Basic notions of Differential Geometry Md. Showkat Ali 7th November 2002

Manifold

Manifold is a multi dimensional space. Riemann proposed the concept of ”multiply extended quantities” which was refined for the next fifty years and is given today by the notion of a manifold.

Definition. A Hausdorff topological space M is called an n-dimensional (topological) manifold (denoted by M n) if every point in M has an open neighbourhood which is homeomorphic to an open set in Rn.

Example 1. A surface R2 is a two-dimensional manifold, since φ : R2 R2 is homeomorphic. −→ Charts, Ck-class and Atlas

Definition. A pair (U, φ) of a topological manifold M is an open subset U of M called the domain of the chart together with a homeomorphism φ : U V of U onto an open set V in Rn. Roughly−→ speaking, a chart of M is an open subset U in M with each point in U labelled by n-numbers.

Definition. A map f : U Rm is said to be of class Ck if every func- tion f i(x1, ..., xn), i = 1, ...,−→ m is differentiable upto k-th order. The map m f : U R is called smooth if it is of C∞-class. −→ Definition. An atlas of class Ck on a topological manifold M is a set (Uα, φα), α I of each charts, such that { ∈ } (i) the domain Uα covers M, i.e. M = αUα, ∪ (ii) the homeomorphism φα satisfy the following compativility condition :

1 the map 1 φα φ− : φβ(Uα Uβ) φα(Uα Uβ) ◦ β ∩ −→ ∩ must be of class Ck.

1 The map φα φ− can be represented by a set of n-functions of n-variables, ◦ β x1 = f 1(y1, ..., yn),

...... xn = f n(y1, ..., yn), that is, all f i, i = 1, ..., n can be differentiable upto k-th order.

If k = , the above definition, the atlas is called smooth atlas. Two Ck at- ∞ lases (Uα, φα) and (Uβ, φβ) are called equivalent if their sum (Uα,Uβ), (φα, φβ) is an{ atlas of C}k. { } { }

A topological manifold M, together with an equivalence class of Ck at- lases, is called a Ck structure on M and M is called Ck manifold and if k = , then M is called smooth manifold. ∞ Smooth manifold

Definition. A smooth manifold M of dimension n is a Hausdorff topo- logical space for which

i) each point p M has a neighbourhood Up which is homeomorphic to an open subset of R∈n.

1 ii) If Up Uq = φ then φp φq− : φq(Up Uq) φp(Up Uq) is a smooth diffeomorphism.∩ 6 (Smooth means◦ : all partial∩ derivatives→ of∩ all orders exist and are continuous.)

Example 1. Let Sn denote the unit sphere in Rn+1, i.e.

Sn = x Rn+1 x2 + ... + x2 = 1 { ∈ | 1 n+1 } is a smooth manifold of dimension n.

2 Example 2. Let Pn denote the set of all lines through the origin in Rn+1 is called the n dimensional real projective space denoted by RPn. So, RPn is a smooth manifold.

Functions on Smooth manifolds

A function f : M R on smooth (C∞) manifold M is called smooth at a point x M if in→ a coordinate chart (U, φ) with x U. The function 1 ∈ ∈ f φ− is smooth in the usual sense. (i.e. Smooth as a function of n-variables x1◦, ..., xn ; n = dim M.)

1 1 Remark 1. f φ− is just a usual function of n-variables, f φ− = fˆ(x1, ..., xn). ◦ ◦

Remark 2. The notion of smoothness of f : M R at x M does not depend on the choice of a chart. (If it is smooth at one→ particular∈ chart (U, φ) it is surely smooth at all others.)

Definition. If a function f : M R is smooth at every point x M, then it is called a smooth function on→ the smooth manifold M. ∈

Differentiable Manifold

A differentiable manifold is a topological space equipped with a collec- tion of charts. Each chart provides a one-to-one correspondence between open subsets of the manifold and open sets of the model space Rn, and charts with overlapping domains provide consistency in differentiability properties.

