Cauchy formula applied to g and induction hypothesis and by Fubini, Z 1 f (w1,z1,...,zn) f (z1,..., zn) = i w −z dw1 2π S1 1 1 a1,r1 Z Z 1 1 1 f (w1,...,wn) = i w −z n−1 ( − )···( − ) dw2 ··· dwndw1 2π S1 1 1 (2πi) w2 z2 wn zn a1,r1 Z 1 f (w) = ( )n w−z dw. 2πi Tn(a,r)

Theorem 2.36. Holomorphic maps f : → Ck, for U ⊂ Cn open, are smooth, i.e. all partial derivatives exist.

Proof. It is enough to consider only the case k = 1. Since by the Cauchy formula Z 1 f (w) f (z) = ( )n w−z dw, 2πi Tn(a,r) we have Z Z ∂ f 1 ∂ f (w) 1 f (w) ∂z (z) = ( )n ∂z w−z dw = ( )n ( − )( − ) dw . j 2πi Tn(a,r) j 2πi Tn(a,r) wj zj w z Thus, by induction, all partial derivatives exist. Holomorphicity follows from the holomorphicity of the integrand.

19 2.3 Introduction to Several Complex Variables

Some more multiindices: Let k = (k1,..., kn). Then, with

k! = k1! ··· kn!, the following can be defined: Z (k) ∂k1+...+kn ∂k1 ∂kn k! f (w) f (z) := = ... f (z) = n + dw k1 ··· kn k1 kn (2πi) n (w−z)k 1 ∂z1 ∂zn ∂z1 ∂zn T (a,r) Further, by k ≥ 0 we denote k ∈ Nn.

Theorem 2.37 (Taylor/Power series expansion). The Taylor series of a holo- morphic function f : D˚ a,r → C converges everywhere to f :

X 1 (k) k f (z) = k! f (a)(z − a) . k∈Nn Remark 2.38. We will see that also the proof of the power series expansion theorem works just like in the one-dimensional case again. But we need the following observation: It is known that the formula for the geometric series holds in C, i.e. for |q| < 1 it is ∞ X 1 qk = . − k=0 1 q Using this, for |z| < |w| we get ∞ 1 1 1 X k = = z . − − z wk+1 w z w 1 w k=0 For the multi-variable version we will make use of the absolute convergence of the geometric series that will allow us to change the order of summation.

n For z, w ∈ C such that zj < wj for all j = 1, . . . , n it is k 1 kn k 1 1 1 X z1 zn X z = ... = k +1 ... kn+1 = k+1 − − − w 1 wn w w z w1 z1 wn zn k≥0 1 k≥0 Proof. We will start with the right-hand side and plug in the definition of f (k)(z), this yields: Z X 1 (k) k 1 X k f (w) k f (a)(z − a) = ( )n (z − a) k+1 dw ! 2πi Tn (w−a) k≥0 k≥0 a,r Ñ é Z 1 X (z−a)k = ( )n f (w) k+1 dw 2πi Tn (w−a) a,r k≥0 Z 1 f (w) = n dw (2πi) n (w−a)−(z−a) Ta,r Z 1 f (w) = n dw (2πi) n (w−z) Ta,r = f (z)

20 Complex Manifolds where we used the observation of remark 2.38 and for the last equality the Cauchy integral theorem.

Corollary 2.39 (Principle of analytic continuation). Let U ⊂ Cn open and connected, ∅ 6= V ⊂ U open and f : U → C holomorphic with f |V = 0. Then f (z) = 0 for all z ∈ U.

Proof. Define \ n o E := z ∈ U | f (k)(z) = 0 k∈Zn then E is a closed subset of U and V ⊂ E. Further, by the power series expansion theorem, E is open. Since U is connected we can deduce that U = ∅ or E = U. Since E 6= ∅ by assumption, it it E = U. Thus the power series theorem implies that f ≡ 0. Remark 2.40. In the one dimensional case we know that it even suffices to know that f (z) = 0 for on set V ⊂ U that has an accumulation point in U. This does not generalize to higher dimensions. An easy counter-example for this is f (z1, z2) = z1. Then f 6= 0 but E = {z1 = 0} certainly has an accumulation point. The power series expansion theorem and the corollary do not hold in a real C∞-setting. There the situation is utterly different. There we can work with a toolbox of functions that are all C∞ but are constant on certain domains.

Toolbox of C∞-functions: Consider f : R → R with

( 0 for x ≤ 0, f (x) = e−1/x for x > 0.

This certainly is C ∞ and so is then

g(x) = f (1 − x2)

and Z x h(x) = g. 0

From h this we can build a smooth function hˆ : R → [0, 1] with hˆ(x) = 1 1 ˆ 1 for x ∈ [− 4 , 4 ] and h(x) = 0 for x ∈ R \ (−1, 1).

21 2.4 Complex Structures and Manifolds

There are also higher dimensional versions: We can define a smooth function ˜ n ˜ ˆ 2 2 h : R → R, h(x) = h(x1 + ··· + xn)

which vanishes outside the unit ball and is constant ≡ 1 inside the 1 ball of radius 2 .

2.4 Complex Structures and Manifolds

Now we are able to define what complex manifold is supposed to be. As already mentioned, most of the definitions translate pretty much directly from the smooth setting.

Definition 2.41 (holomorphically compatible). Two charts (U, ϕ), (V, ψ) on a 2n-dimensional topological manifold M are called holomorphically com- −1 patible if the corresponding coordinate change f = ψ ◦ ( ϕ|U ∩ V) is biholo- morphic.

Definition 2.42 (complex structure). A complex structure on a topological manifold M is a maximal atlas of holomorphically compatible charts.

Definition 2.43 (Complex manifold). An n-dimensional complex manifold M is a 2n-dimensional topological manifold together with a complex structure.

Example 2.44 (Riemann sphere).

22 Complex Manifolds

Conisder the Riemann sphere Cˆ = C ∪ {∞} ' S1 together with the stere- ographic projection σN from the North Pole and σS from the South Pole. 1 −1 1 Then, (σN, σ¯S) form an atlas of S and the transition map σ¯S ◦ σN = z is holomoprhic, hence the equivalence class of mutually compatible maps on Cˆ defines a complex structure on the Riemann sphere. This turns Cˆ into a complex manifold.

Definition 2.45 (Holomorphic map). Let M and M˜ be complex manifolds. Then a map f :M → M˜ is called smooth if for every chart (U, ϕ) of M and every chart (V, ψ) of M˜ the map

ψ ◦ f ◦ ϕ−1 : ϕ( f −1(V) ∩ U) → ψ(V), x 7→ ψ( f (ϕ−1(x)))

is holomorphic.

Definition 2.46 (biholomorphic). Two complex manifolds M, M˜ are called biholomorphic if there is a biholomorphism between them.

Example 2.47.

(i) Any open subset M ⊂ Cn is a complex manifold.

(ii) (Complex Torus) 2n ∼ n Let a1,..., a2n be a real basis of R = C . Define

Γ := {z1a1 + ..., z2na2n | z1,..., z2n ∈ Z} ,

23 2.5 Complex Linear Subspaces

then Γ is called an integer-Lattice in R2n. Then Cn Cn M := Γ := ∼ with z ∼ w :⇔ z − w ∈ Γ together with the quotient topology is a topological manifold. The obvious charts are for U ⊂ Cn such that z − w ∈/ Γ for all z, w ∈ U given by ϕ : U → Cn, ϕ(z) = z. Then coordinate changes are of the form z 7→ z + γ for γ ∈ Γ, thus in particular holomorphic. Taking the maximal atlas turns M into an n-dimensional complex manifold. Complex analysis deals with classifying complex manifolds up to biholo- morphy and studying their properties. In this class we will mainly deal with one complex-dimensional manifolds, thus they get a special name.

Definition 2.48 (Riemann surface). A complex manifold with dimC M = 1 is called a Riemann surface.

As a motivation why one should study Riemann surfaces we already con- sidered elliptic integrals. Doing so, the problem of how to deal with ex- » pressions of the form w = (1 − z2)(1 − e2z2) arouse. If we consider the complex curve ¶ © M = (z, w) ∈ C2 | w2 = (1 − z2)(1 − e2z2) then this will be a Riemann surface. This is due to the implicit function theorem, but to get a feeling why this is true, have a look in the Appendix on the topic of algebraic curves.

2.5 Complex Linear Subspaces

Knowing what a complex manifold is immediately raises the question of how the notion of a submanifold is possible in the complex setting. It turns out, that it works in a similar manner, so that we even find a complex equiv- alent of theorem 2.27.

In the turtorials (cf. Appendix) we have already learned about linear com- plex structures, so we will shortly recall the following.

24 Complex Manifolds

Definition 2.49 (Complex vector space). A complex n-dimensional vector space is a real 2n-dimensional vector space with J ∈ End(V) with J2 = −Id.

Definition 2.50 (Complex subspace). A linear subspace (in the real sense) U of a complex vector space V is called complex subspace if JU ⊂ U, i.e. it is J-invariant.

We will introduce the following notation that will come in handy for our further purposes.

HomR(V, W) = {g : V → W | g linear in the real sense}

Definition 2.51 (Complex linear vector space). Let V, W be complex vector spaces. Then f ∈ HomR(V, W) is called complex linear if JW ◦ f = f ◦ JV.

Denote the set of complex linear homomorphisms as

HomC(V, W) = { f ∈ Hom(V, W) | JW ◦ f = f ◦ JV}.

If V, W are complex vector spaces, then the complex structure JV⊕RW on V ⊕R W, where ⊕R denotes the real direct sum, is given by

JV⊕RW (v, w) = (JVv, JW w). ( ) Moreover, the complex structure JHomC(V,W) of HomC V, W is given by = ◦ = ◦ JHomC(V,W) f JW f f JV.

Theorem 2.52. Let V, W be complex vector spaces. Then f ∈ HomR(V, W) is complex linear if and only if the graph Gf = {(v, f (v)) ∈ V ⊕R W | v ∈ V} of f is a complex subspace.

Proof.

”⇒”: For f ∈ HomC(V, W) we have

J(v, f (v)) = (Jv, J f (v)) = (Jv, f (Jv)) ∈ Gf .

”⇐”: If Gf is a complex subspace, then (Jv, J f (v)) = J(v, f (v)) ∈ Gf , thus J f (v) = f (Jv).

25 2.5 Complex Linear Subspaces

Remark 2.53. Let M ⊂ Rk be a submanifold.

Then M is locally the graph of a function f , p = (x, f (x)). Then we have the ”pre-tangent space of M at p” given by ˆ 0 n Tp M := Gf 0(p) = {v, f (p)v | v ∈ R }.

On the other hand M is locally the zero set of a function g. Then

0 Tˆ p M = kerg (p).

Moreover, M is locally the image of a map ψ, ψ(x) = p. Then

0 Tˆ p M = Image(ψ (p)).

Definition 2.54 (Complex submanifold). A 2n-dimensional submanifold of R2k = Ck is called a complex submanifold if for each p ∈ M the pre-tangent k space Tˆ p M is a complex subspace of C .

Theorem 2.55. Let M ⊂ Ck be a subset. Then the following are equivalent:

a) M is an n-dimensional complex submanifold.

b) Locally M is the graph of a holomorphic function from an open subset U ⊂ Cn into Ck−n.

c) Locally M is the zero set of a holomorphic g into Ck−n of full rank.

d) Locally M is the image of a holomorphic ψ from some open subset U ⊂ Cn with full rank.

For all these equivalences, even in the real case, we rely on f 0 to exist. For the Ck setting this is fine, but in the general case, we are missing an ambient space that allows us to think of tangent vectors. Remedy will be provided by the definition of a tangent space for the general setting.

26 3. The Tangent Bundle

3.1 Tangent Spaces

The space C∞(M) is a real vector space. Define tangent vectors X at p by their directional derivative operators.

Definition 3.1 (Tangent vector). A linear map X : C∞(M) → R is called a tangent vector of M at p if there is a smooth curve γ : (−ε, ε) → M with

γ(0) = p and X f = d f (γ(t)) = ( f ◦ γ)0(0). dt t=0

The tangent space of M at p is the set of tangent vectors of M at p and will be denoted by Tp M, i.e.

∞ ∗ Tp M := {X ∈ C (M) | X is tangent vector of M at p}

∂ For a chart (U, ϕ), U 3 p, ϕ = (x1,..., xn) we define ∂x ∈ Tp M by j p

Ç å ∂ d −1 f := f ◦ ϕ (q + tej) =: ∂e fˆ(q). ∂xj dt t=0 j p | {z } =: fˆ

27 3.1 Tangent Spaces

0 Let X ∈ Tp M defined by γ, γˆ = ϕ ◦ γ, γˆ (0) = (a1,..., an). Then

0 0 0 0 X ∂ X f = ( f ◦ γ) (0) = ( fˆ ◦ γˆ) (0) = fˆ (q)γˆ (0) = aj f . ∂xj j p

This shows that Tp M is a linear subspace and ∂ ∂ Tp M = span{ ,... } . ∂x1 p ∂xn p

∂ That the vectors ∂x are also linearly dependent follows from the follow- j p ing lemma.

n ∂ Lemma 3.2. For each a ∈ R there is a function f such that ∂x f = aj for j p j = 1, . . . , n.

Proof. The existence of a locally defined f is clear — we can just use a chart. Now we need to show that we can extend f to a function defined on M globally. This can be done using a so called bumb function, i.e. a non- negative smooth function ρ : M → [0, 1] which is constantly 1 on a small neighborhood of p and zero outside the chart domain.

Theorem 3.3 (Transformation of coordinate frames). If (U, ϕ) and (V, ψ) are charts with p ∈ U ∩ V, ϕ|U ∩ V = Φ ◦ ψ|U ∩ V. Then for every X ∈ TpM,

X ∂ X ∂ X = ai = bi , ∂xi p ∂yi p

where ϕ = (x1,..., xn), ψ = (y1,..., yn), we have Ü ê Ü ê a1 b1 . 0 . . = Φ (ψ(p)) . . an bn

28 The Tangent Bundle

Proof. Let γ : (−ε, ε) → M such that X f = ( f ◦ γ)0(0). Let γ˜ = ϕ ◦ γ and γˆ = ψ ◦ γ, then a = γ˜ 0(0), b = γˆ 0(0). Let Φ : ψ(U ∩ V) → ϕ(U ∩ V) be the coordinate change Φ = ϕ ◦ ψ−1. Then γ˜ = ϕ ◦ γ = Φ ◦ ψ ◦ γ = Φ ◦ γˆ. In particular, a = γ˜ 0(0) = (Φ ◦ γ˜)0(0) = Φ0(ψ(p))γˆ 0(0) = Φ0(ψ(p))b.

˜ ˜ Definition 3.4 (Derivative as map Tp M → Tf (p) M). Let M and M be smooth manifolds, f :M → M˜ smooth, p ∈ M. Then define a linear map ˜ dp f :TpM → T f (p)M

∞ by setting for g ∈ C (M˜ ) and X ∈ TpM

dp f (X)g := X(g ◦ f ).

Remark 3.5. dp f (X) is really a tangent vector in TpM˜ because, if X corre- sponds to a curve γ : (−ε, ε) → M with γ(0) = p then

d d d

dp f (X)g = (g ◦ f ) ◦ γ = g ◦ ( f ◦ γ) = g ◦ γ˜. dt t=0 dt t=0 | {z } dt t=0 =:γ˜ Notation: The tangent vector X ∈ TpM corresponding to a curve γ : (−ε, ε) → M with γ(0) = p is denoted by X =: γ0(0).

Theorem 3.6 (Chain rule). Suppose g :M → M˜ , f : M˜ → Mˆ are smooth maps, then dp( f ◦ g) = dg(p) f ◦ dpg.

29 3.1 Tangent Spaces

Definition 3.7 (Tangent bundle). The set G TM := TpM p∈M

is called the tangent bundle of M. The map π : TM → M, TpM 3 X → p is −1 called the projection map. So TpM = π ({p}).

Most elegant version of the chain rule: If f :M → M˜ is smooth, then d f :TM → TM˜ where

d f (X) = dπ(X) f (X).

So every X ∈ TM knows ”where it comes from”. With this notation,

d( f ◦ g) = d f ◦ dg.

Theorem 3.8. If f :M → M˜ is a diffeomorphism then for each p ∈ M the map ˜ dp f :TpM → T f (p)M

is a vector space isomorphism.

−1 −1 Proof.f is bijective and f is smooth, IdM = f ◦ f . For all p ∈ M,

−1 IdTpM = dp(IdM) = d f (p) f ◦ dp f .

So dp f is invertible.

Theorem 3.9 (Manifold version of the inverse function theorem). Let f :M → M˜ be smooth, p ∈ M with dp f :M → M˜ invertible. Then there are ˜ open neighborhoods U ⊂ M of p and V ⊂ M of f (p) such that f |U : U → V is a diffeomorphism.

Proof. The theorem is a reformulation of the inverse function theorem.

Definition 3.10 (Submersion). A map g : M → M˜ is called a submersion if dpg is injective for all p ∈ M.

30 The Tangent Bundle

Definition 3.11 (Immersion). A map ψ : M → M˜ is called an immersion if dpψ is surjective for all p ∈ M.

Remark 3.12. Surjectivity/injectivity can be verified via linear indepen- dence, for instance by det, of rows/columns of parts of the derivative. As det is smooth, this means that if a map h : M → M˜ is a submer- sion/immersion for some p ∈ M, then it is one in an open neighborhood of p.

Theorem 3.13 (Submersion theorem). Let f : M˜ → Mˆ be a submersion, i.e. ˜ ˜ ˆ for each p ∈ M the derivative dp f :TpM → T f (p)M is surjective. Let q = f (p) be fixed. Then M:= f −1({q}) is an n-dimensional submanifold of M˜ , where n = dim M˜ − dim Mˆ .

Remark 3.14. The sumbersion theorem is a manifold version of the implicit function theorem.

Proof. Take charts and apply 2.27.

Theorem 3.15 (Immersion theorem). Let f :M → M˜ be an immersion, i.e. for every p ∈ M the differential dp f :TpM → T f (p)M is injective. Then for each p ∈ M there is an open set U ⊂ M with p ∈ M such that f (U) is a submanifold of M˜ .

Proof. Take charts and apply 2.27. Is there a global version, i.e. without passing to U ⊂ M? Assuming that f is injective is not enough.

31 3.2 The tangent bundle as a smooth vector bundle

Figure 3.1: Example of an injective immersion that is no submanifold.

Theorem 3.16. Let M be a compact manifold and f : M → M˜ an injective im- mersion. Then f (M) is a submanifold and f : M → f (M) is a diffeomorphism, i.e. f is an embedding.

Remark 3.17. This time, there is no reason to state the definitions for the complex case again. All definitions and theorems extend to the complex case by replacing smooth by holomorphic.

3.2 The tangent bundle as a smooth vector bun- dle

Let M be a smooth n-manifold, p ∈ M. The tangent space at p is an n- dimensional subspace of (C ∞(M))∗ given by 0 ∞ TpM = {X | ∃γ : (−ε, ε) → M, γ(0) = p, X f = ( f ◦ γ) (0), ∀ f ∈ C (M)} The tangent bundle is then the set G TM = TpM p∈M and comes with a projection

π : TM → M, TpM 3 X 7→ p ∈ M. −1 The set π ({p}) = TpM is called the fiber of the tangent bundle at p. Goal: We want to make TM into a 2n-dimensional manifold.

If ϕ = (x1,..., xn) be a chart of M defined on U 3 p. Then we have a basis

∂ ∂ ,..., of TpM. So there are unique y1(X),..., yn(X) ∈ R such that ∂x1 p ∂xn p

X ∂ X = yi(X) . ∂xi p

Let {(Uα, ϕα)}α∈A be a smooth atlas of M. For each α ∈ A we get an open −1 n set Uˆ α := π (Uα) and a function yα : Uˆ α → R which maps a given vector to the coordinates yα = (yα,1,..., yα,n) with respect to the frame defined by ϕα.

32 The Tangent Bundle

Now, we define −1 n n 2n ϕˆα : π (U) → R × R = R by ϕˆα = (ϕα ◦ π, yα).

For any two charts we have a transition map φαβ : ϕα(Uα ∩ Uβ) → ϕβ(Uα ∩ Uβ)

such that ϕβ = φαβ ◦ ϕα|U ∩ U . The chain rule yields: Uα ∩ Uβ α β 0 yβ(X) = φαβ(ϕα(π(X)))yα(X).

−1 Hence we see that ϕˆ β ◦ ϕˆα is a diffeomorphism. Topology on TM: ¶ © OTM := W ⊂ TM | ϕˆα(W ∩ Uˆ α) ∈ OR2n for all α ∈ A . Exercise 3.18. a) This defines a topology on TM.

b) With this topology TM is Hausdorff and 2nd-countable.

c) All ϕˆα are homeomrophisms onto their image. Because coordinate changes are smooth, this turns TM into a smooth 2n- dimensional manifold.

3.3 Vector Bundles

Now that we have learned about the tangent bundle we will see that it is only a special case of a much more general concept, namely vector bun- dles. We will mainly consider complex one-dimensional vector bundles, also called complex line bundles, throughout this course. But first things first, so we give the following definition.

33 3.3 Vector Bundles

Definition 3.19 (Vector bundle). A smooth vector bundle of rank k is a triple (E, M, π) where E and M are manifolds and π : E → M is smooth such that

−1 (i) The fiber Ep := π ({p}) has the structure of a k-dimensional real vector space.

(ii) There is U ⊂ M open, p ∈ U and a diffeomorphism φ : π−1(U) → k U × R such that π1 ◦ φ = π|U where π1 is the projection on the first k component, i.e. for each q ∈ U the map φq : Eq → R is a vector space Ä ä isomorphism defined by q, φq(ψ) = φ(ψ).

Definition 3.20 (Section). A smooth map ψ : M → E is called a section of E if π ◦ ψ = idM.

We denot the set of all smooth sections of E by

Γ(E) := {ψ | ψ is a section of E}

Note that Γ(E) is a real vector space under pointwise addition. In fact Γ(E) is a module over the ring C∞(M): for f ∈ C∞(M), ψ ∈ Γ(E) define f ψ ∈ Γ(E) by

( f ψ)p := f (p)ψp.

Example 3.21.

(i) (Vector fields:) We have seen that the tangent bundle TM of a smooth manifold is a vector bundle of rank dim M. Its smooth sections were called vector fields.

34 The Tangent Bundle

(ii) (The trivial bundle:) The product M × Rk is called the trivial bundle of rank k. Its smooth sections can be identified with Rk-valued functions. More precisely, if k k π2 :M × R → R , then

k ∞ Γ(M × R ) 3 ψ = (p, f (p)) ←→ f := π2 ◦ ψ ∈ C (M).

From now on we will keep this identification in mind and often con- fuse Γ(M × Rk) with C∞(M, Rk).

(iii)T ∗ M := (TM)∗ is called the cotangent bundle.

(iv) Bundles of multilinear forms with all the E1,..., Er, F copies of TM, T∗ M or M × R are called tensor bundles. Sections of such bundles are called tensor fields.

Ways to make new vector bundles out of old ones The general principle is the following: Any linear algebra operation that gives new vector spaces out of given ones can be applied to vector bundles over the same base manifold.

Example 3.22. Let E be a rank k vector bundle over M and F be a rank ` vector bundle over M.

(i) Then E ⊕ F denotes the rank k + ` vector bundle over M the fibers of which are given by (E ⊕ F)p = Ep ⊕ Fp. (ii) Then Hom(E, F) denotes the rank k · ` vector bundle over M with fiber given by Hom(E, F)p := { f :Ep → Fp | f linear}. ∗ ∗ ∗ (iii)E = Hom(E, M × R) with fibers (E )p = (Ep) .

Let E1, . . . .Er, F be vector bundles over M. ∗ ∗ (iv) Then a there is new vector bundle E1 ⊗ · · · ⊗ Er ⊗ F of rank

rankE1 ··· rankEr · rankF

with fiber at p given by

∗ ∗ E1p ⊗ · · · ⊗ Erp ⊗ Fp = {β :E1p × ... × Erp → Fp | β multilinear}.

35 3.3 Vector Bundles

Definition 3.23 (Vector bundle isomorphism). Two vector bundles E → M, E˜ → M are called isomorphic if there is a bundle isomorphism

f :E → E˜ ∈ Γ Hom(E, F),

that means π˜ ◦ f = π (fibers to fibers) and f | :E → E˜ is a vector space Ep p p isomorphism.

Definition 3.24 (trivial vector bundle). A vector bundle E → M of rank k is called trivial if it is isomorphic to the trivial bundle M × Rk.

Remark 3.25. If E → M is a vector bundle of rank k then, by definition, each point p ∈ M has an open neighborhood U such that the restricted bundle −1 E|U := π (U) is trivial, i.e. each bundle is locally trivial.

Definition 3.26 (Frame field). Let E → M be a rank k vector bundle, ϕ1,..., ϕk ∈ Γ(E). Then (ϕ1,..., ϕk) is called a frame field if for each p ∈ M the vectors ϕ1(p),..., ϕk(p) ∈ Ep form a basis.

Proposition 3.27. A vector bundle E is trivial if and only if it has a frame field.