3 Definition. An n-dimensional differentiable manifold M is a Hausdorff topological space which satisfies the following conditions :

i) for each Ui M there is a homeomorphism φi : Ui Vi, where Vi is an open set in Euclidean⊂ space Rn. →

ii) If Ui Uj = φ, the homeomorphisms φi and φj combine to give a ∩ 6 1 homeomorphism φji = φj φi− of φi(Ui Uj) onto φj(Ui Uj) which is dif- ferentiable map. ◦ ∩ ∩

Example. The sphere S2 = (x, y, z) x2 + y2 + z2 = 1 is a differentiable manifold. { | }

Complex Manifold

Definition. A complex manifold is a smooth manifold locally modelled Cn 1 on the complex Euclidean space, and whose transition function φp φq− of open sets of Cn into Cn are holomorphic (complex analytic). ◦

Example Let Pn denote the set of all lines through the origin in Cn+1 is called the n dimensional complex projective space denoted by CPn. So, CPn is a complex manifold of real dimension 2n.

Submanifold

Definition. Let M be a differentiable manifold of dimension m and let N be a subset of M satisfying the conditions :

i) N is a differentiable manifold of dimension n.

ii) If p is a point of N, there is a local coordinate neighbourhood U of p in M with a local coordinate system φ : U V , where V is an open cell in Euclidean m-space such that φ(N U) is→ the subset of V satisfying ∩ xn+1 = xn+2 = ... = xm = 0, and φ restricted to U N is a local coordinate system for N around p. Then N is called a submanifold∩ of M.

Example. The unit sphere x2 + y2 + z2 = 1 in R3. This is a differential manifold. So the unit sphere in R3 is a 2-dimensional submanifold of R3.

4 Many common examples of manifolds are submanifolds of Euclidean space.

Remark. A submanifold is a subset of a manifold which is itself a man- ifold, but has smaller dimension.

Manifold with boundary

There is an important extension of the idea of a differential manifold as defined. This extension is motivated by the example of a closed disk D. An interior point p of D has a neighbourhood that is an open disk but a point q on the boundary has neighbourhood(in the disk) that is a half disk. Thus, if D is to be thought of as a differentiable manifold, two different kinds of coordinate neighbourhoods will have to be considered, according as the point in question is interior or not.

Definition. A differential manifold of dimension n with boundary is a topological space M with a subspace N and a countable open covering U1,U2, ... with homeomorphisms φ1, φ2, ... satisfying the following conditions :

1) each set Ui of the given covering is either contained in M N, in which − case there is a homeomorphism φ : Ui Vi, where Vi is a solid open sphere → in n-space, or, otherwise there is a homeomorphism φ : Ui Vi where Vi is n 2 → a hemisphere of the form x < 1, xn 0, the set Ui N being mapped Pi=1 i ≥ ∩ on the subset of Vi for which xn = 0.

2) If Ui and Uj are two sets of the given covering and if φi and φj are the 1 homeomorphisms (just described) and if Ui Uj = φ, then φij = φi φ− is ∩ 6 ◦ j a differential map of φj(Ui Uj) onto φi(Ui Uj). ∩ ∩

Since homeomorphisms of the type φij must carry interior points into in- terior points and frontier points into frontier points, it is clear that if those Ui that meet N are restricted to N, they form a covering that defines a structure of an (n 1)-dimensional differentiable manifold on N. − If a manifold contains its own boundary, it is called, not surprisingly, a ”manifold with boundary”. The closed unit ball in Rn is a manifold with boundary, and its boundary is the unit sphere. The concept can be general- ized to manifolds with corners, by definition every point on a manifold has a

5 neighbourhood together with a homeomorphism of that neighbourhood with an open ball in Rn.

Example. The solid sphere and the solid torus are differentiable 3- manifolds with boundaries, the boundaries, of course being the sphere and the torus, respectively.

Tangent Vector and Tangent Vector Space

Let M be a smooth n-dimensional manifold, x M is an arbitrary point, ∈ and x, the set of smooth functions at x. O ~ Definition. A tangent vector, Vx to M at a point x is a linear map ~ Vx : x R O −→ ~ f(x) Vxf −→ which satisfies the two axioms : ~ ~ ~ (1) Vx(αf + βg) = αVxf + βVxg; α, β R f, g x. ~ ~ ~ ∀ ∈ ∀ ∈ O (2) Vx(f.g) = f(x)Vxg + g(x)Vxf; f, g x. ∀ ∈ O ~ ~ If f x, Vx is a tangent vector, then Vxf is called the derivative of f ∈ O ~ along the vector Vx.

Definition. The set of all tangent vectors at a point x denoted by TxM, together with addition and scalar multiplication defined by ~ ~ ~ ~ (αUx + βVx)(f) = αUx(f) + βVx(f) is a vector space called the tangent vector space.

Theorem. Let dim M = n then the dimension of Tx0 M is equal to n at any point xo M. ∈ Proof. [2] or you can see any book of differential geometry.

6 Vector bundle

Definition. A vector bundle of rank k (real or complex) over a manifold M n (called the base) is a manifold E (called the total space) and a projection π : E M such that → 1 (i) p M,Ep := π− (p) is a vector space of dimension k which is iso- ∀ ∈ k k morphic to R or C ; Ep is called the fibre over p.

(ii) p M a neighbourhood Up of p and a homeomorphism ∀ k∈ ∃k 1 k k hp : Up R (or C ) π− (Up) such that q Up, hp : q R or C Eq is a linear× map. → ∀ ∈ { }× →

Example. The product space M Rn is a trivial vector bundle over M. × Tangent Bundle

Definition. Let M be a manifold and any point p M, TpM be the ∈ tangent vector space of M at p, the set of all tangent vector space TpM at p M is called the tangent bundle of M and is denoted by ∈ TM : TM = T M. [ p p M ∈ Tangent bundle is an example of a vector bundle and rank TM = dim M.

Normal bundle

Let M be an n-dimensional submanifold of Rn+k. Then the normal bun- dle N(M) of M in Rn+k is a k-dimensional vector bundle E over M with total space :

n+k E := (x, v) M R v.ω = 0, ω TxM { ∈ × | ∀ ∈ } and projection map π : E M defined by π(x, v) := x. −→ Example. The normal line bundle to the 2-sphere M 2 = S2 R3 is trivial, N(S2) = S2 R. ⊂ ×

7 Section

Definition. A section of a bundle (E, M, π) is a map s : M E such 1 → that the image of each point x M lies in the fibre π− (x) over x, ∈

i.e. π s = idM . ◦ where idM is the identity on M. The set of smooth sections is denoted by Γ(M,E) or H0(M,E) if M and E are holomorphic.

Smooth vector fields

Definition. Let M be a n-dimensional smooth manifold, M the ring of smooth functions. A smooth vector field on M is a map O ~ V : M M O −→ O such that (1) V~ (αf + βg) = αV~ (f) + βV~ (g) ~ ~ ~ (2) V (f.g) = V (f).g + f.V (g); α, β R, f, g M . ∀ ∈ ∀ ∈ O Fact. The set of all smooth vector fields on M is a vector space (V~ + W~ and αV~ are well defined) denoted by Γ(M,TM).

Lie-algebra structure on Γ(TM)

Let M be a smooth n-dimensional manifold, M the ring of smooth func- tions, Γ(TM) the vector space of smooth vectorO fields.

Proposition. There is a well-defined bi-linear map called Lie-bracket or commutator

[, ] : Γ(TM) Γ(TM) Γ(TM) × −→ (X,Y ) [X,Y ] −→ given by [X,Y ]: M M O −→ O f [X,Y ]f := X(Y (f)) Y (X(f)) → − 8 Proof. We have to show that

[X,Y ]: M M O −→ O f [X,Y ]f := X(Y (f)) Y (X(f)) → − satisfies (1) [X,Y ](αf + βg) = α[X,Y ]f + β[X,Y ]g (2) [X,Y ](f.g) = f[X,Y ]g + g[X,Y ]f; α, β R, ; f, g M . ∀ ∈ ∀ ∈ O Remark 1. ”Bilinear” above means (1) [αX1 + βX2,Y ] = α[X1,Y ] + β[X2,Y ] (2) [X, αY1 + βY2] = α[X,Y1] + β[X,Y2].