Proof.

”⇒”: Let E be trivial then there ∃F ∈ ΓHom(E, M × Rk) such that

k Fp : Ep → {p} × R

is a vector space isomorphism for each p. Then, for i = 1, . . . , k define −1 ϕi ∈ Γ(E) by ϕp := F ({p} × ei).

”⇐”: Let (ϕ1,..., ϕk) be a frame field of E, then we can define a bundle isomorphism F ∈ ΓHom(E, M × Rk) as the unique map such that

Fp(ϕi(p)) = {p} × ei

for each p ∈ M.

Example 3.28. A rank 1 vector bundle E (a line bundle) is trivial if and only if there exists a nowhere vanishing ϕ ∈ Γ(E).

36 The Tangent Bundle

Example 3.29. (i) Consider E = S1 × R. Then we can define a section ψ ∈ Γ(E) by 1 setting ψp = (p, 1) for all p ∈ S . As R is only one-dimensional, ψ, as it is nowhere vanishing, defines a frame field of E, thus E is trivial.

(ii) The Mobius¨ band shaped rank 1 vector bundle over S1 below is not trivial, as any non-vanishing (i.e. non-zero) section in Γ(E) cannot be smooth.

(iii) There is a nowhere vanishing vector field X ∈ Γ(TM) for M = T2.

Example 3.30. Let M ⊂ R` be a submanifold of dimension n. Then this admits a rank ` − n vector bundle NM (the normal bundle of M). It is given by ⊥ ` ` Np M = (NM) = (Tp M) ⊂ TpR = {p} × R . As a matter of fact, The normal bundle of a Moebius band is not trivial. Example 3.31. The tangent bundle of S2 is not trivial - a fact known as the hairy ball theorem: Every vector field X ∈ Γ(T S2) has zeros.

37 3.4 The Lie-bracket of Vector Fields

3.4 The Lie-bracket of Vector Fields

Let X ∈ Γ(TM), f ∈ C∞(M). Then

X f : M → R, p 7→ Xp f , is smooth. So X can be viewed as a linear map

C∞(M) → C∞(M), f 7→ X f .

Theorem 3.32 (Leibniz’s rule). Let f , g ∈ C∞(M),X ∈ Γ(TM), then

X( f g) = (X f )g + f (Xg).

Definition 3.33 (Lie algebra). A Lie algebra is a vector space g together with a skew bilinear map [., .] : g × g → g which satisfies the Jacobi identity,

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0.

Theorem 3.34 (Lie algebra of endomorphisms). Let V be a vector space. End(V) together with the commutator [., .] : End(V) × End(V) → End(V), [A, B] := AB − BA forms a Lie algebra.

Proof. Certainly the commutaor is a skew bilinear map. Further,

[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = A(BC − CB) − (BC − CB)A + B(CA − AC) − (CA − AC)B + C(AB − BA) − (AB − BA)C, which is zero since each term appears twice but with opposite sign.

Theorem 3.35. For all f , g ∈ C∞(M),X, Y ∈ Γ(M),

[ f X, gY] = f g[X, Y] + f (Xg)Y − g(Y f )X.

Lemma 3.36 (Schwarz lemma). Let ϕ = (x1,..., xn) be a coordinate chart. ∂ ∂ Then [ , ] = 0. ∂xi ∂xj

Exercise 3.37. Prove Schwarz lemma above.

38 The Tangent Bundle

Theorem 3.38. Γ(TM) ⊂ End(C∞(M)) is a Lie subalgebra.

This theorem can alternatively be stated in an equivalent way, namely

Theorem 3.39. For X, Y ∈ Γ(TM) there is a unique vector field [X, Y] ∈ Γ(TM) such that for all f ∈ C∞(M)

[X, F] f = x(Y f ) − Y(X f ).

Proof. The right hand side can be evaluated at p ∈ U, U ⊂ M open, thus X ∂ X ∂ only depends on X|U, Y|U, f |U. So if X = ai and Y = bj , we i ∂xi j ∂xj get

X(Y f ) − Y(X f ) n Ñ n é n Ñ n é X ∂ X ∂ f X ∂ X ∂ f = ai bj − bi aj i=1 ∂xi j=1 ∂xj i=1 ∂xi j=1 ∂xj Ñ é n n Ç å n 2 2 ! X X ∂bj ∂aj ∂ f X ∂ f ∂ f = ai − bi + aibj − i=1 j=1 ∂xi ∂xi ∂xj i,j=1 ∂xi∂xj ∂xj∂xi Ñ n Ñ n Ç åé é X X ∂bj ∂aj ∂ = ai − bi f i=1 j=1 ∂xi ∂xi ∂xj

The last equality relies on the fact that the second sum is zero, as by lemma 3.36 partial derivatives commute. By defining

n Ñ n Ç åé X X ∂bj ∂aj ∂ [X, Y] := ai − bi i=1 j=1 ∂xi ∂xi ∂xj we yield a unique [X, Y] ∈ Γ(TM).

39 4. Almost Complex Structures

4.1 Tensors

Let E1 ..., El, F be vector bundles over the same manifold M. Further let T be the vector bundle over M with fibers ¶ © Tp = µp : (E1)p × ... × (El)p → F | µp multilinear

Consider µ ∈ Γ(T), then one can define the following:

µˆ : Γ(E1) × ... × Γ(El) → Γ(F) by µˆ(ψ1,..., ψl)p = µp(ψ1(p),..., ψl(p)) ∈ Fp To first define µˆ is often is the most convenient way to define µ, but it is not clear yet under which circumstances such a µˆ really comes from an µ ∈ Γ(T), as most of them don’t. A standard way to check if this is the case is provided by the following theorem.

Theorem 4.1. Let E1 ..., El, F be vector bundles over M and let µ˜ : Γ(E1) × ... × Γ(El) → Γ(F) be real linear in all its arguments. Then the following are equivalent:

∞ (i) For all f ∈ C (M), ψ1 ∈ Γ(E1),...,ψl ∈ Γ(El), µ˜ is tensorial in all arguments, i.e.

µ˜( f ψ1,..., ψl) = ... = µ˜(ψ1,..., f ψl) = f µ˜(ψ1,..., ψl)

(ii) There is µ ∈ Γ(T) such that µˆ = µ˜.

Proof. We only check this for l = 1, i.e. T = Hom(E, F).

40 Almost Complex Structures

”⇐” If µ˜(ψ)p = µp(ψp) then, because

µ˜( f ψ)p = µp( f (p)ψp) = f (p)µp(ψp) = ( f µ˜(ψ))p

it certainly holds that tildeµ( f ψ) = f µ˜(ψ).

”⇒” Assume that µ˜( f ψ) = f µ˜(ψ) for all f ∈ C∞(M), ψ ∈ Γ(E). Let p ∈ M then we claim that µp(ψ) = (µ˜(ψ))p only depends on ψp. Choose a frame field ϕ1,..., ϕl ∈ Γ(E|U) for some U ⊂ M open with p ∈ U. Further choose a bump-function f ∈ C∞(M) such that   f (q) = 0 , q ∈/ U  f (q) = 1 , q ∈ V ⊂ U

V open. We want to show that ψp = ψ˜ p implies that (µ˜(ψ))p = (µ˜(ψ˜)p, or in other words, considering a section ψˆ := ψ − ψ˜, that

ψˆ p = 0 ⇒ (µ˜(ψˆ))p = 0

then by the linearity of µ˜ the claim follows. For an easier notation we will only write ψ instead of ψˆ. So let ψ ∈ Γ(E) with ψp = 0, then locally on U we can express ψ via the frame field as ψ|U = a1 ϕ1 + ... + ak ϕk ∞ with a1,..., ak ∈ C (U). The trick is now, using the tensoriality of ψ˜ to compute

f 2(µ˜(ψ)) = µ˜( f 2ψ)

= µ˜ (( f a1)( f ϕ1) + ... + ( f ak)( f ϕk)) = ( f a1)µ˜( f ϕ1) + ... + ( f ak)µ˜( f ϕk)

∞ where we used that ( f aj) ∈ C (M) and ( f ϕj) ∈ Γ(E) for all j = 1, . . . , k, thus are defined on the whole of M as the bump function assures that they vanish outside of U. Evaluating at p yields, as a1(p) = ... = ak(p) = 0 because ψp = 0,

2 f (p) (µ˜(ψ))p = (µ˜(ψ))p = 0

where we also used that by construction f (p) = 1. By remembering that we denote ψˆ by ψ the claim follows.

Remark 4.2. In the following we keep this identification be tensors and tensorial maps in mind and just speak of tensors.

41 4.2 The Newlander-Nirenberg Theorem

4.2 The Newlander-Nirenberg Theorem

Definition 4.3 (Almost complex structure). An almost complex structure on a manifold M is a section J ∈ ΓEnd(TM) with J2 = −Id.

Definition 4.4 (Almost complex manifold). An almost complex manifold is a manifold together with an almost complex structure.

Remark 4.5.

(i) If M is an almost complex manifold, then all tangent spaces Tp M are complex vector spaces. This means dim M = 2n for some n ∈ N. (ii) All complex manifolds are almost complex. We only need to define for X ∈ Tp M, p ∈ Uα for a chart (Uα, ϕα) in the atlas

−1 JX = dϕ (JCdϕX) This is well-defined because coordinate changes are biholomorphic. A more detailed discussion can be found in the Appendix. (iii) Not every almost complex manifold is a complex manifold. As a special case for C2 this is true, but not in general. We will soon learn about a criterion to check this.

Definition 4.6 (Nijenhuis-tensor). Let M be a manifold, A ∈ ΓEnd(TM). Then the Nijenhuis tensor NA is defined on vector fields X, Y ∈ Γ(TM) as

2 NA(X, Y) := −A [X, Y] − [AX, AY] + A([AX, Y] + [X, AY]).

We have to check that NA really is tensorial: First NA is skew in X and Y, thus it is enough to check that it is tensorial in the first slot: 2 NA( f X, Y) = f NA(X, Y) + A ((Y f )X) + ((AY) f )AX − A((Y f )AX + A((Y) f AX + ((AY) f )X)

= f NA(X, Y). where, for the last equality, we used that A is linar. n Example 4.7. Let M = C and JX = iX then NJ = 0. This can be seen as follows. As NJ is tensorial, we can without loss of generality choose vector fields X, Y that are constant. Then also JX, JY are constant and all Lie-brackets vanish.

42 Almost Complex Structures

Theorem 4.8 (Newlander–Nirenberg). An almost complex manifold (M, J) is complex if and only if its Nijenhuis tensor vanishes, NJ = 0.

Proof. One direction follows directly from the previous example. The other direction (”⇐”) is hard and far beyond the scope of this course.

4.3 Connections on Vector Bundles

If we consider a rank k vector bundle E → M, then a section ψ ∈ Γ(E) is k ∼ k similar to a function ψ : M → R , only that ψ(p) ∈ Ep = R . Now it is only natural that we want to think about how a section changes in E when we vary the point p ∈ M.

The problem that arises with this is that two tangent vectors ψp, ψq ∈ E with different basepoints p and q do, in the first place, have nothing in common as they lie in distinguished vector spaces Ep and Eq.

We will introduce a suitable substitute that will provide remedy in the case that we have a path in M connecting p and q.

Definition 4.9 (Connection). A connection on a vector bundle E → M is a bilinear map ∇ : Γ(TM) × Γ(E) → Γ(E) such that for all f ∈ C ∞(M),X ∈ Γ(TM), ψ ∈ Γ(E),

(i) ∇ f Xψ = f ∇Xψ

(ii) ∇X f ψ = (X f )ψ + f ∇Xψ.

The proof of the following theorem will be postponed until we have estab- lished the existence of a so called partition of unity.

43 4.3 Connections on Vector Bundles

Theorem 4.10. On every vector bundle E there is a connection ∇.

Definition 4.11 (Parallel section). Let E → M be a vector bundle with connection ∇. Then ψ ∈ Γ(E) is called parallel if, for all X ∈ TM, ∇Xψ = 0

Example 4.12. As sections are similar to functions, the main example will be E = M × Rk. Then for a section ψ ∈ Γ(E),

Ä ä k ψp = p, ψˆ(p) for ψˆ : M → R .

Then naturally we define Ä ˆä Ä ˆ ä (∇Xψ)p := p, Xpψ = p, dpψ(X) . This particular connection is called the trivial connection on the trivial bundle and thus is often also denoted by d, so that

∇Xψ = dXψ.

We already know, that there is a connection on every vector bundle, so quickly the question of how many there are arises. We will later see, that there is in fact a unique connection that has particulary nice properties, but the following theorem provides that in general a connection is not unique.

Definition 4.13 (Difference tensor). For two connections ∇, ∇˜ on a vector bundle E → M then the map

A : Γ(TM) × Γ(E) → Γ(E), AXψ = ∇˜ Xψ − ∇Xψ

is called the difference tensor of ∇ and ∇˜ .

To make sure that the difference tensor is well defined, we need to check that it is tensorial in both arguments, then the claim follows by theorem 4.1.It is by the properties of a connection

A f Xψ = ∇˜ f X − ∇ f X = f ∇˜ Xψ − f ∇Xψ = f AXψ

AX ( f ψ) = ∇˜ ( f ψ) − ∇X( f ψ) = (X f )ψ + f ∇˜ Xψ − (X f )ψ − f ∇Xψ = f AXψ thus A is well defined.

44 Almost Complex Structures

Such an A is more or less the same thing (i.e. isomorphic) as a section, meaning A ∈ Γ Hom (TM, End(E)) . Thus given any connection ∇ on E, all other connections on E are of the form ∇˜ = ∇ + A. In particular: On the the trivial bundle M × Rk all connections are of the form ∇ = d + A. In many applications vector bundles E have additional structures and usu- ally connections on E are required to be compatible with these. What this is supposed to mean will be clarified in the following.

Definition 4.14 (Complex connection). Let E be a vector bundle with com- plex structure J ∈ End(E) on E, then (E, J) is a complex vector bundle. A connection ∇ on E is called complex connection if it is compatible with J, i.e. for all X ∈ Γ(TM), ψ ∈ Γ(E)

∇X(Jψ) = J∇Xψ.

A complex connection ∇ is thus complex linear in ψ.

Let E → M be a rank k vector bundle and Sym(E) be the bundle whose fiber at p ∈ M consists of all symmetric bilinear forms Ep × Ep → R. We see that k(k+1) rank(Sym(E)) = 2 .

Definition 4.15 (Metric). A metric on E is a section h., .i of Sym(E) such that h., .ip is a Euclidean inner product for all p ∈ M, i.e. h., .ip is positive definite for all p ∈ M.

Definition 4.16 (Euclidean vector bundle). A vector bundle together with a metric (E, h., .i) is called Euclidean vector bundle.

Definition 4.17 (Riemannian manifold). A Riemannian manifold is a man- ifold M together with a Riemannian metric, i.e. a metric h., .i on TM.

45 4.3 Connections on Vector Bundles

Definition 4.18 (Metric connection). Let (E, h., .i) be a Euclidean vector bundle over M. Then a connection ∇ is called metric if for all ψ, ϕ ∈ Γ(E) and X ∈ Γ(TM) we have

Xhψ, ϕi = h∇Xψ, ϕi + hψ, ∇X ϕi.

Putting these two notions together we get a sensible definition of the com- patibility of complex structures and metrics on vector bundles.

Definition 4.19 (Compatibility of J and h., .i). A euclidean inner product h., .i and a complex structure J on a vector bundle E are said to be compatible if for all u, v ∈ V hJu, vi = − hu, JVi or equivalently if J∗ = −J, i.e. J is skew-adjoint.

Remark 4.20. If a metric h., .i is compatible with the complex structure J on a vector bundle E, then J is automatically orhtogonal, as it holds that ¨ ∂ hJu, Jvi = − u, J2v = hu, vi .

Definition 4.21 (Complex scalar product). We define the complex scalar product by

h., .iC : V × V → C, hu, viC = hu, vi + i hJu, vi .

We see that by definition the following two equalities hold:

hJu, viC = hJu, vi − i hu, vi = −i hu, viC

hu, JviC = hu, Jvi + i hJu, Jvi = hu, Jvi + i hu, vi = i hu, viC

So h., .iC is anti-linear in the left argument and linear in the right argument, thus hermitian. If conversely h., .iC is hermitian, then h., .i := Re h., .iC is compatible with J.

Definition 4.22 (unitary vector bundle). A unitary vector bundle is a com- plex vector bundle together with a compatible euclidean metric h., .i.

46 Almost Complex Structures

Definition 4.23 (Compatible ∇ for unitary bundles). A connection ∇ on a unitary vector bundle is called compatible if it is complex and metric.

Let us denote the set of sections in endomorphisms on E that commute with a given complex structure J on E by

Γ End+(E). The set of sections in endomorphisms on E that anti-commute with a given complex structure J on E is denoted by

Γ End−(E).

Proposition 4.24. If ∇ is a connection on a vector bundle E that is compatible with J, then another connection ∇˜ = ∇ + A on E is compatible with J if and only if AX J = JAX this means AX ∈ Γ End+(E).

Proof. We have

J∇˜ ψ = ∇˜ X(Jψ) ⇔ J∇Xψ + JAXψ = ∇X(Jψ) + AX Jψ ⇔ J∇Xψ + JAXψ = J∇X(ψ) + AX Jψ ⇔ JAXψ = AX Jψ

Proposition 4.25. If ∇ is a connection on a vector bundle E that is compatible with h., .i, then another connection ∇˜ = ∇ + A on E is also compatible with h., .i.

Proof. Again we use the definition and derive that ¨ ∂ ¨ ∂ X hψ, φi = ∇˜ Xψ, φ + ψ, ∇˜ Xφ

⇔ h∇Xψ, φi + hψ, ∇Xφi = h∇Xψ + AXψ, φi + hψ, ∇Xφ + AXφi ⇔ 0 = hAXψ, φi + hψ, AXφi ∗ ⇔ AX = −AX which is a true statement. When is a connection ∇ on TM compatible with the status of TM as the tangent bundle?

47 4.3 Connections on Vector Bundles

Definition 4.26 (affine connection). A connection ∇ on the tangent bundle TM is called an affine connection.

Definition 4.27 (Torsion tensor). Let ∇ be an affine connection, then the map

T : Γ(TM) × Γ(TM) → Γ(TM), (X, Y) 7→ ∇XY − ∇YX − [X, Y]

is called the torsion tensor of ∇.

To assure that T is well defined, we need to check the tensoriality of it. As T is obviously skew it suffices to check that for the left argument. For f ∈ C∞(M) and X, Y ∈ Γ(TM) we have

T( f X, Y) = f ∇XY − (Y f )X − f ∇XY + (Y f )X = f T(X, Y) thus T really is a tensor. Example 4.28. Take a look at our formula for the Lie-brackets expressed in local coordinates. We see that the trivial connection ∇ on TRn has T = 0.

Definition 4.29 (Torsion free). A connection ∇ in TM is called torsion-free if the torsion tensor vanishes, i.e. for all X, Y ∈ Γ(TM)

∇XY − ∇YX = [X, Y].

So we have now introduced a lot of structures on a vector bundle E, but not every bundle admits all of these, for example there exists no complex structure J on TS4. But we already learned, that every vector bundle has a Riemannian metric. Further, we say a complex structure on E is compatible with the metric h., .i if

hu, vi = hJu, Jvi ⇔ J∗ = −J ⇔ J∗ J = I

So is it true, that if we have a vector bundle with both, a complex struc- ture J and a metric h., .i, we can always have them in a way that they are compatible?

Theorem 4.30. Every vector bundle E with a complex structure J has a com- patible metric h., .i.

48 Almost Complex Structures

∼ Proof. Choose h., .i to be any metric on E and define

1 ∼ ∼ hψ, φi := 2 (hψ, φi + hJψ, Jφi ) then 1 ∼ ∼ hJψ, φi = 2 (hJψ, φi − hJψ, Jφi ) = − hψ, Jφi where we used that J∗ = −J. Although we did not actually prove it, we will use theorem 4.8 to give a proof of the following theorem. In one part of the proof we will also make use of the partition of unity, of which we will learn in the next chapter.

Theorem 4.31. An almost complex manifold is complex if and only if there is a torsion-free connection ∇ on the tangent bundle TM that is compatible with J, i.e. ∇J = 0.

Proof.

”⇒”: It is

NJ(X, Y) = [X, Y] − [JX, JY] + J ([JX, Y] + [X, JY]) = ∇XY − ∇YX − J∇JXY + J∇JYX Ä ä + J ∇JXY − J∇Y(X) + J∇XY − ∇JYX = 0

”⇐”: Let (M, J) be complex, then using charts (Uα, ϕα), we locally find compatible connections ∇α. Using a partition of unity, we ”glue” them together to obtain the desired connection.

Theorem 4.32 (Fundamental theorem of Riemannian geometry). On a Riemannian manifold there is a unique affine connection ∇ which is both metric and torsion-free. ∇ is called the Levi-Civita connection.

Proof. Uniqueness: Let ∇ be metric and torsion-free, X, Y, Z ∈ Γ(TM). Then

XhY, Zi + YhZ, Xi − ZhX, Yi

= h∇XY, Zi + hY, ∇XZi + h∇YZ, Xi + hZ, ∇YXi − h∇ZX, Yi − hX, ∇ZYi = h∇XY + ∇YX, Zi + hY, ∇XZ − ∇ZXi + h∇YZ − ∇ZY, Xi = h2∇XY − [X, Y], Zi + hY, [X, Z]i + h[Y, Z], Xi.

49 4.3 Connections on Vector Bundles

Hence we obtain the so called Koszul formula:

1Ä h∇ Y, Zi = XhY, Zi + YhZ, Xi X 2 ä − ZhX, Yi + h[X, Y], Zi − hY, [X, Z]i − h[Y, Z], Xi .

So ∇ is unique. Conversely define ∇XY by the Koszul formula (for this to make sense we need to check tensoriality). Then check that this defines a metric torsion-free connection.

Definition 4.33 (Almost hermitian manifold). An almost complex mani- fold (M, J) together with a metric on TM that is compatible with the almost complex structure J on TM is called an almost hermitian manifold.

Remark 4.34. Almost hermitian manifold are sometimes also called unitary manifolds.

Definition 4.35 (Kahler–manifold)¨ . An almost hermitian manifold for which the Levi-Civita connection ∇ of h., .i is compatible with the almost complex structure J is called a K¨ahler-manifold.

Example 4.36.

(i) On CPn (which is a complex manifold) there is the Fubini-Study-metric h., .i which turns CPn into a Kahler-manifold.¨

(ii) Let M˜ be a complex manifold and M˜ ⊂ M a complex submanifold, then M has an induced J which is also complex.

(iii) Let M˜ be a Riemannian manifold and M˜ ⊂ M a submanifold, then M has an induced metric h., .i.

Exercise 4.37. Show that a complex submanifold of a Kahler-manifold¨ is Kahler¨

Theorem 4.38 (A Version of Kodaira embedding theorem). Every compact K¨ahlermanifold admits a complex embedding (not necessarily isometric) in some CPn.

Proof. The proof is way to difficult to do it in the scope of this lecture, so we will skip it. It may be seen in a lecture about algebraic geometry, as the result is one of the highlights of that lecture.

50 Almost Complex Structures

Consider n CP \ {0} ∼ 1 1 M := z 7→ 2z = S × S . M is called a Hopf -manifold and is a complex manifold.

Theorem 4.39. The Hopf-manifold (M, J) mentioned above as a compex mani- fold has no K¨ahler-metricand therefore cannot be realized in any CPn.

Proof. This proof is not too hard. It may be done in the end of this course, when we established more theory to make use of.

4.4 Partition of unity

Theorem 4.40 (partition of unity). Let M be a manifold, A ⊂ M compact and ∞ (Uα)α∈I an open cover of A. Then there are $1,..., $m ∈ C (M) such that for each i ∈ 1, . . . , m there is αi ∈ I such that supp $i ⊂ Uαi is compact. Moreover m X $i(p) ≥ 0 for all p ∈ M and $i(p) = 1 for all p ∈ A. i=0

Proof. We already know that there is a function g ∈ C∞(Rn) such that g(p) ≥ 0 for all p ∈ Rn and g(p) > 0 if p ∈ D := {x ∈ Rn||x| < 1}. Define

n n Dp := {x ∈ R ||x − p| < 1}, D˜ := {x ∈ R ||x| < 2},

n D˜ p := {x ∈ R ||x − p| < 2}.

For each p ∈ A there is a chart (Up, ϕp) such that ϕp : Up → D˜ p is a diffeomorphism and Up ⊂ Uα for some α ∈ I. If we define −1 Vp := ϕp (Dp) then (Vp)p∈A is an open cover of A. This implies that there are p1,..., pm ∈ A such that A ⊂ Vp ∪ · · · ∪ Vp . 1 m  0 if p ∈/ Vpi Now define for i ∈ {1, . . . , m} $˜i : M → R by $˜i = g ◦ ϕpi if p ∈ Vpi This means ($˜1 + ··· + $˜m)(p) ≥ 0 always holds, and in particular ($˜1 + ··· + $˜m)(p) > 0 for p ∈ A. If now A = M, then we define

$˜i $i := ($˜1 + ··· + $˜m) and we are done. Otherwise, since A is compact, $˜1 + ··· + $˜m attains its minimum in A, i.e. there is an e > 0 such that ($˜1 + ··· + $˜m)(p) ≥ e for all p ∈ A. Consturct

51 4.4 Partition of unity h ∈ C∞(M) such that h(x) > 0 for all x ∈ R and h(x) = x for x ≥ e. Now for i ∈ {1, . . . , m} define

$˜i $i := h($˜1 + ··· + $˜m) then clearly ($˜1 + ··· + $˜m)|A = 1.