Remark 2. [X,Y ] = [Y,X]. − Definition. A Lie-algebra is a vector space V together with a bilinear map

[, ]: V V V × −→ (a, b) [a, b] −→ such that (A1) [a, b] = [b, a] (A2) [[a, b], c]− + [[c, a], b] + [[b, c], a] = 0 a, b, c V . ∀ ∈ Theorem. The pair (Γ(TM), [, ]) is a Lie-algebra.

Proof. From the definition of Lie-bracket [,], so axiom (A1) follows of it. Let us check axiom (A2) X,Y,Z Γ(TM) then by the definition we can easily prove [[a, b], c] + [[c,∀ a], b] + [[b,∈ c], a] = 0.

9 Affine connection

Let M be a smooth manifold, M the ring of smooth functions, Γ(TM) the vector space of vector fields. O

Definition. An affine connection on M, is a map (denoted by ) ∇ : Γ(TM) Γ(TM) Γ(TM) ∇ × −→ (X,Y ) X Y −→ ∇ such that 1) X (Y1 + Y2) = X Y1 + X Y2 ∇ ∇ ∇ 2) X +X Y = X Y + X Y ∇ 1 2 ∇ 1 ∇ 2 3) X (fY ) = X(f)Y + f X Y ∇ ∇ 4) fX Y = f X Y, f M X,Y Γ(TM). ∇ ∇ ∀ ∈ O ∀ ∈ Torsion and Curvature tensor

let M be a smooth n-dimensional manifold, M the ring of smooth func- tions, Γ(TM) the vector space of vector fields. O

Definition. A tensor A of rank (1, p) on M is a multilinear map A : Γ(TM) Γ(TM) ... Γ(TM) Γ(TM) × × × → p copies | −{z } (X1,X2, ..., Xp) A(X1,X2, ..., Xp) −→ which satisfies

A(fX1,X2, ..., Xp) = A(X1, fX2, ..., Xp) = ... = A(X1,X2, ..., fXp) = fA(X1,X2, ..., Xp), for any function f M and X1,X2, ..., Xp Γ(TM). ∈ O ∈ Remark 1. ”Multilinear” means :

A(X1, ..., Xi + Yi, ..., Xp) = A(X1, ..., Xi, ..., Xp) + A(X1, ..., Yi, ..., Xp), i. ∀ Remark 2. An affine connection is not a tensor because in axiom (3) of , the first term is extra, but in axiom∇ (4) of , there is no extra term. ∇ ∇

10 Torsion. The torsion T ∇ of an affine connection , is a map ∇ T ∇ : Γ(TM) Γ(TM) Γ(TM) × −→ (X,Y ) T ∇(X,Y ) −→ where T ∇(X,Y ) := X Y Y X [X,Y ]. ∇ − ∇ − Theorem 1. its torsion T ∇ is a tensor of rank (1, 2). ∀ ∇ Proof. Omitted.

Definition. The curvature, R∇ of an affine connection , is a map ∇ R∇ : Γ(TM) Γ(TM) Γ(TM) Γ(TM) × × −→ (X,Y,Z) R∇(X,Y,Z) −→ where R∇(X,Y,Z) := X Y Z Y X Z [X,Y ]Z. ∇ ∇ − ∇ ∇ − ∇ Theorem 2. its curvature R∇ is a tensor of rank (1, 3). ∀ ∇ Proof. Omitted.

Definition. An affine connection is called integrable if R∇ = 0. ∇

Definition. An affine connection is called flat if T ∇ = R∇ = 0. ∇

References

1) The Geometry of Physics, Theodore Frankel.

2) Modern Differential Geometry, Chris J Isham.

3) Differential Forms and Connection, R.W.R. Darling.

4) Differential Manifolds, Serge Lang.

5) Differential Geometry, Mike Spivak.

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