Theorem 4.41 (partition of unity - general version). Let M be a manifold and (Uα)α∈I an open cover of M, i.e. ∪α∈IUα = M. Then there is a family ∞ ($β)β∈J with $β ∈ C (M) such that

1. for each β ∈ J there is α ∈ I such that supp $β ⊂ Uα is compact.

2. $i(p) ≥ 0 for all p ∈ M. X 3. $β(p) 6= 0 only for finitely many β ∈ J and and $β(p) = 1 for all β∈J p ∈ M.

Proof. This theorem will remain without proof, as we only cite it to em- phasize the existence of a more general version, but do not actually use it much.

Nonetheless we will shortly give some applications of the general partition of unity.

Theorem 4.42. Every manifold has a Riemannian metric.

Proof. We have coordinate charts (Uα, ϕα), i.e. an open cover (Uα)α∈I of ∗ M and a Riemannian metric gα on Uα, where gα = ϕαgRn is the pullback 1 metric . Now we choose a partition of unity ($β)β∈J subordinate to the cover (Uα)α∈I. Define X g := $βgα(β) β∈J which is, as a linear combination of positive definite symmetric bilinear forms with positive coefficients, also positive definite. Then g is a Rieman- nian metric on M.

Remark 4.43. It is not in general true that every manifold has a Pseudo- Riemannian metric, i.e. there is no Lorenz-metric on S2.

1The concept of pullbacks is introduced in differential geometry. We add a short intro- duction on these in the appendix.

52 Almost Complex Structures

Theorem 4.44. Every vector bundle E over M has a connection.

1 m Remark 4.45. If ∇ ,..., ∇ are connections on any bundle E and λ1,..., λm ∈ ∞ C (M) with λ1 + ··· + λm = 1, then ∇ defined by

1 m ∇Xψ := λ1∇Xψ + ··· + λm∇Xψ for X ∈ Γ(TM), ψ ∈ Γ(E), certainly is a connection.

k Proof. Locally (on Uα) E looks like Uα × R , therefore, by pulling pack the α trivial connection, we have a connection ∇ on E|Uα . Choose a partition of unity ($β)β∈J subordinate to the cover (Uα)α∈I, then we can say

X α(β) ∇ = $β∇ β∈J is certainly a connection, as only finitely many of the $ 6= 0.

4.5 Conformal Equivalence

Let (M, h., .i) be a Riemannian manifold and γ : [0, 1] → M a smooth curve on M, then the length of γ is given by

Z 1 0 L(γ) := γ . 0

If we consider another smooth curve η : [0, 1] → M on M such that γ(s) = η(t) for some s, t ∈ [0, 1] then

hγ0(s), η0(t)i cos α := |γ0(s)||η0(t)| where α is the angle between γ and η at p := γ(s) = η(t). This means that a Riemannian metric allows us to measure angles on a manifold M.

53 4.5 Conformal Equivalence

Definition 4.46 (Conformal equivalence). Two Riemannian metrics h., .i ∼ and h., .i on a manifold M are called conformally equivalent if there is func- tion u ∈ C∞(M) such that

∼ h., .i = e2u h., .i .

We see that two Riemannian metrics are conformally equivalent if they only differ by a positive scaling factor for all p ∈ M. Remark 4.47. Conformal equivalence defines an equivalence relation on the set of Riemannian metrics on M.

Definition 4.48 (Conformal structure). A conformal structure on a mani- fold M is an equivalence class of conformally equivalent Riemannian metrics on M. A question that immediately arises is that if we know how to make con- formal changes of a metric, do we also know how the corresponding Levi- Civita connection on M changes? Luckily we do, as the following theorem assures.

∼ Theorem 4.49. Let h., .i = e2u h., .i then the corresponding Levi-Civita con- nections ∇, ∇˜ are related as

∇˜ XY = ∇XY + hX, Gi Y + hY, Gi X − hX, Yi G

where G ∈ Γ(TM) is defined as G = grad u with respect to h., .i, i.e. hG, Zi = du(Z) = Zu.

Proof. It is easy to see that ∇˜ really defines a connection, so we have to ∼ proof that it is torison free and metric with respect to h., .i . We have, for X, Y, Z ∈ Γ(TM)

∇˜ XY − ∇˜ YX = ∇XY + hX, Gi Y + hY, Gi X − hX, Yi G − ∇YX − hX, Gi Y − hY, Gi X + hX, Yi G = ∇XY − ∇YX = [X, Y] thus ∇˜ is torsion free. Further, on the one hand ∼ Ä ä X hY, Zi = X e2u hY, Zi 2u = e (2(Xu) hY, Zi + h∇XY, Zi + hY, ∇XZi) 2u = e (2 hG, Xi hY, Zi + h∇XY, Zi + hY, ∇XZi)

54 Almost Complex Structures and on the other hand

−2u Ĩ ∂ ¨ ∂ä e ∇˜ XY, Z + Y, ∇˜ XZ = h∇XY, Zi + hY, ∇XZi + hX, Gi hY, Zi + hY, Gi hX, Zi − hX, Yi hG, Zi + hX, Gi hZ, Yi + hZ, Gi hX, Yi − hX, Zi hG, Yi

= h∇XY, Zi + hY, ∇XZi + 2 hG, Xi hY, Zi

55 5. Integration on manifolds

5.1 Volume Forms

In the following we will naturally handle differential forms and pullbacks without having them especially defined. A short introduction on both top- ics can be found in the appendix.

In R3 we know that for three vectors X, Y, Z ∈ R3 we have that the volume of the parallelepiped spanned by them has volume |det(X, Y, Z)|. We want to introduce n-dimensional versions of this. To do so, we need the following.

Definition 5.1 (Alternating k-forms). Let V be a vector space, then, for k ≥ 1 we define

Λk(V) := {ω : V × ... × V → R | ω is k-linear and alternating}

i.e. for v1,..., vl ∈ V with i 6= j

ω(v1,..., vi,..., vj,..., vk) = −ω(v1,..., vj,..., vi,..., vk).

Consequently, for k = 0 we set Λ0(V) = R.

56 Integration on manifolds

There are special cases that we will frequently use. The first one is

Λ1(V) = V∗

Further, if V is of dimension n ∈ N then

dim Λn(V) = 1.

This is because V =∼ Rn as a vector space and on Rn we have

n detR ∈ Λn(Rn)

n n with detR 6= 0, where detR is the usual determinant function on Rn. From n linear algebra we know that detR can be defined as the unique element of n n Rn n n Λ (R ) with det (e1,..., en) = 1. In particular, if ω ∈ Λ (R ) with ω 6= 0 then n ω detR = . ω(e1,..., en)

Definition 5.2 (Determinant form). For an n-dimensional vector space V, a non-zero element det ∈ Λk(V) is called a determinant form on V.

Definition 5.3 (Volume form). A volume form on an n-dimensional mani- n fold M is a section det ∈ ΓΛ (M) such that detp 6= 0 for all p ∈ M.

Note that in the upper definition we use the notation

Λn(M) := Λn(TM) for the vector bundle whose fibers at p ∈ M are

n n Λ (M)p := Λ (Tp M).

We now want to introduce the notion of an orientation. For an n-dimensional manifold M this is often defined by the existence of a nowhere vanishing n-form on M, but to emphasize that such a particular form is more of a consequence of an orientation we will define it in a way that allows us to have it as a theorem.

57 5.1 Volume Forms

Definition 5.4 (Orientation).

(i) Two charts (U, ϕ), (V, ψ) of a manifold M are called consistently ori- ented if the coordinate change φ : ϕ(U ∩ V) → ψ(U ∩ V) is orientation preserving, i.e. det φ0 > 0.

(ii) An orientation on a manifold M is an atlas of consistently oriented charts which is maximal in this sense.

(iii) A manifold M is called orientable if M has a consistently oriented atlas.

Theorem 5.5. An n-dimensional manifold M is orientable if and only if M has a volume form ω ∈ Ωn(M).

Proof.

”⇒”: Let (Uα, ϕα)α∈I be a consistently oriented atlas. Choose a partition of unity ($β)β∈J subordniate to the open cover (Uα)α of M. Let ϕ := ϕα = (x1,..., xn), ϕ˜ := ϕα˜ = (y1,..., yn) then on Uα ∩ Uα˜ it is

ϕ˜ = φ ◦ ϕ

Then Ü ê dy1(z) . dϕ˜ : Tp M → Tϕ˜(p), z 7→ . . dyn(z) Thus dϕ˜ = dφ ◦ dϕ

58 Integration on manifolds

which is Ü ê Ü ê dy1(z) dx1(z) . 0 . . = φ (ϕ(p)) . dyn(z) dxn(z)

It is left as an exercise to show that this implies dy1 ∧ ... ∧ dyn = Ä 0 ä det φ ◦ ϕ dx1 ∧ ... ∧ dxn. In short, this is true for every chart and we yield ∗ Rn dx1 ∧ ... ∧ dxn = ϕαdet ∗ Rn dy1 ∧ ... ∧ dyn = ϕα˜ det . Defining X ∗ Rn ω := $β ϕα(β)det . β∈J will then do the trick. Checking that this really defines a volume form is also left as an exercise. ”⇐”: Let ω ∈ Ωn(M) be the volume form of M For each chart (U, ϕ) of M decompose U = U1 ∪ ... ∪ Um with Uj open and connected. As Ωn(M) is 1-dimensional it is n ϕ∗detR = λω ∞ with λ ∈ C (U). Now modify ϕ by past-composing ϕ|Uj by a reflec- n tion of R on each Uj for which λ|Uj < 0. By doing this we get ϕ˜ with λ˜ > 0. Doing so for every chart in the atlas yields an atlas (Uα˜ , ϕα˜ )α∈I of mutually compatible orientation.

5.2 Integration of Forms

Definition 5.6 (support of a section). If E is a vector bundle over M and ψ ∈ Γ(E), then we define the support of ψ as

supp ψ := {p ∈ M|ψp 6= 0}

Remark 5.7. The support is well defined as {p ∈ M|ψp 6= 0}, as the com- plement of the closed set {p ∈ M|ψp = 0}, is open. We define Γ0(M,E) := {ψ ∈ Γ(E) | supp(ψ) compact } and n k Ω0 (M,E) := Γ0Λ (M,E).

59 5.2 Integration of Forms

n n n n Definition 5.8 (Integral of ω ∈ Ω0 (R )). If ω ∈ Ω0 (R ), then we define Z Z ω := f . Rn Rn

n n Theorem 5.9. Given ω, ω˜ ∈ Ω0 (R ) and a diffeomorphism ϕ : supp ω˜ → supp ω such that ω = ϕ∗ω˜ and det ϕ0 > 0, then Z Z ω = ω˜ Rn Rn

Proof. Let ω˜ = f˜ dx1 ∧ · · · ∧ dxn and ω = f dx1 ∧ · · · ∧ dxn with f = ω(X1,..., Xn) where Xk(p) = (p, ek), k = 1, . . . , n. Then we see that ∗ [(ϕ ω˜ )(X1,..., Xn)]p = [ω˜ (dϕ(X1),..., dϕ(Xn))]p ˜ Ä 0 0 ä = f ◦ ϕ(p) det ϕp(e1) ··· ϕp(en) ˜ Ä 0 ä = f ◦ ϕ(p) det ϕp but also ∗ [(ϕ ω˜ )(X1,..., Xn)]p = [ω(X1,..., Xn)]p = f (p) for all p ∈ Rn, i.e. Ä ä Ä ä f = f˜ ◦ ϕ det ϕ0 = f˜ ◦ ϕ| det ϕ0 | where we used det(ϕ0) > 0 in the last equality. We finally yield, using the transormation of coordinates theorem Z Z Z Z Z ω˜ = f˜ = f˜ ◦ ϕ|det(ϕ0)| = f = ω Rn Rn Rn Rn Rn

n Definition 5.10 (Integral of ω ∈ Ω0 (M) with supp ω ⊂ U). If M is oriented such that there is a chart (U, ϕ) that is consistent with the orentation, n ω ∈ Ω0 (M) and supp ω ⊂ U, then we define Z Z ω := (ϕ−1)∗ω M Rn

Remark 5.11. The Integral is well defined, because assume (U˜ , ϕ˜) is another chart such that γ˜ = ϕ˜−1 and the change of coordinates ψ, then γ˜ ∗ω = (γ ◦ ψ)∗ω = ψ∗γ∗ω. By theorem 5.9 and since ψ is orientation preserving, we see that Z Z γ∗ω = γ˜ ∗ω Rn Rn which makes the definition independent of the choice of (U, ϕ).

60 Integration on manifolds

n n Definition 5.12 (Integral over ω ∈ Ω0 (M)). Let ω ∈ Ω0 (M), (Uα, ϕα)α∈I an open cover of M consisting of coodrinate charts and $1,..., $m a partition of unity subordinate to (Uα)α∈I then we define

m Z X Z ω := $iω M M i=1

To ensure that our newly gained integral is well defined, we proof the following theorem.

Z Theorem 5.13. ω thus defined is independet of the choices. M

Proof. Without loss of generality we can assume (Uα)α∈I contains all coor- dinate neighbourhoods, i.e. the independence of (Uα)α∈I is no problem at all. For the independence of the partition of unity let $1,..., $m, $˜1,..., $˜m ∈ C∞(M) be two partitions of unity, then

m m Ñ m é m X Z X Z X X Z $iω = $˜j $iω = $˜j$iω M M M i=1 i=1 j=1 i,j=1

m Ñ m é m X Z X X Z = $i $˜jω = $˜jω M M j=1 i=1 j=1

5.3 Stokes Theorem

61 5.3 Stokes Theorem

Definition 5.14 (Topological manifold with boundary). A topological manifold with boundary is a second-countable, Hausdorff space which is lo- cally homeomorphic to an open subset of

n n H = {(x1,..., xn) ∈ R | x1 ≤ 0} .

Definition 5.15 (General smoothness). Let A ⊆ Rn be any subset, then a map f : A → Rk is called smooth if there is an open subset U ⊆ Rn with k A ⊆ U and a smooth map f˜: U → R with f˜|A = f .

Thus we know how to define a smooth manifold with a boundary.

Definition 5.16 (Smooth manifold with boundary). A smooth manifold with a boundary is a topological manifold with a boundary together with a n maximal smooth atlas (Uα, ϕα)α∈I,Uα ⊂ M open, ϕα : Uα → ϕα(Uα) ⊂ H homeomorphic onto its image where coordinate changes are smooth.

Theorem 5.17 (Stoke’s theorem). Let M be an oriented manifold with bound- n−1 ary and ω ∈ Ω0 (M), then Z Z dω = ω M ∂M

Proof. As we habe the partition of unity as one of our tools, we may assume n n that without loss of generality M = H = {x ∈ R |x1 ≤ 0}. For further simplification we also only consider the case n = 2 as the calculations work in the same manner for higher dimensions. We can write ω = adx + bdy

62 Integration on manifolds then Ç ∂b ∂a å dω = − dx ∧ dy ∂x ∂y By the definition of the integration of forms in Rn and Fubini’s theorem we now see that Z Z ∞ Z 0 ∂b Z 0 Z ∞ ∂a dω = dxdy − dydx H2 −∞ −∞ ∂x −∞ −∞ ∂y Z ∞ = b(0, y) dy − 0 −∞ Z = b dy ∂H2 Z = ω ∂H2 where we used the compact support of ω and the fact that the y-axis suits the induced orientation of ∂H2.

5.4 Fundamental theorem for flat vector bundles

Let E → M be a vector bundle with connection ∇. Then

E trivial ⇐⇒ ∃ frame field Φ = (ϕ1,..., ϕk) with ∇ϕi = 0, i = 1, . . . , k and E flat ⇐⇒ E locally trivial.

Theorem 5.18 (Fundamental theorem for flat vector bundles). A vector bundle (E, ∇) is flat ⇐⇒ R∇ = 0.

Proof.

”⇒”: Let (ϕ1,..., ϕk) be a local parallel frame field. Then we have for i = 1, . . . , k ∇ R (X, Y)ϕi = ∇X∇Y ϕi − ∇Y∇X ϕi − ∇[X,Y] ϕi = 0. Since R∇ is tensorial checking R∇ψ = 0 for the elements of a basis is enough.

”⇐”: Assume that R∇ = 0. Locally we find for each p ∈ M a neighborhood k U diffeomorphic to (−ε, ε) and a frame field Φ = (ϕ1,..., ϕk) on U. Define ω ∈ Ω1(U, Rk×k) by

k X ∇ϕi = ϕjωji. j=1

63 5.4 Fundamental theorem for flat vector bundles

With ∇Φ = (∇ϕ1,..., ∇ϕk), we write

∇Φ = Φω.

Similarly, for a map F : U → Gl(k, R) define a new frame field:

Φ˜ = ΦF−1

All frame fields on U come from such F. We want to choose F in such a way that ∇Φ˜ = 0. So,

0 =! ∇Φ˜ = ∇(ΦF−1) = (∇Φ)F−1 + Φd(F−1) = (∇Φ)F−1 − ΦF−1dF F−1 = Φ(ω − F−1dF)F−1,

where we used that d(F−1) = −F−1dF F−1. Thus we have to solve

dF = Fω.

The Maurer-Cartan Lemma (below) states that such F : U → Gl(k, R) exists if and only if the integrability condition (or Maurer-Cartan equa- tion) dω + ω ∧ ω = 0 is satisfied. We need to check that in our case the integrability condi- tion holds: We have

∇ 0 = R (X, Y)Φ = ∇X∇YΦ − ∇Y∇XΦ − ∇[X,Y]Φ

= ∇X(Φω(Y)) − ∇Y(Φω(X)) − Φω([X, Y]) = Φω(X)ω(Y) + Φ(Xω(Y)) − Φω(Y)ω(X) − Φ(Yω(X)) − Φω([X, Y]) = Φ(dω + ω ∧ ω)(X, Y).

Thus dω + ω ∧ ω = 0.

Lemma 5.19 (Maurer-Cartan). Let

n 1 k×k U := (−ε, ε) , ω ∈ Ω (U, R ), F0 ∈ Gl(k, R), then

∃F : U → Gl(k, R) : dF = Fω, F(0, . . . , 0) = F0 ⇐⇒ dω + ω ∧ ω = 0

64 Integration on manifolds

Remark 5.20. Note that dω + ω ∧ ω automatically vanishes on 1-dimensional domains.

Proof. A general proof for arbitrary dimensions can be found in the differ- ential geometry II script. We will only proof the case for n = 2 here, then the situation is as follows: ∂F ∂F F(0, 0) = F , = FA, = FB 0 ∂x ∂y

”⇒”: By the product rule we have

∂2F ∂F ∂A ∂A = A + F = FBA + F ∂y∂x ∂y ∂y ∂y

∂2F ∂F ∂B ∂B = B + F = FAB + F ∂x∂y ∂x ∂x ∂x By Schwarz theorem we get that both equalities in fact are equal. This can then be written as follows ∂B ∂A − + AB − BA = 0 ∂x ∂y

because F is nonsingular. Compare this to

Ç ∂B ∂A å dω = d(Adx + Bdy) = − dx ∧ dy ∂x ∂y

ω ∧ ω = (Adx + Bdy) ∧ (Adx + Bdy) = (AB − BA) dx ∧ dy Thus dω + ω ∧ ω = 0. This last equation is also called the ”Maurer-Cartan equation” and is sometimes also written as ∂A ∂B − = AB − BA. ∂y ∂x

”⇐”: We use Picard-Lindelof¨ to obtain a unique solution on the x-axis

k,k 0 G : (−ε, ε) → R with G(0) = F0, G (x) = A(x).

We use picard Lindelof¨ again to obtain

F : (−ε, ε)2 → Rk,k with F(x, 0) = G(x).

65 5.4 Fundamental theorem for flat vector bundles

For a fixed x this yields the linear ODE ∂F (x, y) = F(x, y)B(x, y). ∂y

We have to check that for all (x, y) ∈ (−ε, ε)2 it holds that ∂F (x, y) = F(x, y)A(x, y). ∂x ∂F This is equivalent to showing that ∂x − FA vanishes everywhere. By construction we certainly have that ∂F ( − FA)(x, 0) = 0 ∂x for all x. But then also ∂ ∂F ∂2F ∂F ∂A ( − FA) = − A − F ∂y ∂x ∂x∂y ∂y ∂y ∂ ∂F ∂F ∂A = − A − F ∂y ∂x ∂y ∂y ∂ ∂A = (FB) − FBA − F ∂y ∂y ∂F ∂B ∂A = B + F − FBA − F ∂y ∂y ∂y ∂F Ç ∂B ∂A å = B + F − BA − ∂y ∂y ∂y ∂F = B − FAB ∂y Ç ∂F å = − FA B ∂y

∂B ∂A where we used that by Maurer-Cartan ∂y − BA − ∂y = −AB. So we derived that ∂ ∂F Ç ∂F å ( − FA) = − FA B ∂y ∂x ∂y so for fixed x, the function Ç ∂F å y 7→ − FA (x, y) ∂x satisfies the linear ODE above with Ç ∂F å − FA (x, 0) = 0. ∂x

66 Integration on manifolds

As the zero function is a solution, by the uniqueness of the solution we get ∂F − FA = 0. ∂x

67 6. Riemann surfaces

Finally we are done with revising facts and theorems of differential geom- etry and start doing proper complex analysis.

Definition 6.1 (Riemann surface). A Riemann surface is a 2-dimensional manifold M with an almost complex structure J ∈ ΓEnd(TM).

A Riemann surface is not to be confused with the already known Rieman- nian surface that was defined as follows.

Definition 6.2 (Riemannian surface). A Riemannian surface is a 2-dimensional manifold with a Riemannian metric h., .i.

Theorem 6.3.

a) A Riemann surface is oriented: There is a unique orientation such that for all X ∈ TM,X 6= 0,X, JX is positively oriented.

b) On a Riemann surface there is a unique conformal structure such that for each X ∈ TM the vectors X, JX are orthogonal and have the same length (with respect to any given metric compatible with J). Equivalently: J = −J∗.

c) Given an oriented Riemannian surface, i.e. a Riemannian manifold of dimension 2, then there is a unique almost complex structure J on M such that a) and b) apply.

Remark 6.4. Note that for any endomorphism

hAX, Yi = hX, A∗Yi holds, thus (.)∗ does not depend on the conformal factor.

68 Riemann surfaces

By the theorem above we see that for a 2-dimensional manifold being a Riemann surface is really the same thing as having a conformal structure and an orientation.

Proof. Choose some Riemannian metric h., .i∼ on M and define

1 ∼ ∼ hX, Yi := 2 (hX, Yi + hJX, JYi ).

Then 1 ∼ ∼ hJX, Yi = 2 (hJX, Yi − hX, JYi ) = −hJY, Xi and thus J∗ = −J. Now define, σ ∈ Ω2(M), by

σ(X, Y) = hJX, Yi then this defines a volume form and hence an orientation on M. Further

σ(X, JX) = hJX, JXi = |X|2 > 0 menaing that X, JX are positively oriented. The uniqueness-part is left as an exercise. This proves a) and b). For c) we choose J to be the 90-degree- rotation in a positive sense.

Example 6.5. Let f : M → R3 be an oriented immersed surface then its Gauss map N defines a complex structure J by

d f (JX) := N × d f (X).

The example above raises the question if the almost complex structure we get is also complex.

Remark 6.6. Let V with h., .i, dim V = 2, J ∈ End(V), J2 = −1, J∗ = −J. Then, if B ∈ End(V), B∗ = −B, we have B = λJ for some λ ∈ R.

Theorem 6.7. Let M be an oriented Riemannian surface. Then M is K¨ahler, i.e. ∇J = 0 for the Levi-Civita connection ∇ of M.

69 Proof. Let X ∈ Γ(TM) and A ∈ ΓEnd(TM). Then (∇A)∗ = ∇A∗ i.e. (.)∗ is parallel. This is because from hAX, Yi = hX, A∗Yi we get

h∇(AX), Yi + hAX, ∇Yi = h∇X, A∗Yi + hX, ∇(A∗Y)i by differeniating both sides of the equation. Let B = ∇X J. Then ∗ ∗ ∗ B = (∇X J) = ∇X J = −∇X J = −B. Thus B = λJ for some function λ. Moreover, J2 = −1 yields

0 = J∇J + ∇JJ = JB + BJ = 2λJ2 = −2λ .

Thus ∇J = B = 0.

Corollary 6.8. The Nijenhuis tensor of a Riemannian surface vanishes.

Topologically, the list of compact Riemann surfaces is as follows:

We will later proof the uniformization theorem which implies that the fol- lowing two surfaces are isometric.

Interestingly enough this does not generalize to surfaces of higher genus, for example the following two are not isometric.

70 Riemann surfaces

This is also true for a fat and a thin torus.

Given J and a volume form σ such that X, JX is positively oriented. Then

hX, Yi := σ(X, JY) defines a Riemannian metric. To see this consider a 2-dimensional real vector space V with determinant det. Let A ∈ End(V). Then

det(X, AY) − det(Y, AX) = trA det(X, Y)

For an almost complex structure J Cayley-Hamilton yields

0 = J2 − tr(J)J + det(J)I = (det(J) − 1)I − tr(J)J.

Thus det(J) = 1 and tr(J) = 0. as I and J are linearly independent. In particular J is skew with respect to det. Same holds for σ and J above. Thus h., .i is symmetric. Moreover,

hX, Xi = σ(X, JX) > 0.

We make this a theorem.

Theorem 6.9. Let (M, J) be a Riemann surface and σ a volume form on M, such that σ(X, JY) > 0 for X 6= 0, then

hX, Yi = σ(X, JY) defines a Riemannian metric in the conformal class of (M, J).

6.1 Holomorphic line bundles over a Riemann sur- face

Definition 6.10 (Complex line bundle). A Complex line bundle over a man- ifold M is a rank-2 vector bundle L with complex structure J ∈ ΓEnd(L).

Remark 6.11. By definition, in a complex line bundle all fibers are 1-dimensional complex vector spaces, i.e. complex ”lines”.

71 6.1 Holomorphic line bundles over a Riemann surface

Example 6.12. We gather some examples of complex line bundles over a Riemann surface: (i) L = M × C, then Γ(L) = C∞(M; C). (ii) TM itself is a complex line bundle.

(iii) T∗ M, then Γ(T∗ M) = Ω1(M; C).

Definition 6.13 (complex connection on a complex vector bundle). Let E be a complex vector bundle. A connection ∇ on a complex vector bundle is called complex is for all ψ ∈ ΓE we have

∇(Jψ) = J∇ψ

or in other words ∇J = 0.

Proposition 6.14. Locally every complex vector bundle is isomorphic to U × Ck

Proof. Choose a real frame field φ1,..., φ2k, without loss of generality we can assume that φ1,p, Jφ1,p,..., φk,p, Jφk,p is a basis of Ep. Then, by con- tinuity, this will hold on a neighborhood of p, which yields a complex frame.

Definition 6.15 (complex trivial). A complex vector bundle (with connec- tion) is complex trivial if it is isomorphic as a complex vector bundle (with connection) to M × Ck.

Consider two complex connections ∇ and ∇˜ on (E, J), then we already know that ∇˜ = ∇ + A, meaning that they differ by an End(E)-valued 1- form A = ∇˜ − ∇ ∈ Ω1(E, End(E)). We observe that J∇ψ + AJψ = ∇(Jψ) + A(Jψ) = ∇˜ (Jψ) = J∇˜ ψ = J(∇ψ + Aψ) = J∇ψ + JAψ thus AJψ = JAψ meaning that in fact even 1 A ∈ Ω (E, End+(E)). Conversely, if ∇ is a complex connection and A ∈ Ω1(E, End(E)), then ∇ + A is a complex connection.

72 Riemann surfaces

Theorem 6.16. Any complex vector bundle has a complex connection.

Proof. Use the local triviality and a partition of unity.

Definition 6.17 (Curvature tensor). The curvature tensor R∇ of a connec- tion on a vector bundle E over a manifold M is defined as the map

R∇ : Γ(TM) × Γ(TM) × Γ(E) → Γ(E),

(X, Y, ψ) 7→ ∇X∇Yψ − ∇Y∇Xψ − ∇[X,Y]ψ

The curvature tensor R∇ of a connection ∇ on a vector bundle can be viewed as an End(E)-valued 2-form: R∇ ∈ Ω2(M; End(E)) .

Proposition 6.18. Let E be a complex vector bundle and ∇ be a complex con- nection on E. Then if ∇ commutes with J, so does R∇, i.e.

∇ 2 ∇J = J∇ ⇒ R ∈ Ω (M; End+(E)).

Proof. For X, Y ∈ Γ(TM) and ψ ∈ Γ(E) it is

R(X, Y)(Jψ) = ∇X∇Y(Jψ) − ∇Y∇X(Jψ) − ∇[X,Y](Jψ) = JR(X, Y)ψ.

6.2 Poincar´e-Hopfindex theorem

Throughout this section M denotes a compact oriented 2-dimensional man- ifold (without J) and L denotes a rank 2 vector bundle over M with J, i.e. a complex line bundle. The endomorphism bundle End(L) splits into com- plex linear End+(L) and complex anti-linear part End−(L),

End(L) = End+(L) ⊕ End−(L), End±(L) = {A ∈ End(L) | AJ = ±JA} .

In particular, End+(L)p = span{Ip, Jp} and Lp is a complex vector space setting (α + iβ)ψ := (αI + βJ)ψ . So, for a complex line bundle L we have ∼ End+(L) = M × C and therefore for a complex connection ∇ on L there is Ω ∈ Ω2(M; C) such that for all X, Y ∈ Γ(TM), ψ ∈ Γ(L) R∇(X, Y)ψ = −Ω(X, Y)ψ .

73 6.2 Poincar´e-Hopfindex theorem

Definition 6.19 (Curvature 2-form). The unique Ω ∈ Ω2(M, E) such that for all X, Y ∈ Γ(TM), ψ ∈ Γ(L) R∇(X, Y)ψ = −Ω(X, Y)ψ holds is called the curvature 2-form of R∇.

If dim M = 2 and M is oriented we can compute Z Ω ∈ C . M Interestingly this number is independent of the particular choice of the connection, although Ω depends on R∇. We can see this by the following computation: Let ∇˜ = ∇ + α for α ∈ Ω( M, C). then we get ∇˜ ˜ ˜ R (X, Y)ψ = ∇X(∇Yψ) − ∇Y(∇Xψ) − ∇[X,Y] − α([X, Y])ψ ∇ = R (X, Y)ψ + ∇(α(Y)ψ) + α(X)α(Y)ψ + α(X)∇Yψ − ∇Y(α(X)ψ) − α(Y)α(X)ψ − α([X, Y])ψ = R∇(X, Y)ψ + (Xα(Y) − Yα(X)) ψ − α([X, Y])ψ = R∇(X, Y)ψ + dα(X, Y)ψ

˜ So if ∇˜ = ∇ + α for α ∈ Ω1(M; C), then R∇ = R∇ + dα. Hence Ω˜ = Ω − dα hence by Stokes’ theorem Z Z Ω = Ω˜ . M M

Definition 6.20 (degree of a complex line bundle). If L is a complex line bundle over an oriented compact surface, then Z 1 deg(L) := i Ω 2π M is called the degree of L.

Remark 6.21. The Poincare-Hopf´ index theorem will tell us that deg(L) is actually an integer. Bur for now, the intermediate goal will be to show that it is real. Choose a euclidean fiber metric h., .i on L that is compatible with J. We know that there is one such and that all other possible choices are of the form h., .i∼ = e2uh., .i. Given a complex connection ∇, then we define

(∇Xh., .i)(ϕ, ψ) := X hϕ, ψi − h∇X ϕ, ψi − hϕ, ∇Xψi .

74 Riemann surfaces

This is not yet positive definite, but already (∇Xh., .i) is symmetric, bilinear and invariant under J. So ∇Xh., .i has all properties os h., .i except for that is not necessarily positive definite. Consequently we find an ω ∈ Ω1(M, R) such that ∇h., .i = ωh., .i. What happens to ∇h., .i of we change ∇ to ∇˜ = ∇ + α? By the definition we have, using α = β + iγ, that

(∇˜ Xh., .i)(ϕ, ψ) = (∇Xh., .i)(ϕ, ψ) − hα(ψ), ϕi − hψ, α(X)ϕi = (∇Xh., .i)(ψ, ϕ) − 2β(X) hψ, ϕi 1 ˜ Thus if we choose β(X) = 2 ω we get that ∇ is a metric connection in L. So without loss of generality we can assume that ∇ is compatible with J and h., .i. As deg does not depend on ∇ we can just choose it like this.

So as a quick summary, consider a complex line bundle L with h., .i com- patible with J and a complex connection ∇ that is metric with respect to h., .i. Then we want to show that R∇ = −Ω with Re(Ω) = 0. This means that the curvature 2-form of R∇ is purely imaginary.

Theorem 6.22. If, on a complex line bundle L, ∇ is complatible with J and h., .i, then Ω is purely imaginary.

Proof. By a straightforward calculation we yield ¨ ∂ hR(X, Y)ψ, ϕi = ∇X∇Yψ − ∇Y∇Xψ − ∇[X,Y]ψ, ϕ

= X h∇Yψ, ϕi − h∇Yψ, ∇X ϕi − Y h∇Xψ, ϕi ¨ ∂ h∇Xψ, ∇Y ϕi − [X, Y] hψ, ϕi + ψ, ∇[X,Y] ϕ

= XY hψ, ϕi − X hψ, ∇Y ϕi − Y hψ, ∇X ϕi + hψ, ∇Y∇X ϕi − yX hψ, ϕi + Y hψ, ∇x ϕi + X hψ, ∇Y ϕi − hψ, ∇X∇Y ϕi ¨ ∂ − [X, Y] hψ, ϕi + ψ, ∇[X,Y] ϕ = − hψ, R(X, Y)ϕi In fact, for general vector bundles with metric and metric connection ∇ it holds that for R(X, Y) ∈ Γ(E) it is R(X, Y)∗ = −R(X, Y) , i.e. R takes values in the skew-adjoint endomorphisms. In our case, with Ω = α + Jβ ∈ C we get R(X, Y)ψ = −Ω(X, Y)ψ = (α(X, Y) + β(X, Y)J)ψ

75 6.2 Poincar´e-Hopfindex theorem so R(X, Y)∗ = −R(X, Y) ⇔ α = 0.

Thus Ω˜ is purely imaginary and we get the following corollary.

Corollary 6.23. Z 1 deg(L) = i Ω ∈ R . 2π M The original claim was that L is even an integer and to proof this we are going to ”count” it. But before, we will ”steal” the following result from differential topology, that are hard to proof, but easy to believe. So we will save the time and omit the proofs.

Lemma 6.24 (Sard). Let U ⊂ Rn open, if f : U → Rk ∈ C∞, then the set n o y ∈ Rk | ∃x ∈ U with f (x) = y and rank f 0(x) < k of critical values of f has measure zero.

With some work one can use Sard’s lemma to proof the following theorem.

Theorem 6.25 (Transversality theorem). Call two submanifolds M1, M2 ⊂ M˜ transversal if for all p ∈ M1 ∩ M2 we have Tp M1 + Tp M2 = Tp M.˜ Given ∞ M1, M2 ⊂ M˜ one can perturb M1 slightly such (refering to C -topology) such that afterwards M1 and M2 are transversal.

If we apply the transversality theorem to  ˜ = ˜ = M total space of L, dim M 4  ¶ © M1 = 0p | p ∈ M graph of the zero section   ¶ © M2 = ψp midp ∈ M graph of some section ψ ∈ Γ(L) Then we now there is some section ψ transversal to the zero section. This means that locally M = U ⊂ R2 (including orientation), L = U × C then ψ can be viewed as a map ψ : U → C

76 Riemann surfaces

0 In this case we have that M1 is transversal to M2 if and only if det ψ (p) 6= 0.

Definition 6.26 (Winding number). We define the winding number of a section ψ ∈ Γ(L) that is transversal to the zero section as

 0 1, det ψ (p) > 0 indpψ := 0 . −1, det ψ (p) < 0

Remark 6.27. The winding number indpψ is independetn of oriented charts.

More generally, if p is an isolated zero of ψ (seen as a map ψ : U → C) define 1 Z dψ ind ψ = p 2πi ψ |z−p|=$ which is known from the complex analysis 1 lecture.

Theorem 6.28 (Poincare-Hopf´ index theorem). Let L be a cmplex line bundle over a compact oriented surface and ∇ a complex connection with curvature 2- 2 form Ω ∈ Ω (M, C) and ψ ∈ Γ(L) with isolated zeroes p1,..., pn. Then

n 1 Z X Ω = indpi ψ ∈ Z. M 2πi i=1

Remark 6.29. Note that by the compactness of the surface ψ can only have finitely many isolated zeroes.

Remark 6.30 (Contemplate consequences).

77 6.2 Poincar´e-Hopfindex theorem

i) deg(L) ∈ Z and depends only on L as a oriented rank2 real vector bundle, i.e. is independent of J and ∇. (A change of the orientation changes the sign of deg(L))

ii) From the fact that it does not depend on ψ, counted with multiplicity, all sections with only finitely many zeros ψ ∈ Γ(L) have the same number of zeroes.

Definition 6.31 (Euler characteristic). Given an oriented compact surface M, define the Euler characteristic χ(M) as

χ(M) := deg(TM).

Remark 6.32. If we change the orientation of M, also the orientation of Tp M is changed, therefore indpX is unchanged. Hence χ(M) is independent of the orientation.

Theorem 6.33 (Gauß-Bonnet). If M is an oriented compact Riemannian sur- face with volume form det and curvature form Ω = iK det of the Levi-Civita Connection ∇, for the Gaussian curvature K ∈ C∞(M), then Z K det = 2πχ(M). M

Example 6.34. Consider M = S2, then the Gaussian curvature K ≡ 1 and Z area(S2) = 1 det = 4π = 2π · 2 S2 as χ(S2) = 2.

Remark 6.35. The setup for the Poincare-Hopf´ index theorem is the fol- lowing: M is a compact, oriented 2-dimensional manifold (without any complex structure J), L is a complex line bundle, i.e. rank2-vector bundle

78 Riemann surfaces over M with almost complex structure J. Now we make choices  Choose a connection ∇ on L with ∇L = 0

Choose a connection ψ ∈ Γ(L) with isolated singularities p1,..., pn ∈ M

By Poincare-Hopf,´ for the volume 2-form Ω ∈ Ω2(M, C) of the curvature tensor R∇ we have

n 1 Z X Ω = indpj ψ =: deg(L) ∈ Z M 2πi j=1

We notice that the left hand side of the equation does not depend on the choice of ψ and the middle part does not depend on the connection ∇, hence the right hand side deg does neither depend on the choice of ψ, nor on the choice of ∇.

Proof. We choose open neighborhoods U1,..., Un ⊂ M of p1,..., pn respec- tively such that

(i) there is ϕ ∈ Γ(L|Uj ) without zeros, i.e. there exists a 1-dimensional frame field on Uj

2 (ii) there is a chart φj : Uj → R such that φ(pj) = 0 and ¶ 2 2 2 2© Bε(0) = (x, y) ∈ R | x + y ≤ ε ⊂ φj(Uj) for 0 < ε < 1.

(iii) Uj ∩ Uk = ∅ for j 6= k

79 6.2 Poincar´e-Hopfindex theorem

We define  n   [ −1  Mε := M \ φj (Bε(0)) j=1  and see that this defines a compact manifold with boundary. Further ψ|Mε has no zeros. 1 Define a 1-form η ∈ Ω (Mε, C) by ∇ψ = ηψ. For X, Y ∈ Γ(TM) we compute −Ω(X, Y)ψ = R∇(X, Y)ψ

= ∇X∇Yψ − ∇Y∇Xψ − ∇[X,Y]ψ

= ∇Xη(Y)ψ − ∇Yη(X)ψ − η([X, Y])ψ = (Xη(Y) − Yη(X) − η([X, Y])) ψ + η(Y)η(X)ψ − η(X)η(Y)ψ = dη(X, Y)ψ where we used that η(Y)η(X)ψ − η(X)η(Y)ψ = 0 as η takes values in C and the multiplication on C is commutative. So we have that Ω = −dη. ε −1 This immediately calls for stokes theorem! First we define Bj := φj (Bε(0)), then n Z Z Z X Z Ω = − dη = − η = η M M ∂M ∂Bε ε ε ε j=1 j Note that for the last equation, the orientation yields a sign-flip.

For fixed ε > 0 and j we define

ψ| ε = γϕ . ∂Bj j

80 Riemann surfaces

ε 1 Further we assume that ∂Bj = S . We use the fact that we can compute the winding number using the integral that counts zeroes and see that with stokes theorem 1 Z dγ indp ψ = j 2πi S1 γ 1 Z = (d log |γ| + idarg(γ)) 2πi S1 1 Z = darg(γ) 2π S1

ε For X ∈ Tq∂Bj ⊂ TM we have

∇Xψq = dγ(X)ϕj(q) + γ(q)∇X ϕj(q) = dγ(X) + ϕj(q) + γ(q)ω(X)ϕj(q) 1 where ω ∈ Ω (Uj) is 1-form defined by

ω(X)ϕj(q) = ∇X ϕj(q). On the other hand

∇Xψq = η(X)ψq = η(X)γ(q)ϕj(q).

As ϕj(q) appears everywhere we can cancel it and derive dγ γη = dγ + γω ⇔ η = + ω. γ That is fortunate, because then Z Z dγ Z Z η = + ω = ind ψ2πi + ω. ε ε ε pj ε ∂Bj ∂Bj γ ∂Bj ∂Bj So it remains to show that Z lim ω = 0. ε→0 ε ∂Bj This is again an application of stokes. As ω is also smooth on the interior of Uj we have Z Z ω = dω → 0 ε ε ∂Bj Bj for ε → 0.

81 6.2 Poincar´e-Hopfindex theorem

For now consider a 2-dimensional manifold M and equip the tangent bun- dle TM with a J. This turns M into a Riemann surface. We now will apply the Poincare-Hopf´ index theorem to L = TM. Remark 6.36. Recall that the Euler characteristic χ(M) is the index sum of any vector field with only isolated zeroes. In particular, χ(M) is indepen- dent of J. Another consequence of the transversality theorem is that on every man- ifold there is a height function h ∈ C∞(M) such that dh vanishes only at isolated points. For the case dim M = 2 consider an immersion f : M → R3 with normal field N : M → S2. We can manufacture a height function h as above as follows: 2 Choose some regular value a ∈ S of N (i.e. N(p) = a ⇒ dp N has full rank) and define h := h f , ai .

In the sketch we have 8 critical points of gradh, that is when gradh is verti- cal, hence  1 if h has a min or max at p. indp gradh = −1 if h has a saddle point at p.

Example 6.37. Consider the function h(x, y) = x2 − y2. This h has a saddle at (0, 0). The level sets look more or less like this

Corollary 6.38. The Euler characteristic of a Riemann surface can be expressed as χ(M) = |min| + |max| − |saddles|.

82 Riemann surfaces

Definition 6.39 (k-cell).

i)A 2-cell F in a surface M is a subset diffeomorphic to a convex polygon in R2.

ii)A 1-cell e in a surface M is a subset diffeomorphic to [0, 1].

iii)A 0-cell p in a surface M is a point in M.

Definition 6.40 (Cell-decomposition). A cell-decomposition of a complex surface with boundary M is a collection C1,..., Cm of 0−, 1−, or 2-cells such that m [ Ci = M i=1 and for each i, j ∈ {1, . . . , m} with i 6= j there is k ∈ {1, . . . , m} such that ¶ © Ci ∩ Cj = Ck and dim Ck < max dim Ci, dim Cj .

Definition 6.41 (triangulation). A triangulation of a complex surface M is cell-decompostion of M in cells of dimesnion at most 2.

Given a triangulation of a compact oriented surface with F being the set of 2-cells (”faces”), E the set of 1-cells (”edges”) and V the set of 0-cells (”vertices”) one can construct X ∈ Γ(TM) with

 a source at each face center  a sink at each vertex  a saddle at each edge center

83 6.2 Poincar´e-Hopfindex theorem

This yields the more popular formula for the Euler characteristic

χ(M) = V − E + F.

84 7. Classification of Line Bundles

We want to classify line bundles up to isomorphisms. The final result will be that deg is the only invariant possible, hence L =∼ L0 if and only if 0 deg L = deg L . Further we show that for all z ∈ Z there is a bundle Lz such that deg Lz = z. But first it will be necessary to briefly revise some topics of (multi-)linear algebra.

7.1 Tensor products of vector spaces and bundles

Definition 7.1 (tensor product of vector spaces). Let V, W be finite di- mensional K-vector spaces with K ∈ {R, C}. Then

V ⊗ W := {β : V∗ × W∗ → K | β bilinear}

This leads to the infamous bra-ket notation known from e.g. quantum mechanics. For α ∈ V∗ and v ∈ V we define

α(v) := hα|vi .

From this, to emphasize that α is a dual-vector we can also write hα|, anal- ogously for a vector v we write |vi. See also the correspondence to the earlier used musical isomorphisms [ and ]. In total we have the following identity D E [ [ [ hu, vi = u v = u (v) = u v.

Definition 7.2 (tensor product). Let v ∈ V and w ∈ W, then for αinV∗, β ∈ W∗ we define v ⊗ w ∈ V ⊗ W by

v ⊗ w(α, β) := α(v) · β(w) = hα|vi hβ|wi .

85 7.1 Tensor products of vector spaces and bundles

If a1,..., an is a basis of V and b1,..., bm is a basis of W then

ai ⊗ bj, 1 ≤ i ≤ n, 1 ≤ j ≤ m is a basis of V ⊗ W which implies that dim V ⊗ W = dim V · dim W.

Remark 7.3. The tensor product of line bundles yields again a line bundle.

Let for now L1 and L2 be two line bundles over a manifold M, then L1 ⊗ L2 is a line bundle over M as well.

Proposition 7.4. If L˜ 1 is isomorphic to L1 and L˜ 2 is isomorphic to L2 then L˜ 1 ⊗ L˜ 2 is isomorphic to L1 ⊗ L2.

Proof. The proof should be clear.

So the tensor product is well defined on isomorphism-classes of line bun- dles over M.

Now consider the trivial bundle M × C, then there is an isomorphism be- tween (M × C) × L → L. This can be seen as follows: ∗ ∗ Let βin(M × C)p × Lp, then for ω ∈ C and η ∈ Lp it is β(ω, η) ∈ C and

∗ ∗ f (β) = β(ω, ·) ∈ (Lp) = Lp. as f (β) only has an input for elements of the second slot (Note that ω(1) = 1 hence ”ω = 1∗”). So (M × C) × L is canonically isomorphic to L itself. Similarly, for β ∈ (L˜ ⊗ L)p we construct

f (β) ∈ (L ⊗ L˜ )p by ( f (β))(ω˜ , ω) := β(ω˜ , ω). So switching arguments yields an isomorphsim.

To summarize this: The isomorphism classes of complex (or real) line bun- dles over a connected manifold M form an abelian group under the tensor product. The trivial bundle (or any bundle isomorphic to it) is the neutral element and L∗ =: L−1 is the inverse element of L.

86 Classification of Line Bundles

7.2 Line Bundles on Surfaces

The intermediate goal will be to proof that the map deg is an isomorphism of abelian groups between the set of equivalence classes of line bundles with ⊗ and (Z, +). Some preparation is needed beforehand.

Lemma 7.5. If M is a connected manifold and U ⊂ M open, p ∈ M. Then there is a diffeomorphism f : M → M such that f (U) ⊂ U and f (p) ∈ U.

Proof. Since M is connected, there is a smooth γ : [0, 1] → M with γ(0) = p and γ(1) =: q ∈ U. By the transversality theorem, γ can be chosen as an immersion with transversal self-intersections. Resolve the self-intersections to get a collection of embedded curves and delete loops to obtain an em- bedded γ.

By the ”tubular neighborhood theorem” we find a diffeomorphism  g(0, 0) = p g : [−ε, 1 + ε] × Dn−1 → V ⊂ M with . g(1, 0) = q

87 7.2 Line Bundles on Surfaces

Now construct a diffeomorphism

f˜: [−ε, 1 + ε] × Dn−1 → [−ε, 1 + ε] × Dn−1 such that

 Ä − ä  f (x) = x, for x in some neighborhood of ∂ [−ε, 1 + ε] × Dn 1  f (0, 0) = (1, 0)

Then define  − g ◦ f˜ ◦ g 1(r) for r ∈ V f (r) := r else then f is smooth and does the trick.

Proposition 7.6. If M is a smooth and connected n-dimesnional manifold and n p1,..., pm ∈ M, then there is a U ⊂ M diffeomorphic to D such that p1,..., pm ∈ U.

Proof. We will proof this by induction on m ∈ N. For m = 1 the statement is clear. By the induction hypothesis p1,..., pm−1 are already contained in some U˜ . Use the above Lemma to find a diffeomorphism f : M → M with U := f (U˜ ) ⊂ U˜ and f (pm) ∈ U˜ . Then p1,..., pm ∈ U, hence the claim is proven.

Theorem 7.7. Let E be a vector bundle over the closed n-dimensional unit disc Dn, then E is trivial.

Proof. Choose a basis φ1,..., φk of E0 and any connection ∇ on E. Then there are unique sections ϕ1,..., ϕk ∈ Γ(E) such that ϕj(0) = φj such that n ∇X ϕj = 0 when Xp = p is the position vector field on D . Then ϕ1,..., ϕk form a frame field. Now we are able to proof the desired theorem.

Theorem 7.8. The map [L] 7→ deg L ∈ Z is an isomorphism of abelian groups.

88 Classification of Line Bundles

Proof. We will give two proofs for the homeomorphism property:

i) Let ψ ∈ Γ(L) and ψ˜ ∈ Γ(L˜ ) with isolated zeros. Then we immediately get indp(ψ ⊗ ψ˜) = indp ψ + indp ψ˜. This is because the lefthand side has a zero if either of the two vec- tor fields has a zero and a multiplication of winding numbers yields ”winding number of γ · γ˜ = winding number of γ + winding number of γ˜”.

ii) The second proof uses curvature. Choose complex connections ∇ on L and ∇˜ on L˜ . Then there is a unique connection ∇ˆ on L ⊗ L˜ such that for all ψ ∈ Γ(L), ψ˜ ∈ Γ(L˜ ) we have

∇ˆ (ψ ⊗ ψ˜) = (∇ψ) ⊗ ψ˜ + ψ ⊗ (∇˜ ψ˜).

Then

ˆ R∇(X, Y)(ψ ⊗ ψ˜) ˆ ˆ ˆ ˆ ˆ = ∇X∇Y(ψ ⊗ ψ˜) − ∇Y∇X(ψ ⊗ ψ˜) − ∇[X,Y](ψ ⊗ ψ˜) Ä ä = ∇ˆ X (∇Yψ) ⊗ ψ˜ + ψ ⊗ (∇˜ Yψ˜) (ψ ⊗ ψ˜) ˆ Ä ˜ ä ˆ − ∇Y (∇Xψ) ⊗ ψ˜ + ψ ⊗ (∇Xψ˜) (ψ ⊗ ψ˜) − ∇[X,Y](ψ ⊗ ψ˜)

= ∇X∇Yψ ⊗ ψ˜ + ∇Yψ ⊗ ∇˜ Xψ˜ + ∇X ⊗ ∇˜ Yψ˜ + ∇˜ X∇˜ Yψ˜

− ∇Y∇Xψ ⊗ ψ˜ − ∇Xψ ⊗ ∇˜ Yψ˜ − ∇Y ⊗ ∇˜ Xψ˜ − ∇˜ Y∇˜ Xψ˜ ˜ − (∇[X,Y]ψ) ⊗ ψ˜ − ψ ⊗ (∇[X,Y]ψ˜) ˜ = R∇(X, Y)ψ ⊗ ψ˜ + ψ ⊗ R∇(X, Y)ψ˜

which implies that Ωˆ = Ω + Ω˜ and yields deg(L ⊗ L˜ ) = deg L + deg L˜ .

It remains to show that deg is surjective and injective. We will start with the anterior. To show that deg is surjective we use the ”skyscraper construction” to con- struct a vector bundle of arbitrary degree d. Choose a point p ∈ M and take (M \ {p}) × C, i.e. the trivial bundle over M \ {p}. also, for some open neighborhood U of p take U × C. Consider now a [(M \ {p}) × C] [U × C]

= {(0, q, z) | q ∈ M \ {p} , z ∈ C} ∪ {(1, q˜, w) | q˜ ∈ U, w ∈ C} .

89 7.2 Line Bundles on Surfaces

We define an equivalence relation ∼ on it by

(0, q, z) ∼ (1, q˜, w) :⇔ q = q˜, w = ϕ(q)dz. where vφ is a diffeomorphism ϕ : U → V ⊂ R2 with 0 ∈ V.

Note that q 6= p hence ϕ(q)d 6= 0. Let now

L := {equivalence classes} then  π[(0, q, z)] := q π : L → M, π[(1, q˜, w)] := q˜ is well defined, hence L is a line bundle. Now define ψ ∈ Γ(L) by  [(0, q, 1)], forq 6= p ψp = [(1, q, ϕ(q)d)] forq ∈ U then ψ has a zero of degree d at p, hence L has deg L = d > 0 and deg L−1 = −d which yields the surjectivity. It remains to show that deg is injective and since it is already clear that it is a homeomorphism of groups, it suffices to show the following statement:

deg L = 0 ⇔ L has a nowhere vanishing section ψ ∈ Γ(L).

By the transversality theorem we can choose a section ψ ∈ Γ(L) with iso- lated zeros p1,..., pn. Then by assumption

indp1 ψ + ... + indpn ψ = 0.

By the preparation we did beforehand we can apply a diffeomorphism n M → M such that p1,..., pn ∈ U for some U diffeomorphic to D . In particular this can also be done such that this is true for some U1 ⊂ U.

90 Classification of Line Bundles

n Because U is diffeomorphic to D we can choose some section ϕ ∈ Γ(L|U) ∞ without zeros, as L|U is trivial. Then there is g ∈ C (U, C) such that

ψ|U = g · ϕ.

As the total number of zeros is zero, just as in the proof of the Poincare-´ Hopf index theorem we get that the winding number

Z dg 0 = ∂U1 g which means that Z dg = 0 γ g ∞ for all closed curves γ in U \ U1. S0 there exists f ∈ C (u \ U1, C) with

g = e f .

Let now σ ∈ C∞(R) with σ(x) = 0 for x < 2 and σ(x) = 1 for x > 3. Define

f˜: U → C by f˜(p) = σ(|z(p)|) f (p).

Then a smooth section ψ˜ ∈ Γ(L) of L can be obtained by setting

 ˜ e f (p) ϕ for p ∈ U ψ˜(p) = p ψp else

Further ψ˜ has no zeros, hence is a frame.

7.3 Combinatorial Topology

The setup for the first part of this section will be a compact oriented smooth surface M. We have already learned what a cell-decomposition of such a surface is. In order to proof the existence of one such we need to work off the following steps:

91 7.3 Combinatorial Topology

i) Choose a Riemannian metric. As M is compact, this gives a distance fucntion d : M × M → R.

ii) Choose p1,..., pn ∈ M such that d(pi, pj) < ε for all i, j and a given ε which is sufficiently small.

iii) Define so called Voronoi-cells by

¶ © Φj := q ∈ M | d(pj, q) ≤ d(pi, q) for all i ∈ {1, . . . , n} .

These cells form the faces of the cell-decomposition.

The points p1,..., pn are the vertices of the dual cell-decomposition. Vertices of the original cell-decomposition correspond 1 − 1 to faces of the dual cell- decomposition and vice versa.

Given a cell-decomposition of M, define E˜ to be the set of oriented edges.

92 Classification of Line Bundles

This gives a bijective map $ : E˜ → E˜ without fixed points that assigns to each edge ”the same” edge with differ- ent orientation. It is easy to see that

2 $ = $ ◦ $ = IdE˜ hence $ is an involution. Additionally we have a permutation

σ : E˜ → E˜ which is defined as follows:  find the face ϕ which is to the left of e define σ(e) as the oriented edge following e in the edge-cycles of ϕ.

So in summary, a cell-decomposition of a compact oriented surface M gives: i) finite set E˜

ii) $ a fixed point free involution of E˜

iii) σ a permutation of E˜ Ä ä Remark 7.9. For every such triple E˜, $, σ there is a compact oriented sur- face with a cell-decomposition (includes di- or one-gons) which is unique up to a diffeomorphism. We have the following 1 − 1 correspondences:

F := {faces of cell-decomposition} ↔ {cycles of σ}

E := {edges of cell-decomposition} ↔ {cycles of $}

Example 7.10. Consider E˜ = {e+, e−}, then $(e+) = e− and $(e−) = e+. Then there are two possible cases of how this surface looks like:

i) σ = IdE˜ :

93 7.3 Combinatorial Topology

ii) σ = $:

˜ To avoid such degenerate examples one can ask for E ≥ 3. We see that the composition $ ◦ σ rotates an edge e clockwise around its starting vertex. Thus we obtain the set

V := {vertices of cell-decomposition} ↔ {cycles of σ ◦ $}

In particular we know the cardinality of V, E and F which leads to the following theorem.

Theorem 7.11. For a compact oriented surface, the Euler-characteristic χ = |V| − |E| + |F| is even.

Proof. We know that sgn $ = (−1)|E|. If now σ˜ is a cycle of length k, then k+1 sgn σ˜ = (−1) . The face ϕj has kj edges. Then

˜ sgn σ = (−1)(k1+1)+...+(k|F|+1) = (−1)|E|+|F| = (−1)|F| where the last equality inherits from the fact that E˜ must be even. Similarly we get sgn σ ◦ $ = (−1)|V|.

As sgn is a homeomorphism from Sn → Z/2Z we get

(−1)|V| = sgn σ ◦ $ = sgn σsgn $ = (−1)|F|(−1)|E| which means that

|V| = |F| + |E| = −|F| + |E| mod 2

94 Classification of Line Bundles

Let us for now restrict our attention to a solely discrete oriented surface

M = (E˜, $, σ) with a finite set E˜, bijective maps $, σ : E˜ → E˜ whereby $ has no fixed points. We obtain the sets

E = {cycles of $} , F = {cycles of σ} , V = {cycles of $ ◦ σ} .

Definition 7.12 (dual surface). The dual discrete surface M∗ is given by Ä ä M∗ = E˜, $ =: $∗, $ ◦ σ =: σ∗ .

Definition 7.13 (subsurface). A finite set Eˆ ⊆ E˜ is called a subsurface of M of $(Eˆ) = Eˆ and σ(Eˆ) = E.ˆ

ˆ Ä ˆ ä Remark 7.14. Then the M = E, $|Eˆ , σ|Eˆ is also a discrete surface.

Definition 7.15 (connected discrete surface). A discrete surface M is called connected if it has no proper subsurfaces.

Remark 7.16. Every discrete surface is the disjoint union of its connected components.

Theorem 7.17. A discrete surface M is connected if and only if its dual surface M∗ is connected.

Proof. The invariance of a subsurface Eˆ of M under $ and σ gives

σ∗(Eˆ) = $ ◦ σ(Eˆ) = $(Eˆ) = Eˆ as well as $∗(Eˆ) = $(Eˆ) = Eˆ. This conclusion is symmetric in M and M∗ which yields the claim.

95 7.4 Discrete Forms

7.4 Discrete Forms

Having discrete surfaces defined it is only natural to aim for a definition of discrete forms as in the smooth case these are the natural choice to integrate over a surface. For a discrete surface M just like in the previous section we define

Ω0(M) := { f : V → R} ¶ © Ω1(M) := ω : E˜ → R Ω2(M) := {η : F → R}

But given these definitions, what does it mean to have a discrete form? When M comes from a cell decomposition, then a discrete 2-form η is just like a smooth 2-form where the only information we keep is

Z η ϕ for each fache ϕ ∈ F. Hence, for each face ϕ ∈ F there is a real number R. For a discrete 1-form ω this can be motivated by only keeping the informa- tion Z ω. e Here, as we find many forms that agree on e with both orientations, discrete 1-forms correspond to equivalence classes of smooth 1-forms that agree on all edges in both possible orientations.

In the smooth setup, on a connected, oriented surface M an ω ∈ Ω1(M) is exact if and only if for every closed curve γ : [0, 1] → M with γ(0) = γ(1) we have Z Z 1 0 ω = ωγ(t)γ (t) dt = 0. γ 0

96 Classification of Line Bundles

We aim for a discrete analogue of this well-known theorem. For an oriented edge e ∈ E˜ we define  start(e) := orbit of $(e) under $ ◦ σ end(e) := orbit if e under $ ◦ σ

Definition 7.18 (discrete path).

i) A discrete path in M is a sequence γ = (e1,..., en) with ei ∈ E˜ such that end(ei) = start(ei+1) for i = 1, . . . , n − 1.

ii) A path γ is called closed if end(en) = start(e1).

Remark 7.19. Note that a path γ is a sequence of edges, not vertices!

0 Definition 7.20 (discrete exterior derivative d0). For a 0-form f ∈ Ω (M) define d f ∈ Ω1(M) by

d f (e) := f (end(e)) − f (start(e)) .

Theorem 7.21. M is connected if and only if for all pairs of points p, q ∈ V there is a path γ in M with start(γ) = p and end(γ) = q.

Proof. The proof will be left as an exercise.

Definition 7.22 (integral of a discrete 1-form). Let γ = (e1,..., en) be a path in M and ω ∈ Ω1(M), then

n Z X ω = ω(ei). γ i=1

Now we can proof the desired analogue of the smooth case.

Theorem 7.23. On a connected M we have that a discrete 1-form ω ∈ Ω1(M) is exact if an only if Z ω = 0 γ for all closed paths γ in M.

97 7.5 Poincar´eDuality

Proof. ”⇒”: Let ω be exact, i.e. ω = d f for some f ∈ Ω0(M) and γ a closed path in M. Then n n n Z X X X ω = ω(ei) = d f (ei) = ( f (end(ei)) − f (start(ei))) = 0 γ i=1 i=1 i=1 as γ is assumed to be a closed path.

”⇐”: Fix a vertex p ∈ V. For q ∈ M define Z f (q) = ω γ

where start(γ) = p and end(γ) = q. This is well defined as we assume M to be connected and by the hypothesis. Now it remains to check that d f = ω.

7.5 Poincar´eDuality

If M is a compact oriented n-dimensional manifold, then we have a natural map Ωk(M) × Ωn−k(M) → R given by Z (ω, η) 7→ ω ∧ η. M This is a non-degenerate pairing of vector spaces in the sense that Z ω ∧ η = 0 for all η ∈ Ωn−k(M) ⇒ ω = 0 M and Z ω ∧ η = 0 for all ω ∈ Ωk(M) ⇒ η = 0. M Consider the case that both, ω and η are closed, i.e. dω = dη = 0. For any other (n − k − 1)-form α ∈ Ωn−k−1(M) we obtain Z Z Z Z ω ∧ (η + dα) = ω ∧ η − (−1)k d (ω ∧ η) = ω ∧ η M M M M where the last equality is due to Stoke’s theorem. Denoting the exterior derivative d Ωk(M) −→k Ωk+1(M)

98 Classification of Line Bundles

Definition 7.24 (k-th cohomology of M). We define a real vector space called the k-th cohomology of M by

Hk(M) := ker dk . Imdk−1

Further Z h[ω], [η]i := ω ∧ η M is well-defined. Later we will see that

k βk(M) := dim H < ∞ which will be the k-th betty number of M. We will see that βn−k = βk which is then the Poincare´ duality. The problem to define an analogue in the discrete setting is that there is no ”good” wedge product for discrete forms. But still there is a sensible notion of a Poincare-duality.´

For surfaces the solution will be to bring the dual surface into the picture. There is indeed a good discrete definition of

Z ω ∧ η M for ω ∈ Ωk(M) and η ∈ Ωn−k(M∗). The strategy will be to stick to the original primal cell-decomposition but to give a suitable interpretation for discrete k-forms on M∗.

0 ∗ View f ∈ Ω (M ) as a function taking a fixed value fϕ ∈ R on the interior of each face ϕ ∈ F. Then actually f is the limit of smooth functions which we can be imagined as seen in the sketch.

99 7.5 Poincar´eDuality

Recall that for ω ∈ Ω2(M) we considered an equivalence class of 2-forms where only Z ωϕ = ω ϕ mattered (in the sense that we only consider equivalence classes of smooth 2-forms which are equal on oriented edges). Now we pair a primal 2-form and a dual 0-form to Z h f , [ω]i := f ω. M From the purely discrete viewpoint this is

 0 ∗  f ∈ Ω (M ) gives fϕ ∈ R for each ϕ ∈ F. 2 ω ∈ Ω (M) gives ωϕ ∈ R for each ϕ ∈ F.

How do we do this the other way round?

Consider η ∈ Ω2(M∗) as the limit of smooth 2-forms on M. A discrete f ∈ Ω0(M) is an equivalence class of smooth functions where only the values f (v) on the vertices v ∈ V matter. Then if

Z lim ηt = ηv ∈ R t→0 Uv and supp ηt shrinks to v we have

Z X lim f ηt = f (v)ηv. t→0 M v∈V

Discrete 1-forms ω ∈ Ω1(M∗) are limits of smooth 1-forms such that

Z lim ωt = ωe. t→0 γ for all curves γ from the right of e to its left.

100 Classification of Line Bundles

1 1 ∗ Consider a smooth η ∈ Ω (M) and ω ∈ Ω (M ). With ωt = ft(x, y)dy and η = adx + bdy we obtain

Z ε lim ft(x, y)dy = ωe t→0 −ε for all x ∈ [0, 1] hence

Z Z 1 Z ε Z 1 Z 1 lim η ∧ ωt = lim a(x, y) ft(x, y) dydx = a(x, 0)ωe = ωe η = [η]e t→0 M t→0 0 −ε 0 0

1 ∗ From the discrete point of view that is: For ω ∈ Ω (M ) given by ωe ∈ R ˜ 1 for each e ∈ E with ω$(e) = −ωe and a smooth η ∈ Ω (M) we set

Z 1 X h[η], ωi := ηeωe = ” η ∧ ω”. 2 M e∈E˜

So we can think of ω as kind of ”delta-distribution form”.

7.6 ”Baby Riemann Roch Theorem”

In this section we will use M to denote a smooth, oriented, compact and Ä ä connected manifold. We fix a cell decomposition Mˆ of M by E˜, $, s . Note that we change the notation from σ to s (as in ”shift”). M∗ will denote the discrete surface dual to Mˆ . On M we have

101 7.6 ”Baby Riemann Roch Theorem”

d1 d0 Ω2(M) Ω1(M) Ω0(M)

Ω0(M) Ω1(M) Ω2(M) d0 d1 where ↔ means that these are paired in the sense that for ω ∈ Ωk(M) and η ∈ Ωn−k(M) we have a non-degenerate Z hω|ηi := ω ∧ η. M 0 1 For f ∈ Ω (M) and ω ∈ Ω (M) we can pair ω and d0 f to Z Z hω|d0 f i = ω ∧ d0 f = −d0 f ∧ ω M M Z Z = −d1( f ω) + f d1ω = f d1ω = hd1ω| f i M M where we used the product rule and Stoke’s theorem. This shows that in the smooth picture ∗ d1 = d0. For some f ∈ Ω0(M) and ω ∈ Ω1(M) we see that Z Z h f |d1ωi = f d1ω = d1( f ω) − d0 f ∧ ω = − hd0 f |ωi M M hence ∗ d1 = −d0. Remark 7.25. One can change the pairings in a way such that the − sign in the last equation vanishes. In the discrete setting the diagram becomes

∂1 ∂0 Ω2(M∗) Ω1(M∗) Ω0(M∗)

Ω0(Mˆ ) Ω1(Mˆ ) Ω2(Mˆ ) d0 d1

The definition was that for f ∈ Ω0(Mˆ )

d f (e) = fend(e) − fstart(e) hence d f (e) = 0 ⇔ f = const.

102 Classification of Line Bundles

In particular, dim ker d0 = 1. From the theorem that M∗ is also connected if Mˆ is connected we immedi- ately get that also dim ker ∂0 = 1. The kernel thus consists of the constant functions. An interesting conse- quence is the following:

Theorem 7.26. Let σ ∈ Ω2(Mˆ ) be a discrete 2-form. Then there is a discrete 1-form ω ∈ Ω1(Mˆ ) such that dω = σ (i.e. σ is exact) if and only if Z σ = 0. M

Proof. The first implication follows directly from Stoke’s theorem, which holds also in the discrete setting. The contrary implication follows from ∗ ⊥ linear algebra facts, because im d1 = (ker d1) . This gives ß Z ™ 2 im d1 = σ ∈ Ω (Mˆ ) | σ = h1|σi = 0 M so that we are done. Remark 7.27. The theorem also holds for smooth manifolds, but the proof requires knowledge about elliptic operators which is beyond the scope of this lecture.

∗ ∗ Theorem 7.28. It holds that d0 = ∂1 and d1 = −∂0.

Proof. The first identity will be a homework exercise. For the second iden- tity we compute on the one hand that ∗ X X X h f |dωi = fϕdωϕ = fϕ ωe. ϕ∈F ϕ∈F left(e)=ϕ On the other hand we have ∗ 1 X Ä ä − h∂ f |ωi = − 2 fright(e) − fleft(e) ωe e∈E˜ 1 X 1 X = 2 fleft(e)ωe − 2 fright(e)ωe e∈E˜ e∈E˜ 1 X X 1 X X = 2 fϕωe − 2 fright($(e))ω$(e) ϕ∈F left(e)=ϕ ϕ∈F right($(e))=ϕ X X = fϕωe ϕ∈F left(e)=ϕ

103 7.6 ”Baby Riemann Roch Theorem”

So we have already found out that by the dimension formula

dim im d1 = |F| − 1 = |E| − dim ker d1.

This gives dim ker d1 = |E| − |F| + 1 which is the number of closed 1-forms on Mˆ . Analogously one can derive that dim im d0 = |V| − dim ker d0 = |V| − 1 where we sue that dim ker d0 = 1. Combining these we get Å ã β (Mˆ ) := dim H1(Mˆ ) = dim ker d1 = 2 − χ(M). 1 im d0

Definition 7.29 (First Betti-number). We define the first Betti-number of a discrete surface Mˆ to be

1 β1(Mˆ ) := dim H (Mˆ ).

Remark 7.30. Note that by the Poincare-Hopf´ index theorem, χ(Mˆ ) = χ(M) and therefore there is no typo in the equations above and the Betti- number does not depend on the cell-decomposition of M.

Theorem 7.31 (”Baby” Riemann-Roch). The first Betti-number of Mˆ is given by β1(Mˆ ) = 2 − χ(M).

Proof. This is the result of the observations above. To obtain a smooth version of the Riemann-Roch theorem we need to get rid of the ˆ in β1(Mˆ ). Consider the map

Ωk(M) −→π Ωk(Mˆ ), ω 7→ ωˆ .

Then

d0 d1 Ω0(M) Ω1(M) Ω2(M)

π0 π1 π2

Ω0(Mˆ ) Ω1(Mˆ ) Ω2(Mˆ ) dˆ0 dˆ1

104 Classification of Line Bundles which is by Stoke’s theorem a commutative diagram. By some diagram- chasing we get that π1(im d0) ⊂ im dˆ0 and π1(ker d1) ⊂ ker dˆ1 and thus a well defined map

ker dˆ $ : ker d1 −→ 1 , [ω] 7→ $([ω]) := [ωˆ ]. im d0 im dˆ0 | {z } | {z } =H1(M) =H1(Mˆ )

Since π1 is surjective the map $ is surjective.

Lemma 7.32. The map $ is also injective.

Proof. Let ω ∈ Ω1(M) and dω = 0. As we assumed the cell-decomposition to be diffeomorphic to convex sets there are (by the Poincare´ Lemma)

gv : Uv → R such that dgv = ω|Uv . These are unique up to an additive constant. Further $([ω]) = 0 means that ωˆ is exact, i.e. ωˆ = d fˆ for some fˆ ∈ Ω0(Mˆ ). We can interpret fˆ as a function on M which is constant on the dual faces Uv. As an exercise check that ω = d(g − f ) which then yields that [ω] = 0, hence yields the claim.

7.7 A Natural Complex Structure on Dual Spaces

We want to answer the following question:

”Let V be a vector space with complex structure J. What is the natural complex structure on V∗?”

More generally, if f : V → W is an isomorphism of vector spaces, how do linear forms ω ∈ V∗ transform under f ?

We are looking for f˜: V∗ → W∗ such that ¨ ∂ f˜(α), f (v) = hα, vi .

105 7.8 Holomorphic Structures on Vector Bundles

By rewriting the equality using the adjoint map we get ¨ ∂ ¨ ∂ f ∗ ◦ f˜(α), v = f˜(α), f (v) = hα, vi − which implies that f˜ = ( f ∗) 1.

Applying the observations above to f = J yields that the natural complex structure on V∗ is given by − Ä ä∗ (J∗) 1 = J−1 = −J∗.

Definition 7.33 (Hodge-star). For a Riemann-surface M, we have a complex structure J on TM. This gives rise to a complex structure ∗ = −J∗ on T∗ M, the so called Hodge-star.

Remark 7.34. i) We have that ∗ ∈ ΓEnd(T∗ M) and if ω ∈ Ω1(M), then for X ∈ Γ(TM) it holds that −ω(JX) = ∗ω(X), because ∗ω(X) = h∗ω|Xi = h−J∗ω|Xi = − hω|JXi = −ω(JX)

ii) There are many papers (including many that originate from Berlin, or the C-seminar by Franz Pedit) which use what we refer to as the ”Berlin convetntion” which is ∗ω(X) = ω(JX), i.e. ∗ = J∗. But this does only work for surfaces and causes trouble in higher dimensions.

7.8 Holomorphic Structures on Vector Bundles

∗ We have a natural splitting of HomR(TM, C) = T M ⊗ C into

HomR(TM, C) = K ⊕ K, where

K := {ω ∈ HomR(TM, C) | ω(JX) = Jω(X) for all X ∈ Γ(TM)} = {ω ∈ HomR(TM, C) | ∗ω = −Jω} and

K := {ω ∈ HomR(TM, C) | ω(JX) = −Jω(X) for all X ∈ Γ(TM)} = {ω ∈ HomR(TM, C) | ∗ω = Jω} .

106 Classification of Line Bundles

Definition 7.35 (canonical bundle). The bundle K that arises from the above splitting is called the canonical bundle of M.

∗C For a line bundle L we denote the bundle whose fiber at p ∈ M is Lp by L−1. Here ∗C has to be understood as the complex dual space, i.e. HomC(L, C). As the line bundle takes values in C, any nonzero z ∈ C has a natural inverse z−1. Also we note that the L−1 is indeed the same we know from the classification of line bundles. In this notation one often comes across (in particular in pure complex analysis literature) the term TM = K−1.

Definition 7.36 (canonical bundle splittings). For a complex vector bundle E over a Riemann surface M we define

KE := {ω ∈ HomR(TM, E) | ω(JX) = Jω(X) for all X ∈ Γ(TM)}

and

KE := {ω ∈ HomR(TM, E) | ω(JX) = −Jω(X) for all X ∈ Γ(TM)} .

Using the canonical bundle splitting we can decompose any ω ∈ Ω1(M, E) into ω = ω0 + ω00 |{z} |{z} ∈Γ(KE) ∈Γ(KE) so that more generally

Ω1(M, E) = Γ(KE) ⊕ Γ(KE).

If E is further equipped with a complex connection ∇ then we can introduce a splitting of the 1-form ∇ψ into

∇ψ = ∂ψ + ∂ψ¯ .

Thus in the notation above ∂ψ corresponds to ω0 and ∂ψ¯ to ω00 respectively. To summarize: Let M be a Riemann surface and E a complex line bundle over M with connection ∇. Then we can define ∂¯ : Γ(E) → Γ(KE¯ ) by

¯ 00 1 ∂ψ := (∇ψ) := 2 (∇ψ − J ∗ ∇ψ) . In particular, for the trivial bundle E = M × C with trivial connection ∇ = d we have Γ(M × C) = C∞(M, C) (”more or less the same thing as”) and 1 1 d f = 2 (d f + J ∗ d f ) + 2 (d f − J ∗ d f ) .

107 7.8 Holomorphic Structures on Vector Bundles

We check that these are indeed contained in the rightful spaces by comput- ing

1 ∗∂ f = 2 (∗d f − Jd f ) = −J∂ f ∗∂¯ f = ... = J∂¯ f .

Definition 7.37 (holomorphic sections). Let ∇ be a complex connection on a complex vector bundle E. Then ψ ∈ Γ(E) is called holomorphic if ∇ψ ∈ Ω1(M, E) is complex linear at all points, i.e. ∂ψ¯ = 0.

For f ∈ C∞(M, C) holomorphicity means

d f (JX) = id f (X) which, spelled out, is nothing more than the Cauchy-Riemann equations.

Definition 7.38 (holomorphic structure). A complex linear map

∂¯ : Γ(E) → Γ(KE)

is called a holomorphic structure (or ”∂¯-operator”) if

∂¯( f ψ) = (∂¯ f )ψ + f ∂ψ¯

holds for all f ∈ C∞(M) and ψ ∈ Γ(E).

Remark 7.39. Two things have to be noted. Sometimes one finds the nomen- clature ∂¯-operator instead of holomorphic structure. Also we solely need to ask for real valued C∞(M) functions f to satisfy the product rule. We get the complex valued functions f ∈ C∞(M, C) for free as we can compute for f = u + iv that

∂¯( f ψ) = ∂¯(uψ) + J∂¯(vψ) = (∂¯u)ψ + u∂ψ¯ + J(∂¯v)ψ + Jv(∂ψ¯ ) = ∂¯(u + iv)ψ + (u + iv)∂ψ¯ = (∂¯ f )ψ + f ∂ψ¯ .

It remains to check if ∂¯ operator that arise from complex connections are indeed holomorphic structures.

Proposition 7.40. If ∂¯ comes from a complex connection ∇ on E, then ∂¯ is a holomorphic structure.

108 Classification of Line Bundles

Proof.

¯ 1 ∂( f ψ) = 2 (∇( f ψ) − J ∗ ∇( f ψ)) 1 = 2 (d f ψ + f ∇ψ − J ∗ (d f ψ + f ∇ψ)) 1 1 = 2 (d f − J ∗ d f ) ψ + f 2 (∇ψ − J ∗ ∇ψ) = (∂¯ f )ψ + f (∂ψ¯ )

Ä ä Theorem 7.41. If ∂¯ is a holomorphic structure on E and ω ∈ Γ KEndCE then ∂¯ + ω is another holomorphic structure and all holomorphic structures on E arise in this way.

Proof. The simple computation

(∂¯ + ω)( f ψ) = (∂¯ f )ψ + f (∂ψ¯ ) + f ωψ = (∂¯ f )ψ + f (∂¯ + ω)ψ yields the first part of the theorem. If ∂¯1 and ∂¯2 are two holomorphic struc- tures on E then one can compute that

(∂¯2 − ∂¯1)( f ψ) = f (∂¯2 − ∂¯1)ψ which shows that (∂¯2 − ∂¯1) is a tensor. ∼ In particular for a line bundle L with holomorphic structure ∂¯ it is EndCL = M × C. So ∂ω¯ for some ω ∈ ΓK parametrizes all holomorphic structures on L.

7.9 Elliptic Differential Operators

Definition 7.42 (first order differential operator). Let E, F be vector bun- dles over a manifold M. Then a linear map D : Γ(E) → Γ(F) is called a first order differential operator, if there is A ∈ ΓHom(TM∗, Hom(E, F)) such that, for all ψ ∈ Γ(E), f ∈ C∞(M),

D( f ψ) = Ad f ψ + f Dψ .

A ∂¯-operator is a first order differential operator: Here A is the projection on the complex anti-linear part, i.e.

00 A : ω 7→ aω = ω .

109 7.9 Elliptic Differential Operators

Let us look at this locally: Choose a local frames ϕ1,..., ϕk on E and ˜ ϕ˜1,..., ϕ˜k˜ on E, say on the open neighborhood U. Then a section ψ ∈ Γ(E) can be written locally as X ∞ ψ = yi ϕi, yi ∈ C (U) .

Let further x` denote some coordinate functions of M on U. Then X X X X Dψ = (A ϕ + y Dϕ ) = ( ∂yi A ϕ + y Dϕ ) = ( ∂yi b + y a )ϕ˜ dyi i i i x` dx` i i i x` ij` i ij j i ` i,j,` A is uniquely determined by D and we make the definition:

Definition 7.43 (symbol of a differential operator). Consider a first order differential operator D, then the uniquely determined map A as from above is called the symbol of D.

Let M be a compact oriented n-dimensional manifold and E a vector bundle over M. For ψ ∈ Γ(E) and ω ∈ Ωn(M, E∗) we define Z hhω | ψii := hω|ψi . M This yields a non-degenerate pairing.

Definition 7.44 (adjoint map). If we have non-degenerate pairings between V˜ and V and between W˜ and W and we have a linear map B : V → W, then a linear map B˜ : W˜ → V˜ is called an adjoint of B if for all v ∈ V and w ∈ W˜ we have

¨ ˜ ∂ hw|Bvi = Bw v . Note that B˜ (if it exists) is unique. So we write B˜ = B∗.

In gereral one can prove the following theorem—which is basically integra- tion by parts.

Theorem 7.45. If E, F are vector bundles over a compact oriented manifold M and D : Γ(E) → Γ(F) is a first order differential operator, then D has an adjoint

D∗ : Ωn(M, F∗) → Ωn(M, E∗)

and D∗ is again a first order differential operator.

Fortunately this theorem is almost never needed in practice because D∗ can be written down explicitly in the concrete case.

110 Classification of Line Bundles

Definition 7.46 (elliptic operator). A first order differential operator

D : Γ(E) → Γ(F)

∗ is called elliptic if for all 0 6= ω ∈ Tp (M), p ∈ M the linear map

Aω : Ep → Fp

is a vectorspace isomorphism.

Theorem 7.47 (The elliptic theorem). Let E, F are vector bundles over a com- pact oriented manifold M and D : Γ(E) → Γ(F) is a elliptic first order differen- tial operator. Then:

i) dim ker D < ∞.

ii)D ∗ : Ωn(M; F∗) → Ωn(M; E∗) is also elliptic and im D = (ker D∗)⊥. In particular, (im D)⊥ = ker D∗.

Definition 7.48 (dual pairing). Let M be a compact oriented n-dimensional manifold and E, E˜ be vector bundles over M. Then a dual pairing between E and E˜ is a tensorial bilinear map

h.|.i : Γ(E˜) × Γ(E) → Ωn M

such that hω|ψip = 0 for all ψ implies that ωp = 0 and vice versa.

In particular Z hhω|ψii := hω|ψi M defines a non-degenerate pairing between Γ(E˜) and Γ(E), so Γ(E˜) may be viewed as (Γ(E))∗.

Given a dual pairing between E˜ and E yields a vector bundle isomorphism

E˜ → Ωn(M; E∗), ω 7→ ωˆ where hωˆ (X1,..., Xn)|ψi = hω|ψi(X1,..., Xn) . Example 7.49. E = ΛkTM∗ and E˜ = Λn−kTM∗ have a dual pairing

hω|ηi = ω ∧ η .

111 7.9 Elliptic Differential Operators

On a Riemannian manifold the Hodge-star operator

∗ : ΛkTM∗ → Λn−kTM∗ is given by ω ∧ ∗η = hω, ηi det . It satisfies ∗∗ = (−1)k(n−k). The Hodge-star yields a dual pairing of ΛkTM∗ with itself. This pairing extends to the exterior algebra E = ΛTM∗ such that forms of different degree pair to zero.

Let ΩM = ΓE. The exterior derivative

d : ΩM → ΩM is a first order differential operator with symbol

Ad f (ω) = d f ∧ ω, hence d( f ω) = d f ∧ ω + f dω . It has an adjoint which is d∗ = δ = ± ∗ d∗ where the sign depends on dimension. E.g. in case n = 2 one may check that d∗ = δ = − ∗ d∗. Moreover, δ( f ω) = εk ∗ (d f ∧ ∗ω) + f δω .

Theorem 7.50. D : ΩM → ΩM,D = d + δ is a self-adjoint elliptic first order operator, called the Hodge-Dirac operator.

Proof for n = 2.D ( f ψ) = f Dψ + d f ∧ ψ − ∗(d f ∧ ∗ψ). Thus

Ad f ω = d f ∧ ω − ∗(d f ∧ ∗ω) .

∗ For 0 6= η ∈ Tp M ,

Ö f è Ö− ∗ (η ∧ ∗ω)è η ∧ ω = 0 f = 0 Aη ω = f η − g ∗ η ⇒ η ∧ ∗ω = 0 ⇒ ω = 0 g det η ∧ ω f η − g ∗ η = 0 g = 0

Thus Aη bijective.

Theorem 7.51. Let M be a connected n-dimensional compact oriented manifold. Then Z σ ∈ Ωn M exact ⇐⇒ σ = 0 . M

112 Classification of Line Bundles

Z Proof. ⇒ follows from Stokes’ theorem. ”⇐”: Let σ = 0. If ( f , ω, g det) ∈ M ker D, then Z 0 = (d f − ∗dg) ∧ (∗d f + dg) = kd f k2 + kdgk2 M and hence f and g are constant. In particular, (0, 0, σ) ∈ (ker D)⊥ = im D. Thus σ is exact. From now on let M denote a compact Riemannian surface, E a complex vector bundle over M and L a complex line bundle over M. Let ∂¯ be a holomorphic structure on E:

∂¯ : ΓE → ΓKE¯ , ∂¯( f ψ) = (∂¯ f )ψ + f ∂ψ¯ .

∗ 00 00 If ω ∈ T M then Aωψ = ω ψ. In particular, if ω 6= 0, then ω 6= 0 and hence ψ 7→ Aωψ is bijective. Example 7.52. 1. If ∇ is a connection on E, then ∇ = ∇0 + ∇00 and ∂¯∇ = ∇00 is a ∂¯-operator.

2. Let E = M × C, ∇ = d, then ∂¯∇ = ∂¯ is the canonical ∂¯ on C∞(M; C).

3. Consider E = TM and let I be the identity (tautological 1-form). Then d∇ I = T∇ – the torsion of the connection. Suppose a complex con- nection ∇ on TM is torsion-free. Then

¯∇ 1 1 ∂X Y = 2 (∇XY + J∇JXY) = 2 (∇XY − ∇YX + J[JX, Y]) 1 = 2 ([X, Y] + J[JX, Y]) . The last example is worth to be made into a theorem.

Theorem 7.53. All torsion-free complex connections ∇ on TM have the same ∂¯∇ =: ∂¯ given by ¯ 1 ∂XY = 2 ([X, Y] + J[JX, Y]) .

What about KL = Hom+(TM, L) if L comes with a complex connection ∇? Again, let ∇M be a torsion-free complex connection on TM. Then

ˆ M (∇Xω)(Y) := ∇X(ω(Y) − ω(∇X Y) defines a complex connection. Then

¯∇ˆ 1 Ä M M ä (∂X ω)(Y) = 2 ∇X(ω(Y) − ω(∇X Y) + J∇X(ω(Y) − Jω(∇X Y) ¯∇ ¯ = ∂X (ω(Y)) − ω(∂XY) .

113 7.9 Elliptic Differential Operators

The exterior derivative of ω ∈ Ω1(M; L) is given by

∇ d ω(X, Y) = ∇Xω(Y) − ∇Yω(X) − ω([X, Y]).

For Y ∈ ΓTM we have d∇ω(., Y) ∈ Ω1(M; L) and so (d∇ω(,.Y))00 ∈ ΓKL¯ . Thus

∇ 0 1 Ä (d ω(., Y)) (X) = 2 ∇Xω(Y) − ∇Yω(X) − ω([X, Y]) ä + J(∇JXω(Y) − ∇Yω(JX) − ω([JX, Y])) ¯∇ ¯ = ∂X (ω(Y)) − ω(∂XY) .

In particular, if L = M × C and ∇ = d:

∇ 00 (∂¯ Xω)(Y) = ∂¯ X(ω(Y)) − ω(∂¯ XY) = (d ω(., Y)) (X) .

Behind this fact, there is some identification: To each σ ∈ Ω2(M; L) we can Z assign σˆ Γ(K¯(KL) given by

1 σˆX(Y) = 2 (σ(X, Y) + Jσ(JX, Y)) . One may check that σˆ is of type K in the second slot:

1 1 σˆX(JY) = 2 (σ(X, JY) + Jσ(JX, JY)) = 2 (−σ(JX, Y) + Jσ(X, Y)) = JσˆX(Y) .

Theorem 7.54. The map

Φ : Λ2(M, L) → K¯(KL), σ 7→ σˆ

is an isomorphism of complex line bundles such that Φ ◦ d∇ = ∂¯.

The fact that Φ ◦ d∇ = ∂¯ is equivalent to saying that the following diagram commutes:

∂¯ KL K¯ (KL)

d∇ σ 7→ σˆ

Λ2(M, L)

Theorem 7.55. ω ∈ ΓK is closed ⇔ ω holomorphic.

Remark 7.56. Note that if L is a holomorphic line bundle (i.e. L comes with ∂¯ ), then we automatically get ∂¯ on KL by the identification above.

114 Classification of Line Bundles

Theorem 7.57. If L, L˜ are holomorphic line bundles. Then there is a unique holomorphic structure on L ⊗ L˜ such that

∂¯(ψ ⊗ ψ˜) = ∂ψ¯ ⊗ ψ˜ + ψ ⊗ ∂¯ψ˜

Proof. If locally f , g ∈ C∞(U), f · g = 1, then... Let M be a compact Riemann surface of genus 1 g = 1 − 2 χ(M) and L → M be a holomorphic line bundle.

Definition 7.58.

(i) A section ψ ∈ Γ(L) is called holomorphic if ∂ψ¯ = 0.

(ii)H 0(L) := ker ∂¯ denotes the set of all holomorphic sections of L.

(iii)h 0(L) := dim H0(L).

Theorem 7.59 (Riemann–Roch theorem). Let M be a compact Riemann sur- 1 face of genus g = 1 − 2 χ(M) and L → M be a holomorphic line bundle. Then

h0(L) − h0(KL−1) = deg L + 1 − g .

All zeros of a holomorphic section have positive index. Hence we immedi- ately obtain the following theorem.

Theorem 7.60. If deg L < 0, then h0(L) = 0.

Corollary 7.61. deg L > 2g − 2, then h0(L) = deg L + 1 − g.

Let us first prove the Riemann–Roch theorem for the case that L is the trivial bundle: C∞(M, C)

∂¯

ΓK ΓK¯

d

Ω2(M, C)

115 7.9 Elliptic Differential Operators

We have the diagramm above where the dashed arrows implicate identifi- cation. For σ ∈ Ω2(M; C), f ∈ C∞(M; C) we have a pairing Z hσ| f i = f σ . M Similarly, for ω ∈ ΓK, η ∈ ΓK¯ we have a pairing Z hω|ηi = ω ∧ η . M Since

ω ∧ η(X, JX) = ω(X)η(JX) − ω(JX)η(X) = −2Jω(X)η(JX). we find that this pairing is non-degenerate.

Theorem 7.62. If M is connected, then h0(M × C) = 1, i.e. only constant functions are holomorphic.

The proof uses the following hermitian inner product on ΓK: Z 1 hω, ηi = i ω¯ ∧ η . 2 M This means that

hλω, ηi = λ¯ hω, ηi and hω, ληi = λ hω, ηi and we see that Z Z 1 1 hω, ηi = − i ω ∧ η¯ = i η¯ ∧ ω = hη, ωi . 2 M 2 M Proof. Clearly constant functions are holomorphic. We show that any holo- morphic function is constant. So suppose that ∂¯ f = 0. Then

0 = d2 f = d(∂ f + ∂¯ f ) = d∂ f .

Using Stokes’ theorem, we get Z Z Z 1 1 1 h∂ f , ∂ f i = i ∂ f ∧ ∂ f = i ∂ f ∧ d f = i f d∂ f = 0 2 M 2 M 2 M Hence ∂ f = 0, so d f = 0 and f must be constant.

Z Theorem 7.63. σ ∈ Ω2(M; C) exact ⇐⇒ σ = 0. M

116 Classification of Line Bundles

Proof.

σ ∈ im (d : ΓK → Ω2(M; C)) ⇐⇒ σ ⊥ ker (∂¯ : C∞(M; C) → ΓK¯) = C · 1 Z ⇐⇒ 0 = hσ|1i = σ . M

Theorem 7.64 (Elliptic theorem - a Riemann-Roch version for L = M × C on ∂¯-problem). Let η ∈ ΓK.¯ Then the ∂¯-problem

∂¯ f = η is solvable if and only if η ⊥ H0(K).

Proof. To see this note that, for f ∈ C∞(M; C) and ω ∈ ΓK, Z Z Z h∂¯ f |ωi = ∂¯ f ∧ ω = d f ∧ ω = − f dω . M M M

Hence d = −∂¯∗. The minus sign is not important for image and kernel. Again, let us consider the following diagram:

C∞(M, C)

∂¯

ΓK ΓK¯

d

Ω2(M, C)

The pairings are given by Z hhω|ηii = ω ∧ η, ω ∈ ΓK, η ∈ ΓK¯ , M Z hh f |σii = f σ, f ∈ C∞(M; C), σ ∈ Ω2(M; C) . M We have Z Z Z Z hh f |dωii = f dω = − d f ∧ ω = − ∂¯ f ∧ ω = ω ∧ ∂¯ f = hhω|∂¯ f ii . M M M M Thus d = ∂¯∗ .

117 7.9 Elliptic Differential Operators

Let us keep in mind the following calculation: On M = C we have dz ∈ ΓK. 1 dz = dx + idy. Then dz ∧ dz = 2idx ∧ dy and so dx ∧ dy = 2i dz ∧ dz¯. Then a positive hermitian product h., .i on ΓK¯ is given by

1 hhω, ηii = 2i hhω|ηii . By the elliptic theorem we have that

Im ∂¯ = (H0K)⊥, Im d = (C1)⊥ .

Recall that ker(d : Ω1(M; R) → Ω2(M; R)) 2g = dimR . im(d : C∞(M; R) → Ω1(M; R)) To each α ∈ Ω1(M; R) there corresponds ω ∈ Γ(K¯) given by 1 00 ¯ ω = 2 (α + iα ◦ J) = α ∈ ΓK . The harmonic forms are those which are closed and co-closed:

harm (M) := {α ∈ Ω1(M; R) | dα = d ∗ α = 0} .

Obviously, α ∈ harm (M) ⇔ α00 ∈ H0 M and ω ∈ H0 M ⇔ Reω ∈ harm (M).

Theorem 7.65 (Hodge–theorem). Let η ∈ Ω1(M; R) be closed, dη = 0. Then there is a unique α ∈ harm (M) and f ∈ C∞(M;R) (unique up to a constant) such that η = α + d f .

0 Proof. Let ω1,..., ωk be a basis of H (K) such that ω¯1,..., ω¯k ∈ ΓK¯ are X ˜ = 00 − hh ¯ ii ¯ h ¯ ˜i = j orthonormal. Then η : η j ωj, η ωj will satisfy ωj, η 0 for all , i.e. η˜ ⊥ ker d, and by the elliptic theorem we get η˜ ∈ im ∂¯. So η˜ = ∂¯ f for X f ∈ C∞(M C) 00 = hh ¯ 00ii ¯ 00 ; . Let α : j ωj, η ωj. To α corresponds a real-valued harmonic form α ∈ harm (M). Now, 00 ¯ 00 1 1 η = ∂ f + α = 2 (d f − i ∗ d f ) + 2 (α − i ∗ α) . If we take the real part then we get, with f = u + iv,

η = du + ∗dv + α .

Since dη = 0, we obtain d ∗ dv = 0 and hence v must be a constant: Z Z 0 = − vd ∗ dv = dv ∧ ∗dv = kdvk2 M M

118 Classification of Line Bundles

In other words: each homology class [η] ∈ H1(M) contains exactly one harmonic representative α. Thus

1 0 2g = dimR H (M) = dimR harm (M) g = dimC H (M) ; where 2g − 2 = deg K.

Theorem 7.66. Every complex holomorphic line bundle of degree zero over M has a flat connection ∇ with ∇00 = ∂¯.

Proof. Choose one connection ∇ with ∇00 = ∂¯ and look for ω ∈ ΓK such that ∇˜ = ∇ + iω has curvature R˜ = R + idω = i(dω − Ω) = 0, i.e. dω = Ω, Z which is solvable by elliptic theorem because 0 = 2πdeg (L) = Ω. M

Corollary 7.67. Every holomorphic line bundle over a (not necessarily compact) Riemann surface M locally has a holomorphic section without zeros.

Proof. Glue the bundle locally in a degree zero bundle and use the previous theorem.

Theorem 7.68. Any almost complex surface is complex.

Proof. Locally there is a holomorphic basis section ω ∈ ΓK¯, dω = 0. Since closed forms are locally exact, there is a locally defined complex-valued function z such that ω = dz. z is a holomorphic chart.

Corollary 7.69. Let L be a holomorphic line bundle over a Riemann surface M. Then, expressed in a local holomorphic chart z : U → C and using a local holomorphic basis section ϕ, any holomorphic section ψ is of the form ψ = f (z)ϕ for some holomorphic function f : z(U) → C.

Thus things locally behave as expected. We can define meromorhic sections, removable singularities, etc. also for holomorphich sections of holomorphic line bundles.

Theorem 7.70 (Riemann–Roch). Let L be a holomorphic line bundle over a compact Riemann surface M of genus g. Then

h0(L) − h0(KL−1) = deg L − g + 1 .

This is a special case of the Atiyah–Singer index theorem on the index of an elliptic operator. Recall again the situation.

119 7.9 Elliptic Differential Operators

ΓL

∂¯

ΓKL−1 ΓK¯L

d∇

Ω2(M; L)

We break this into several lemmata. If p ∈ M there is a holomorphic line bundle (p) of degree 1 which has a canonical section ψ ∈ H0(L) with only one simple zero at p—the skyscraper bundle. If L and L˜ are two holomorphic line bundles with sections ψ ∈ H0L and ψ˜ ∈ H0L˜ with only one simple zero p. Then ρ(zψ) = zψ˜ determines an isomorphism of holomorphic bundles—on all of M.

Lemma 7.71. Riemann–Roch holds for L ⇔ Riemann–Roch holds for L(p).

Which, by h0(L(p)) − h0(KL−1(−p)) = deg L + 1 − g + 1, itself follows from

Lemma 7.72. h0(L(p)) − h0(L) = h0(KL−1(−p)) − h0(KL−1) + 1

Proof. Let ϕ ∈ Γ(p) be the canonical holomorphic section of (p). If ψ is a meromorhic section of L with only a simple pole at p and otherwise holomorphic. Then ψϕ ∈ H0(L(p)). Conversely, if ψ˜ ∈ H0(L(p)), then ψ˜/ϕ has at most a simple pole at p. Thus

0 0 h (L(p)) = dim{ψ ∈ H (L|M\{0}) | at most a simple pole at p} .

0 0 0 If h (L(p)) > h (L), then there is ψ ∈ H (L(p)) with ψp 6= 0—otherwise division by ϕ would yield an injective map H0(L(p)) → H0(L). Now suppose we would have h0(KL−1(−p)) > h0(KL−1). Then we would have an L−1-valued meromorphic 1-form ω with a simple pole in p.

0 0 h (L(−p)) = dim{ψ ∈ H (L) | ψp = 0} .

0 −1 0 −1 If ψp ∈ H (L), ψp = 0, then ψϕ ∈ H (L(−p)), where ϕ denotes the canonical section of (−p). Then hω|ψi meromorphic C-valued 1-form on M with a single simple pole at p, which cannot exists by the residue theorem (residues must sum up to zero — follows immediately from Stokes’ theorem). Hence we have

120 Classification of Line Bundles h0(KL−1(−p)) = h0(KL−1). So, if h0(L) goes up by tensoring in (−p), then h0(KL−1) does not. Now

L˜ := KL−1(−p), L˜ (p) = KL−1, KL˜ −1 = KK−1L(p) = L(p), KL˜ −1(−p) = L .

The same arguments as before applied to L˜ , yields then that if h0(KL−1) does not go up, then h0(L) goes up. The dimension can increase at most by one (why?). So we have

h0(L(p)) − h0(L) ∈ {0, 1}, h0(KL−1) − h0(KL−1(−p)) ∈ {0, 1} .

Furthermore,

h0(L(p)) − h0(L) = 1 ⇔ not (h0(KL−1(−p)) − h0(KL−1) = 1).

It follows that h0(L(p)) − h0(L) = 1 − h0(KL−1) + h0(KL−1(−p)). If we keep tensoring in (p), the bundles on the right hand side become of negative degree. Hence after a while, h0(L(kp)) > 0. Hence L has a meromorphic section with a k-order pole at p.

Theorem 7.73. Every holomorphic line bundle over a compact Riemann surface M has a meromorphic section.

Tensoring in the point bundles corresponding to the zeros and poles of this meromorphic section we arrive at the trivial bundle. So we can apply Riemann–Roch for this case.

121 8. Appendix

The aim of this appendix is to give a short overview over additional top- ics of interest concerning complex analysis and Riemann surfaces. In the first place the plan will be to summarize the main ideas that are discussed within the turtorials. Some of the presented summaries are due to Theo Braune who volunteered to take on the burden to type them.

On the Stereographic Projection

2 Vividly, the stereographic projection from the north pole σN : S \{N} → C, 2 or the south pole σS : S \{S} → C respectively, provides us with a bijec- tion of the punctured Riemann sphere to C. It now may be interesting to have an explicit formula for the map for various reasons, one of which will be pointed out soon.

Consider for example σN, then we can parameterize the straight line along which we project, say γ, by

Ö0è Öxè γ(t) = tN + (1 − t)P = t 0 + (1 − t) y . 1 z

−z 1 Then γ(t) = 0 yields t = 1−z thus 1 − t = 1−z . Hence we see that in cartesian coordinates the map is given by

Öxè x y σ : S2 \{N} → C, y 7→ + i . N − z − z z 1 1

In a similar manner we can conclude that x y σ (x, y, z) = + i . S 1 + z 1 + z 122 Appendix

The respective inverses of σN and σS are given by

Å ( ) ( ) −| |2 ã σ−1(z) = 2Re z , 2Im z , − 1 z N 1+|z|2 1+|z|2 1+|z|2

2 Considering the two maps σN, σS as charts of S then we can ask ourselves how coordinate changes look like. As an example we take the coordinate change from σN to σS, then for z ∈ C \{0} it is 1 σ ◦ σ−1(z) = . S N z¯

It turns out that the same holds for the coordinate change from σS to σN. 1 Moreover, if we choose σ¯S instead of σS, then (σN, σ¯S) form an atlas of S −1 1 and the transition map is σ¯S ◦ σN = z and in particular holomoprhic. As 1 z 7→ z is a holomorphic map, coordinate changes on the Riemann sphere S2 = C¯ are holomorphic what turns it into a Riemann surface.

On Topological Manifolds

In the second chapter, we said that we want the kind of topological spaces that we deal with to have nice properties and therefore demanded them to be Hausdorff, second countable and to locally ”look like” euclidean space. To understand why all three of these properties are reasonable to ask for, we now want to come up with examples of topological spaces that fail to fulfill exactly one of them. Hausdorff: A second countable, locally euclidean space that is not Haus- dorff is for example given by the line with two origins. Define [ X = R × {a} R × {b}

123 and define an equivalence relation (x, a) ∼ (x, b) :⇔ x 6= 0.

The resulting space X∼ then looks like the real line just with two different zeros 0 and 00.

It is easy to imagine that if we consider both zeros, any open neigh- bourhoods of these will intersect. The details are left as an exercise. 2nd countable: A locally euclidean Hausdorff space that does not fulfill the second axiom of countability is given by X = (R, D) × (R, S) where D is the discrete topology and S the standard topology on R. A basis of X is given by {{x} × (a, b) | x, a, b ∈ R, a < b}. Using the insight that this indeed is a Basis of X immediately gives an idea how to get a contradiction if we assume the existence of a countable basis of X. Again the details are left as an exercise. locally euclidean: A standard example for a space that is not locally eu- clidean is given by a curve with one end arbitrary close to its own trajectory, for example consider  (0, |t|) , t ∈ (−2, 0]  ( ( πt) + ( πt)) t ∈ ( 3 ) (− ) → R2 7→ cos 2 1, sin 2 , 0, 4 γ : 2, 1 , t 3 !  −t 3  1 + 4 , 1 , t ∈ [ , 1)  1 4 4 The trajectory of the curve looks like this:

124 Appendix

Again, the details are left as an exercise, but it is easily imaginable that it is not locally euclidean at (0, 1).

On Linear Complex Structures

In the lecture we have learned, that a complex structure is determined by the linear map J : V → V. This is because a map f : R2n → R2m is defined to be holomorphic if it is continuously differentiable in the real sense and it is complex-linear, i.e. d f (Jv) = Jd f (v) holds. Although it looks clear what is going on, the term

d f (JR2n v) = JR2m d f (v) has the potential to provide trouble, because, as we now have completely written it in detail, we see that the abbreviation AJ = JA defrauds that in particular we have to deal with two different complex structures, one for the domain and one the range, that have to fit together for holomorphicity.

Consider for example C =∼ R2. If we want to complexify R2, we have to find a linear map J : R2 → R2 such that J2 = −I. Then we can define

(a + ib)v := (a + bJ)v.

As we already know what the complex structure on C is, namely i which is geometrically nothing more than a 90◦-rotation, we can simply make use of the 90◦-rotation matrix that we know from linear algebra and define

ñ0 −1ô J := , 2 1 0 then this is clearly linear and it satisfies J2 = −I. Also we see the expected analogy to the actual complex number, meaning as x + iy ↔ [x y]T we have the identification

ñxô ñ−yô i(x + iy) = −y + ix ↔ J = . 2 y x

Note that also J˜2 := −J2 would do the trick. This simply correspnds to changing the orientation of C. But as a matter of fact, these are the only possible choices for the case C =∼ R2 if we demand our complex structure to preserve angles and length. Check for example that for b ∈ R

ñ0 −bô Jb := 1 b 0 125 also defines a complex structure, but apparently lacks of this property if b 6= 1.

Let us consider a higher dimensional example: C2 =∼ R4. The question of how many possible orthogonal 90◦-rotations there possibly are is much harder to answer in this setup. It turns out that there in fact are S2-may possibilities.

Our first approach to get this insight is based on quaternions. We know that R4 =∼ H where

H := {x0 + ix1 + jx2 + kx3 | x0, x1, x2, x3 ∈ R} with i2 = j2 = k2 = ijk = −1. Choose for example 1 ∈ H, then the orthog- onal complement 1⊥ =∼ R3, because h1, ii = h1, ji = h1, ki = 0. As we aim for orthogonal, the 2-sphere S2 ⊂ R3 are all quaternions of unit length that correspond to a orthogonal 90◦-rotation.1

Another suitable Approach that may acquire intuition is illustrated by the picture below.

Having chosen v ∈ R3, assume |v| = 1 then the orthogonal complement v⊥ of v is the plane where the equator of S2 with respect to v is inscribed. So any map J that maps v to the equator is an orthogonal 90◦-rotation. But note that for our actual setup v ∈ R4, thus S2 becomes S3 and the the equator, as it is an S1 becomes an S2, thus we have S2 many possible orthogonal 90◦-rotations, meaning that

¶ Ä 4ä 2 © ∼ 2 J ∈ EndR R | J = −1 = S .

Here again, allowing scaling factors leads to non-orthogonal complex struc- tures of which we can have much more. Moreover we can characterize the invertible complex linear maps in terms of the real ones, namely

2n AutC(R ) = GLn(C) = {A ∈ GL2n(R) | AJ = JA} 1Consider the set of all unit length quaternions S3 ⊂ R4, then we see that the subset of all purely imaginary ones, meaning an equator of S3, is a 2-sphere.

126 Appendix

Example 8.1. Define a map

ñ x + z ô F : R4 → R2, (x, y, z, w) 7→ . y + w

2 A complex structure for R is given by J2 from the observations above and for R4 ñ ô J2 0 J4 := 0 J2 defines a complex structure.

Differentiating F yields ñ1 0 1 0ô dF = 0 1 0 1 hence F is clearly continuously differentiable in the real sense. Further we have for v ∈ R4 ñ ô −v2 − v1 J2dF(v) = v1 + v3 hence F is holomorphic dF(Jv) yields the same result. Fortunately it is

ñ ô −v2 − v1 dF(J4v) = , v1 + v3 thus F is holomorphic.

Note that changing the complex structure on a vector space corresponds to a change of basis. Therefore if we want to check if our map F is holomor- phic with respect to another J we would have change the matrix represen- tation of d f according to this change of basis as well. We will not need to do this, as the following theorem holds.

Theorem 8.2. Let V be a real even dimensional vector space and J1, J2 : V → V two different complex structures. Then the following holds:

(i)J 1 and J2 are conjugate to each other, i.e. there is a vector space isomor- −1 phism T : V → V , such that J1 = T J2T.

(ii)f : V → V is holomorphic with respect to J1 if and only if it is holomorphic with respect to J2.

Proof. The proof will be a homework exercise.

127 On Algebraic Curves

In this part we want to consider the zero set of complex polynomials and their relation to Riemann surfaces. To deal with these we will recall the implicit function theorem. Here we will use a slightly stronger version than the usual one for the Rk, but is assures that we can locally represent the zero set of some holomorphic function in two variables as the graph of holomorphic map.

Theorem 8.3 (Implicit function theorem for holomorphic functions). 2 Suppose that (z0, w0) is a point in X = {(z, w) ∈ C |p(z, w) = 0} and

p ∂ 6= 0. ∂w (z0,w0)

Then there is a disc D1 centered at z0 and D2 centered at w0 and a holomorphic map φ : D1 → D2 with φ(z0) = w0, such that:

X ∩ (D1 × D2) = {(z, φ(z))|z ∈ D1}.

Affine Curves

Let p : C2 → C be a polynomial and X = {(z, w) ∈ C2|P(z, w) = 0} the zero set. Suppose that at least one of the partial derivatives does not vanish. We will now show that X is a Riemann surface.

∂p Case 1): Suppose that we have (z0, w0) ∈ X with ∂w 6= 0. Then we consider the set Uα = (D1 × D2) ∩ X with (z0, w0) ∈ D1 × D2. We denote the projection map onto the first coordinate by

π1 : D1 × D2 → D1 : (x, y) → x.

Then φα := π1|Uα is a coordinate chart of Uα.

128 Appendix

∂p Case 2): Now suppose that we have (z1, w1) ∈ X with ∂z 6= 0. Then we can define the set Uβ = (B1 × B2) ∩ X, such that (z1, w1) ∈ B1 × B2. This time, we consider the projection onto the second component and

denote it by π2. We define the chart φβ := π2|Uβ .

It remains to show that the defined carts are compatible.

It suffices to consider the case Uα ∩ Uβ 6= ∅, because in the case of Uα ∩ Uα0 6= ∅ we just obtain the identity map as coordinate change. So let z ∈ Uα ∩ Uβ be arbitrary. Then we observe:

−1 φ φβ z 7−→α (z, f (z)) 7−→ p(z)

This is clearly a holomorphic map since p was a polynomial! Thus X is a Riemann surface. But note that it is not necessarily compact.

Projective Curves Recall: The complex projective space CPn is defined as

Cn+1 \{0} ∼ where v ∼ w :⇔ v = λw for λ ∈ C \ {0} .

Figure 8.1: One possible way to visualize about RP2. We can think of CPn in a similar manner.

n Let [z0, ..., zn] = [λz0, ..., λzn] ∈ CP be arbitrary. We call these coordinates homogeneous coordinates. Due to the fact that we consider equivalence classes we can scale with our factor λ in such a way that that we can have 1 or 0 in our first component. This gives us the opportunity to transfer a problem in n variables in Cn into CPn, which can be useful in many applications.

129 Definition 8.4 (Homogeneous polynomial). A polynomial p is called ho- mogeneous polynomial of degree d if

X ( ) = i0 · · in p z0, ..., zn ai0,...,in z0 ... z i0,...,in

Pn where j=0 ij = d and ik ∈ N.

Let us now focus on the case that n = 2.

We consider a homogeneous polynomial p(z0, ..., z2) of degree d. We as- sume that z0 does not divide p. We define the set

2 X¯ = {[z0, z1, z2] ∈ CP |p(z0, z1, z2) = 0}.

Now we consider a slightly different polynomial p˜(w, z) = p(1, w, z). We note that p˜(w, z) defines an affine curve in CP2. If additionally we have ∂p˜ ∂p˜ , 6= 0, ∂w ∂v we can use our theory from above. We can conclude that the set

2 X0 = X¯ ∩ {[z] ∈ CP |z0 6= 0} is a Riemann surface. If we put the ideas together, we obtain the following theorem:

Theorem 8.5. Suppose p(z0, z1, z2) is a homogeneous polynomial of degree d ≥ 1 and the only solution of the equation ∂p = ∂p = ∂p = 0 is (0, 0, 0). Then ∂z0 ∂z1 ∂z2 the solutions of the equation p = 0 in CP2 form a compact Riemann surface.

Proof. The proof is left as a homework exercise.

On Connections on Immersed Surfaces in R3

In the lecture we defined a connection as a substitute for a derivative on vector bundles, but apart from the trivial connection on the trivial bundle, which mimics the usual derivative, we lack of examples.

We will show a way of how to gain a canonical connection, the so called co- variant derivative on immersed surfaces f : U → R3 and convince ourselves that this really defines a connection for the case of M = S2 ⊂ R3.

130 Appendix

So consider the unit sphere S2 ⊂ R3. On R3, seen as a vector bundle, we have the trivial connection denoted by d. A smooth map XS2 → R3 is a vector field on S2 if and only if ¨ ∂ Xp, p = 0 for all p ∈ S2, where h., .i is the usual euclidean scalar product on R3.

We define a connection on S2 by ¨ ∂ (∇XY)p := dpY(Xp) − dpY(Xp), p p which is nothing more than the projection of the usual derivative on R3 to the respective tangent plane of S2.

Proposition 8.6. The map ∇ as defined above is a connection on S2.

Proof. The bilinearity is clear as both, d and h., .i are bilinear. Let f ∈ C∞(S2) and X, Y ∈ Γ(TS2) then

∇ f XY = dY( f X) − hdY( f X), pi p = f X · Y − f hX · Y, pi p = f (dY(X) − hdY(X), pi p)

= f ∇XY so ∇ is tensotial in X. Further

∇X( f Y) = d( f Y)X − hd( f Y)X, pi p = X · ( f Y) − hX · ( f Y), pi p = (X f )Y + f dY(X) − h(X f )Y, pi p − f hdY(X), pi p

= (X f )Y + f ∇XY

131 where we used that h(X f )Y, pi = 0 as Y is a vector field on S2. Thus ∇ satisfies the product rule in the Y-component. The last thing to check is 2 that ∇ is well defined, that is that ∇XY really is a vector field on S . We have

h∇XY, pi = hdY(X) − hdY(X), pi p, pi = hdY(X), pi − hdY(X), pi hp, pi = 0 as hp, pi = 1 because p ∈ S2. As already mentioned, the above construction yields a connection for any immersed surface in R3.

On Differential Forms

We start with the following basic observation:

Consider a smooth manifold M and some point p ∈ M. Identify a coor- dinate chart ϕ = (x1,..., xn) that is defined in a neighborhood of p with x = (x1,..., xn).

Then for a tangent vector u ∈ Tp M we have

n X ∂ up = ui(p) ∂xi p i=1 and as the chart provides a frame-field in a neighborhood around p we have n X u = u ∂ . i ∂xi i=1 For some function f ∈ C∞(M) we get n X u f = u ∂ f = d f (u). i ∂xi i=1

132 Appendix

An important special case fo this is f = xj where xj is the j-th coordinate projection. Then the upper forumula yields

n X ∂x ux = u j = u = dx (u) j i ∂xi j j i=1 and evaluating at p gives

n X ∂xj (uxj)p = ui(p) = dxj (up) ∂xi p p i=1 → R ∈ ∗ thus as dxj p : Tp M is linear and dxj p Tp M we more generally get that n X d f ∈ T∗ M, d f = ∂ f (p)dx . p p p ∂xi i p i=1 Use the frame field for the chart (U, x) to get sections of T∗U the so called cotangent bundle.

Remark 8.7. Let V be a Hilbert space. By the Riesz-representation theorem for each x∗ ∈ V∗ and y ∈ V there is x ∈ V such that

x∗y = hx, yi .

This leads to the definition of the so called musical isomorphisms which turn a vector into a covector and a covector into a vector. This works as follows

] : T∗ M → TM, v∗ 7→ (v∗)] = v

[ : TM → TM, v 7→ v[ = v∗ where the equality has to be understood in the sense of Riesz-represetnation above. In the lecture we defined

Λk(V) := {ω : V × ... × V → R | ω is k-linear and alternating} .

If one has ω ∈ Ωk(M) and η ∈ Ω`(M), then is there some kind of ”multi- plication” that gives us a k + `-form from ω ane η?

Naturally one could try and use the usual tensor product defined by

(ω ⊗ η)(X1,..., Xk, Y1,..., Y`) := ω(X1,..., Xk)η(Y1,..., Y`) but the problem is, that in general

ω ⊗ η ∈/ Ωk+`(M).

133 Remedy is provided by the following alternation map which is defined as follows:

1 X Alt(ω )(X ,..., X ) := sgn σ ω (X ,..., X p 1 k k! p σ(1) σ(k) σ∈Sk which acts as an orthogonal projection in the space of tensors onto the space of alternating multilinear forms. Thus we can define the following

Definition 8.8 (Wedge product). Let ω ∈ Ωk(M) and η ∈ Ω`(M), then a k + `-form ω ∧ η ∈ Ωk+`(M) on M is defined by

(k + `)! ω ∧ η(X ,..., X +`) := Alt(ω ⊗ η)(X ,..., X +`). 1 k k!`! 1 k

Remark 8.9. Λk can also be seen as a quotient on the tensor space with the equivalence relation that is given by

ω ⊗ η ∼ ω˜ ⊗ η˜ :⇔ Alt(ω ⊗ η) = Alt(ω˜ ⊗ η˜).

Example 8.10.

dx ∧ dy = 2 Alt(dx ⊗ dy) = dx ⊗ dy − dy ⊗ dx

Thus if we plug in vector fields X and Y we get

dx ∧ dy(X, Y) = dx(X)dy(Y) − dx(Y)dy(X) = X1Y2 − X2Y1 which is the usual formula for the derivative. The following is majorly copied from the Differential Geometry 2 script where differential forms in general are usually introduced. Therefore most of the things are in a much more general setup then we will use them, because as we are working with Riemann-surfaces there will only be 1-, or 2-forms.

Bundle-Valued Differential Forms

Definition 8.11 (Bundle-valued differential forms). Let E → M be a vector bundle. Then for ` > 0 an E-valued `-form ω is a section of the   bundle Λ`(M, E). We write Ω`(M, E) := Γ Λ`(M, E) . Further, define Λ0(M, E) := E. Consequently, Ω0(M, E) := Γ(E).

134 Appendix

Definition 8.12 (Exterior derivative). Let E → M be a vector bundle with connection ∇. For ` ≥ 0, define the exterior derivative

d∇ : Ω`(M, E) → Ω`+1(M, E)

for vectors X0,..., X` ∈ Γ(TM) as follows: ∇ X i ˆ d ω(X0,..., X`) := (−1) ∇Xi (ω(X0,..., Xi,..., X`)) i X i+j + (−1) ω([Xi, Xj], X0,..., Xˆ i,..., Xˆ j,..., X`) i

`-forms: Let M ⊂ Rn be open and consider again

E = M × R.

∧ · · · ∧ ∈ `( ) Then for i1,..., i` define dxi1 dxi` Ω M by

Ü ( ) ··· ( )ê dxi1 X1 dxi1 X` ∧ · · · ∧ ( ) = . .. . dxi1 dxi` X1,..., X` : det . . . .

dxi` (X1) ··· dxi` (X`) = 6= ∧ · · · ∧ = Note: If iα iβ for α β, then dxi1 dxi` 0 and if

σ : {1, . . . , `} → {1, . . . , `}

is a permutation, we have

dx ∧ · · · ∧ dx = sign σ dx ∧ · · · ∧ dx . iσ1 iσ` i1 i`

⊂ Rn ` ∧ · · · ∧ ≤ Theorem 8.13. Let U be open. The -forms dxi1 dxi` for 1 ` ` i1 < ··· < i` ≤ n are a frame field for Λ (U), i.e. each ω ∈ Ω (U) can be uniquely written as

= X ∧ · · · ∧ ω ai1···i` dxi1 dxi` 1≤i1<···

135 ⊂ Rn = X ∧ · · · ∧ Theorem 8.14. Let U be open and ω ai1···i` dxi1 1≤i1<···

n ∂a = X X i1···i` ∧ ∧ · · · ∧ dω dxi dxi1 dxi` . ∂xi 1≤i1<···

Proof. By Theorem 8.13 it is enough to show that for all 1 ≤ j0 < ··· < j` ≤ n we have Ä ∂ ∂ ä dω ,..., ∂xj0 ∂xj` n ∂a ∂ ∂ = X X i1···i` ∧ ∧ · · · ∧ Ä ä dxi dxi1 dxi` ,..., ∂xi ∂xj ∂xj 1≤i1<···

σ = a1dx2 ∧ dx3 + a2dx3 ∧ dx1 + a3dx1 ∧ dx2.

Let σ = dω with ω = v1dx1 + v2dx2 + v3dx3. Then Ä ∂ ∂ ä ∂ Ä ∂ ä ∂ Ä ∂ ä ∂vj ∂v dω , = ω − ω = − i . ∂xi ∂xj ∂xi ∂xj ∂xj ∂xi ∂xi ∂xj Thus we get that a = curl(v). The proofs of Theorem 8.13 and Theorem 8.14 directly carry over to bundle- valued forms. Theorem 8.16. Let U ⊂ Rn be open and E → U be a vector bundle with connection ∇. Then ω ∈ Ω`(U,E) can be uniquely written as

= X ∧ · · · ∧ ∈ ( ) ω ψi1···i` dxi1 dxi` , ψi1···i` Γ E . 1≤i1<···

n ∇ = X XÄ∇ ä ∧ ∧ · · · ∧ d ω ∂ ψi1···i` dxi dxi1 dxi` . ∂xi 1≤i1<···

136 Appendix

Pullback

Motivation: A geodesic in M is a curve γ without acceleration, i.e. γ00 = (γ0)0 = 0.

But what a map is γ0? What is the second prime?

0 0 We know that γ (t) ∈ Tγ(t)M. Modify γ slightly by

γc0(t) = (t, γ0(t)) meaning that γc0 ∈ Γ(γ∗TM).

Right now γ∗TM is just a vector bundle over (−ε, ε). If we had a connection ∇c then we can define

00 0 γ = ∇c ∂ γc. ∂t

Definition 8.17 (Pullback of forms). Let f : M → M˜ be smooth and ω ∈ Ωk(M,˜ E). Then define f ∗ω ∈ Ωk(M, f ∗E) by ∗ ( f ω)(X1,..., Xk) := (p, ω(d f (X1),..., d f (Xk)))

for all p ∈ M,X1,..., Xk ∈ TpM. For ψ ∈ Ω0(M,˜ E) we have f ∗ψ = (Id, ψ ◦ f ).

For ordinary k-forms ω ∈ Ωk(M˜ ) =∼ Ωk(M,˜ M˜ × R):

∗ ( f ω)(X1,..., Xk) = ω(d f (X1),..., d f (Xk)).

Let E → M˜ be a vector bundle with connection ∇˜ , f :M → M.˜

Theorem 8.18. There is a unique connection

∇ =: f ∗∇˜

∗ ∗ on f E such that for all ψ ∈ Γ(E),X ∈ TpM we have ∇X( f ψ) = Ä ˜ ä p, ∇d f (X)ψ . In other words ∇( f ∗ψ) = ( f ∗∇˜ )( f ∗ψ) = f ∗(∇˜ ψ).

137 Proof. For uniqueness we choose a local frame field ϕ1,..., ϕk around f (p) defined on V ⊂ N and an open neighborhood U ⊂ M of p such that f (U) ⊂ V. ∗ ∞ Then for any ψ ∈ Γ(( f E)|U) there are g1,..., gk ∈ C (U) such that ψ = X ∗ ∗ gj f ϕj. If a connection ∇ on f E has the desired property then, for j X ∈ TpM, XÄ ∗ ∗ ä ∇Xψ = (Xgj) f ϕj + gj∇X( f ϕj) j XÄ ∗ ˜ ä = (Xgj) f ϕj + gj(p, ∇d f (X) ϕj) j XÄ ∗ X ä = (Xgj) f ϕj + gj (p, ωjk(X)ϕk) j k XÄ X ä = (p, (Xgj)ϕj ◦ f + gj ωjk(X)ϕk ◦ f ) , j k where ˜ X ∇d f (X) ϕj = ωjk(X)ϕk ◦ f , k 1 with ωjk ∈ Ω (U). For existence check that this formula defines a connec- tion.

Theorem 8.19. Let ω ∈ Ωk(M, U), η ∈ Ω`(M, V) and ∗ ∈ Γ(U∗ ⊗ V∗ ⊗ W), then f ∗(ω ∧ η) = f ∗ω ∧ f ∗η.

Proof. Trivial.

Theorem 8.20. Let E be a vector bundle with connection ∇ over M˜ , f :M → M˜ , ω ∈ Ωk(M,˜ E), then

∗ d f ∇( f ∗ω) = f ∗(d∇ω).

Proof. Without loss of generality we can assume that M˜ ⊂ Rn is open and that ω is of the form = X ∧ · · · ∧ ω ψi1···ik dxi1 dxik . 1≤i1<···

138 Appendix

Hence

∗ ∇ = X ∗(∇ ) ∧ ∗ ∧ · · · ∧ ∗ f d ω f ψi1···ik f dxi1 f dxik 1≤i1<···

Exercise 8.21. Consider the polar coordinate map f : {(r, θ) ∈ R2 | r > 0} → R2 given by f (r, θ) := (r cos θ, r sin θ) = (x, y). Show that

f ∗(x dx + y dy) = r dr and f ∗(x dy − y dx) = r2 dθ.

Theorem 8.22 (Pullback metric). Let E → M˜ be a Euclidean vector bundle with bundle metric g and f :M → M˜ . Then on f ∗E there is a unique metric f ∗g such that ( f ∗g)( f ∗ψ, f ∗φ) = f ∗g(ψ, φ) and f ∗g is parallel with respect to the pullback connection f ∗∇.

Exercise 8.23. Prove Theorem 8.22.

On Stokes Theorem

We want to give a proof of Stokes theorem in n-dimensions.

Proof. The proof will consist of three steps. At first we consider the case n n−1 n M = H , then for ω ∈ Ω0 (H ) there exists some R > 0 such that

supp ω ⊂ [−R, 0] × [−R, R]n−1.

We can write X ω = ωi dx1 ∧ ... ∧ d‘xi ∧ ... ∧ dxn i and taking the cartan-derivative gives

X i−1 ∂ωi dω = (−1) dx1 ∧ ... ∧ dxn. i ∂xi

139 Evaluating the lefthand side of the equation gives

Z n Z R Z R Z 0 X i−1 ∂ωi dω = ... (−1) dx1 . . . dxn Hn −R −R −R i=1 ∂xi Using Fubini, for i = 2, . . . , n we integrate first with respect to i which yields Z R ∂ωi R = ωi|−R = 0 −R ∂xi as ω is compactly supported. So with this Z Z R Z R dω = ... ω(0, x2,..., xn) dx2 . . . dxn. Hn −R −R On the righthand side we get n Z X Z R Z R ω = ... ωi(0, x2,..., xn) dx1 ... d‘xi . . . dxn Hn −R −R ∂ i=1

As dx1|∂M = 0 since x1 ≡ 0 at the boundary we get that the lefthand- and the righthand-side are equal. As the second step let M be a manifold with boundary and supp ω ⊂ U for a chart (U, ϕ). Then Z Z Z Z Z dω = (ϕ−1)∗dω = d((ϕ−1)∗ω) = (ϕ−1)∗ω = ω. M ϕ(U) ϕ(U) ∂Hn∩ϕ(U) ∂M

As third step let supp ω ⊂ U1 ∪ ... ∪ Um, then choose a partition of unity subordinate to the cover (Ui). We get m m m Z X Z X Z X Z ω = $iω = d($iω) = d$i ∧ ω + $idω M M M M ∂ i=1 ∂ i=1 i=1 Ñ m é m Z X Z X Z = d $i ∧ ω + $idω = dω. M M M i=1 i=1

On the Definition of Vector Bundles

Recall the definition of a vector bundle.

Definition 8.24 (Vector bundle). A smooth vector bundle of rank k is a triple (E, M, π) where E and M are manifolds and π : E → M is smooth such that

−1 (i) The fiber Ep := π ({p}) has the structure of a k-dimensional real vector space.

140 Appendix

(ii) There is U ⊂ M open, p ∈ U and a diffeomorphism φ : π−1(U) → k U × R such that π1 ◦ φ = π|U where π1 is the projection on the first k component, i.e. for each q ∈ U the map φq : Eq → R is a vector space Ä ä isomorphism defined by q, φq(ψ) = φ(ψ).

In the literature (e.g. the script on Riemann surfaces of Prof. Bobenko) one may stumble across a different definition of a vector bundle. We want to show that this in fact is an equivalent definition to ours. We will formulate it as a theorem.

Theorem 8.25. Let E be a rank k vector bundle. Then there are unique holomor- phic mappings gij : Ui ∩ Uj → GLk(C) such that for the transition functions −1 k k φij = φj ◦ φi : (Ui ∩ Uj) × C → (Ui ∩ Uj) × C it holds that φij(p, e) = (p, gij(p)e). Furthermore the cocycle-realtion holds, i.e. gijgjk = gik.

The proof of this theorem should be clear by now. The more interesting implication is the converse of the theorem yielding equivalence of the defi- nitions.

Theorem 8.26. Suppose M is a Riemann surface, U = (Ui)i∈I an open cover- ing of M and (gij)i,j∈I endomorphism-valued functions subordinate to U with gij(p) ∈ GLn(C) for all p ∈ Ui ∩ Uj, which satisfy the cocycle-relation. Then there exists some holomorphic vector bundle (E, M, π) of rank n and a holomor- ¶ n © phic atlas φi : EUi → Ui × C | i ∈ I of E whose transition functions are the given gij’s

Proof. Let E0 := M × C2 × I. The plan is to define an equivalence relation ∼ on E0 such that E0/ ∼ is the desired vector bundle. Equip E0 with the induced topology by M, Cn and I with the discrete topology. We introduce the equivalence relation 0 0 0 0 (p, e, i) ∼ (p , e , j) :⇔ p = p and e = gij(p)e. This is an equivalence relation, because the transitivity follows from the cocycle-relation and reflexivity and symmetry are clear. Define E := E0/ ∼ with the quotient topology, then E is Hausdorff as we have that E0 and M are Hausdorff and

ϕ E0 E

0 π1 π

M

141 holds with all continuous maps. The equivalence relation ∼ is compatible 0 −1 with π1 which implies that π1 is continuos. Further π1 (p) has vector space structure as

−1 n ∼ n π1 (p) = {p} × C × i = C

By the map id × gij × i 7→ j we have an isomorphsim

{p} × Cn × i =∼ Cn.

We still need to check if E is locally trivial. This is given as

−1 0 n π1 (Ui) = (ϕ ◦ π1)(Ui) = ϕ(Ui × C × I) = EU−i.

| n ϕ Ui×C ×{i} is a homeomorphism onto EUi . Now define n φi : EU−i → Ui × C as the inverse of ϕ, then by construction

φi(p, e) = (p, gij(p)e).

On The Tangent Bundle of CPn

Now our aim is to show that

Ç n+1 å n C \ {0} T[x]CP = Hom L, L where L = π−1([x]). Further note that the quotient space in this example is the usual one defined by the equivalence relation x ∼ y :⇔ x − y ∈ L. n So let X ∈ T[x]CP be arbitrary. Then we know from the definition of a tangent vector that there is a curve γ˜ : (−ε, ε) → CPn such that

X f = ( f ◦ γ˜)0(0) for all f ∈ C∞(CPn)

Now we consider a lift of γ˜ this is some curve γ : (−ε, ε) → Cn+1\{0} such that π ◦ γ = γ˜. We can illustrate this in the following commuting diagram

Cn+1\{0} γ π (−ε, ε) CPn γ˜

142 Appendix

If we fix some p ∈ L we can define the tangent vector γ˙ (0) ∈ Cn+1\{0} at p. Then we obtain d π(γ˙ ) = X This definition does not depend on the choice of the lift. To see this let γ, γˆ be two lifts of γ˜ with γ(0) = p = γˆ(0). Then there is some function f : (−ε, ε) → C\{0} with f (0) = 1. This yields

d 0 ˙ γ(t) = f (0) · p + f (0) ·γˆ(0) dt t=0 | {z } | {z } ∈L =1

Thus we can define for a curve γ : (−ε, ε) → Cn+1\{0} with γ(0) = p the homomorphism " # n+1 d C \ {0} X : L → L : p 7→ γ dt t=0 At first this might seem a bit weird but with a closer look one can see that these definitions of X coincide and this yields the claim.

On the Curvature 2-form

After deriving the curvature 2-form Ω of a Riemannian curvature tensor one may ask the question of how to compute it explicitly.

Let L be a complex line bundle over M. Then we know that we have local −1 trivializations (Uα, φα)α∈I with φα : π (Uα) → Uα × C. How do we express a connection ∇ in these terms?

−1 Choose a frame sα = φα (p, 1). Then every section S ∈ Γ(L) can locally be expressed as

s|Uα = fα · sα for fα : Uα → C. Further, on Uα ∩ Uβ we have that

sα = gαβsβ.

We can express a connection ∇ in terms of the trivial connection d and a tensor A by ∇ = d + A. Locally ∇ can be expressed as

(∇S) = ∇( fαsα) = d( fαsα) + fα∇sα, Uα 143 hence 1 ∇sα := Aαsα ∈ Ω . Note that on the one hand

∇sα = Aαsα = Aαgαβsβ and on the other hand

∇sα = ∇(gαβsβ) = d(gαβsβ) + gαβ∇sβ = d(gαβsβ) + gαβ Aβsβ.

Hence the compatibility condition on Aα and Aβ is given by −1 −1 Aβ = gαβ Aαgαβ − gαβ dgαβ. Further, from the homework we know that

R∇ = d∇d∇ . Γ(L)

This gives, for ψ ∈ Γ(L) with ψ = f · s on Uα the local expression R∇ψ = d(d∇ψ) = d(∇ψ) = d(∇( f s)) = d((d f )s + f ∇s) = d((d f )s + f (As)) = d((d f + f A)s) = d(d f + f A)s − (d f + f A) ∧ As = (dd f +d f ∧ A + f dA)s − (d f + f A) ∧ ∇s |{z} |{z} =0 =As = ( f dA + f A ∧ A) s = (dA + A ∧ A) f s = (dA + A ∧ A) ψ Since L is a line bundle A ∧ A = 0 hence

Ωα = dAα.

We ca now do the same thing on Uβ and check that Ω is well defined by checking that Ωα = Ωβ on Uα ∩ Uβ.

 −1  Ωβ = dAβ = d aα − gαβ dgαβ −1 = dAα − (dgαβ ) ∧ dgαβ −1 −1 = dAα + (gαβ dgαβgαβ ) ∧ dgαβ | {z } =...dgαβ∧dgalphaβ=0

= dAα = Ωα An interesting observation we can make on the derivation of the formula dA = Ω is that R∇ = dA + A ∧ A. There is an even more general formula from which this statement follows directly.

144 Appendix

Lemma 8.27. Let ∇ = d + A be a connection on a vector bundle E over M. Then R∇ = Rd + dA + A ∧ A.

Proof. For X, Y ∈ Γ(TM) and ψ ∈ Γ(E) we get

∇ R (X, Y)ψ = ∇X∇Yψ − ∇Y∇Xψ − ∇[X,Y]ψ

= ∇X (dYψ + AYψ) − ∇Y (dXψ + AXψ) − d[X,Y]ψ − A[X,Y]ψ

= dXdYψ + AXdYψ + dX(AYψ) + AX AYψ − dYdXψ + AYdXψ + dY(AXψ) + AY AXψ

− d[X,Y]ψ − A[X,Y]ψ = Rd(x, Y)ψ + dA(X, Y)ψ + A ∧ A(X, Y)ψ

Applying this theorem to the above case yields Rd = 0 as C =∼ R2 has no curvature with respect to the trivial connection and A ∧ A = 0 as it is complex one dimensional. This immediately yields R∇ = dA.

On the curvature form of the sphere with radius R

In definition 6.19 we have seen the definition of the curvature 2-form. Now it is our aim to use this definition and calculate this form for the sphere. As in the last section one might think that it is a constructive way to write our Levi-Civita connection ∇ = d + A where A is a endomorphism valued form. Here we have for X, Y ∈ Γ(S2)

(AYX)p = −hdY(X), pip

We have seen that Ω = dA, but in fact this is for our concrete example not a good way to go. For a fixed radius R we consider spherical coordinates

ÖR cos(ϕ) cos(ϑ)è f (ϕ, θ) = R sin(ϕ) cos(ϑ) R sin(ϑ)

Ä π π ä 1 where ϕ ∈ [0, 2π), ϑ ∈ − 2 , 2 and N = R f . We obtain for the partial derivatives Ö− sin(ϕ) cos(ϑ)è Ö− cos(ϕ) sin(ϑ)è ∂ f ∂ f = R cos(ϕ) cos(ϑ) , = R − sin(ϕ) sin(ϑ) ∂ϕ ∂ϑ 0 cos(ϑ)

145 We choose as a frame

Ö− sin(ϕ)è Ö− cos(ϕ) sin(ϑ)è e1 = cos(ϕ) and e2 = − sin(ϕ) sin(ϑ) . 0 cos(ϑ)

Hence we can write 1 ∂ 1 ∂ e = , e = 1 R cos(ϑ) ∂ϕ 2 R ∂ϑ for our frame. This yields for the dual frame

[ [ e1 = R cos(ϑ)dϕ, e2 = R · dϑ

In particular

∇e1 e1 = d(e1)e1 − hd(e1)e1, NiN Ö− sin(ϕ)è 1 ∂ = cos(ϕ) R cos(ϑ) ∂ϕ 0 ±Ö− sin(ϕ)è Öcos(ϕ) cos(ϑ)èª 1 − cos(ϕ) , sin(ϕ) cos(ϑ) N R cos(ϑ) 0 sin(ϑ) ÖÖ− sin(ϕ)è è 1 = cos(ϕ) − cos(ϑ)N R cos(ϑ) 0 Öcos(ϕ)(cos2(ϑ) − 1)è 1 = sin(ϕ)(cos2(ϑ) − 1) R cos(ϑ) cos(ϑ) sin(ϑ) Ö− cos(ϕ) sin(ϑ)è sin(ϑ) = − sin(ϕ) sin(ϑ) cos(ϑ) cos(ϑ) tan(ϑ) = − e R 2

Ö− cos(ϕ) sin(ϑ)è ±Ö− sin(ϕ)è ª 1 ∂ sin(ϑ) ∇ e = − sin(ϕ) sin(ϑ) − cos(ϕ) , N N e1 2 R cos(ϑ) ∂ϕ R cos(ϑ) cos(ϑ) 0 − tan(ϑ) = e R 2

146 Appendix

Ö− cos(ϕ) sin(ϑ)è 1 ∂ ∇ e = − sin(ϕ) sin(ϑ) − hde (e ), Ni N = 0 e2 2 R ∂ϑ 2 2 (ϑ) | {z } cos − 1 | {z } R 1 − R N

∇e2 e1 = 0

This leads us to the definition of the so called Christoffel symbols. We write

k ∇ej ei = Γijek where we use the Einstein sum convention. Thus the Christoffel-symbols are just special scalar valued functions. We could write equivalently

j ∇ej ei = ωi (ei) ⊗ ej, where ω is a matrix valued connection 1-form. This yields

Ü tan(ϑ) ê [ Ç å 0 e1 0 sin(ϑ) ω = R = dϕ tan(ϑ) − sin(ϑ) 0 − e[ 0 R 2 We can conclude for our curvature form Ç 0 cos(ϑ)å Ω = dω = dϑ ∧ dϕ = J cos(ϑ)dϑ ∧ dϕ − cos(ϑ) 0

Note that this is consistent with our observation in theorem 6.22, where we showed that the curvature form is purely imaginary. If we interpret our tangent space as a one-dimensional line bundle we see that J ”becomes” i. We can use this to calculate the integral of the Gaußian curvature over the sphere. We have

K = hR(e1, e2)e2, e1i = − cos(ϑ)dϑ ∧ dϕ(e1, e2) Ç 1 ∂ 1 ∂ å 1 = − cos(ϑ)dϑ ∧ dϕ , = R cos(ϑ) ∂ϕ R ∂ϑ R2

Thus 1 Z 1 Z 1 Z Ω = KdA = 1 dA = 2 = χS2 2πi RS2 2π RS2 2πR2 RS2 We can draw an important consequence.

Theorem 8.28. (Hairy ball theorem) 2 2 Let X ∈ ΓTS . Then there is some p ∈ S with Xp = 0

147 Proof. Assume that there is a vector field X ∈ ΓTS2 with no zeros. The Poincare-Hopf-Index´ theorem yields

X 1 Z 1 Z 0 = indp X = Ω = K dA = 2 i 2 2 i 2πi S 2π S and therefore a contradiction.

On Homology

In the Riemann-Roch theorem it is our aim to calculate the Betti-numbers. These are defined as dimensions of special quotient spaces. To get a better understanding to these we will now discuss the basic ideas of homology and cohomology.

In our setting let M be a smooth, oriented, connected and compact man- ifold. Note that many of the concepts of homology and cohomology also work for more general manifolds. But we aim for the Riemann-Roch The- orem; thus these assumptions are necessary Let (ni)i∈I ∈ R be real coeffi- cients. Then we define X • C0 as the set of 0-chains p = ni pi.This can be understood as the i formal sum of points pi ∈ M. X • C1 as the set of 1-chains γ = niγi. We can think of it as the formal i sum of curves γi : [ai, bi] → M X • C2 as the set of 2-chains D = niDi. Here D is a formal sum of i singular triangles on M

Note that a singular triangle is a continuous map

2 ϕ : {(t1, t2) ∈ R : t1, t2 ≥ 0, t1 + t2 = 1} → M. | {z } :=∆

Here in our special setting we even see that our k-chains form vector spaces over R. For curves we could think of the inverse element under formal addition as the element that reverses the orientation. Now we can define a boundary operator on our chains. So let D be a singular triangle, γ be a curve and p a fixed point in M. Then

• ∂2 : C2 → C1 : D = (P1, P2, P3) 7→ (P1, P2) + (P2, P3) + (P3, P1)

148 Appendix

• ∂1 : C1 → C0 : γ = (P1, P2) 7→ P2 − P1 We define important subgoups (here even subspaces) of our chains:

• A k-chain γ is called a cycle, if ∂kγ = 0. We set Zk(M) = {c ∈ Ck(M) | ∂c = 0} = ker(∂k) • A k-chain γ is called a boundary, if there is a k + 1-chain D such that ∂k+1D = γ. We define Bk(M) = {c is boundary } = im(∂k+1) 2 It is trivial to see that ∂ = 0. Thus Bk(M) ⊂ Zk(M). This leads to the definition Zk(M) ker ∂ Hk(M) = = k Bk(M) im ∂k+1 of the k-th homology. Remark 8.29. This type of homology is also known as singular homology.

Definition 8.30 (homologous). Let γ1,γ2 ∈ H1(M). They are called ho- mologous if there exists a D such that

γ1 − γ2 = ∂D.

By the definition of Hk(M) we see that homologous curves are in Hk(M) considered as the same thing. This leads to the natural question which element lie in Hk(M).

It is easy to believe that on the 2-sphere all boundary curves of faces are 2 homologous to each other. Thus one can believe that dim Hk(S ) = 0. But on the torus we can see that the curves γ1 and γ2 are not homologous. Nevertheless it is easy to believe that every other boundary curve of a face is homologous to a formal linear combination of γ1 and γ2. Our intuition 2 is that dim(Hk(T )) = 2. In order to proof this, remember that the k-th cohomology is defined as

Hk(M) = ker (dk) . im(dk−1) We will now make use of the theorem of de Rham in the special case that k Hk(M) and H (M) are vector spaces.

149 Theorem 8.31. (de Rham) The map Å Z ã k ∗ I : H (M) → (Hk(M)) : ω 7→ c 7→ ω c is an isomorphism of vector spaces.

We know from the Riemann-Roch-Theorem that the dimension of the k-th cohomology is always finite. Thus by the De-Rham theorem we obtain that dim(Hk(M)) = βk(M). We calculated last time that deg(TS2) = χ(S2) = 2. A similar calculation shows that deg(T(T2)) = χ(T2) = 0. The Riemann-Roch-theorem yields 2 2 2 • dim(H1(S )) = β1(S ) = 2 − χ(S ) = 0 2 2 2 • dim(H1(T )) = β1(T ) = 2 − χ(T ) = 2 This shows in fact that our considerations above on the homology were in fact right.

On a Discrete Gauss-Bonnet Theorem

From now on we will only consider immersed simplicial surfaces M with vertex set V, edges E and faces F. We first of all need to clarify what the curvature of such a simplicial surface is. We define the unit normal Nijk of the triangle ijk by 2 N : F → S , ijk 7→ Nijk 3 where Nijk is the unit normal of the oriented plane in R which is uniquely determined by three vertices of the triangle and their orientation. Analo- jk gously the interior angle θi at the corner i of ijk is defined as

jk θ : F → R, ijk 7→ θi

jk where again θi is the corresponding angle of the triangle as indicated in the sketch.

150 Appendix

Remark 8.32. Consider a simplex ijk ∈ F, then its unit normal, as well as the interior angles, are determined by the positions of the simplex ijk in R3. jk We can explicitly compute Nijk and θi as functions of i, j, k with (k − i) × (j − i) N = ijk |(k − i) × (j − i)| and Æ ∏ jk k − i j − i θ = arccos , . i |k − i| |j − i|

Definition 8.33 (angle defect). The angle defect of an interior vertex i ∈ M◦ is given by the sum X jk Ωi := 2π − θi . ijk∈F

The area of a spherical triangle with interior angles α1, α2, α3 is given by

A = α1 + α2 + α3 − π thus for a spherical polygon with n ∈ N vertices and interior angles α1,..., αn, we yield n n X X A = (2 − n)π + αi = 2π − (π − αi) i=1 i=1 by triangulating the polygon and consecutive application of the formula for a spherical triangle. With some geometric observations we see that the exterior angles of the spherical polygon N(St(i)) given by the Gauss-image jk of a vertex star St(i) are given by θi , for ijk ∈ F. Thus we yield that

X Å Å jkãã X jk AN(St(i)) = 2π − π − π − θi = 2π − θi ijk∈F ijk∈F meaning that the angle defect corresponds to the area of the spherical tri- angle that is determined by the Gauss-map of the unit normals to each face that contains the vertex i. With the link to the smooth theory where a well known interpretation of the Gaussian curvature is as the ratio of the area enclosed by the Gauss map, to surface area of a neighbourhood around a vertex i ∈ V, the following definition makes perfect sense.

Definition 8.34 (discrete Gaussian curvature). The Gaussian curvature of an interior vertex i ∈ M is given by the angle defect and set to zero for

151 boundary vertices.

The total Gaussian curvature of a simplicial surface M is therefore given by X K(M) = Ωi. i∈M How well the characteristics of curvature are satisfied by the discretiza- tion of Gaussian curvature is also seen by the fact that the broadly known Gauss-Bonnet theorem also translates to our discrete setting. We use that the Euler characteristic of a simplicial surface M is given by

χ(M) = |V| − |E| + |F|.

Theorem 8.35 (Gauss-Bonnet for simplicial surfaces). Let f : M → R3 be a simplicial surface without boundary, then X Ωi = 2πχ(M). i∈V

Proof. By the Definition of Ωi, we have Ñ é X X X jk Ωi = 2π − θi i∈V i∈V ijk∈F Ñ é X X X jk = 2π − θi i∈V i∈V ijk∈F Ñ é X X jk = 2π|V| − θi i∈V ijk∈F

Since the interior angles of every face ijk ∈ F add up to π we yield Ñ é X X jk X θi = (3 − 2)π = π (2|E| − 2|F|) i∈V ijk∈F ijk∈F where we have used that every edge is shared by 2 faces. Thus X Ωi = 2π|V| − π (2|E| − 2|F|) = 2πχ(M) i∈V

Remark 8.36. The theorem above extends also to simplicial surfaces with a boundary ∂M. In a similar manner it is possible to proof that X X Ωi + Γi = 2πχ(M) i∈M◦ i∈∂M

152 Appendix where Γi denotes the discrete geodesic curvature of a boundary vertex i ∈ ∂M defined by X jk Γi := π − θi . ijk∈F

153