Differential Geometry

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Vorlesung aus dem Wintersemester 2012/13

Differential Geometry

Prof. Bernhard Leeb, Ph.D.

TEXed by Florian Gartner & Florian Stecker

1 Manifolds and differential forms

16.10.2012

1.1 Reminder from calculus in several variables

Differentiability. The map F : U −→ Rn is called differentiable in x0 ∈ U if F is near

x0 well approximable by a linear map in the following sense: There exists a linear map

L : Rm −→ Rm _{s.t. F (x}

0 + h) = F (x0) + L(h) + r(h) with an error r (defined in a

neighbourhood) which is of smaller order than h, lim

h→0

kr(h)k khk = 0. We also write F (x0+ h) = F (x0) + L(h) + o(khk).

If F is differentiable in x₀ then L is uniquely determined and called the differential of

F in x0

dF (x0) ∈ Hom(Rm, Rn).

If for v ∈ Rm the limit ∂vF (x0) := lim

t→0

F (x0+tv)−F (x0)

t exists, it is called directoral

derivative of F in direction of v. Partial derivatives are defined as

∂F ∂xi

(x0) := ∂eiF (x0).

if F is differentiable in x0, then all directorial derivatives in x0 exist and ∂vF (x0) =

dFx0(v). Relative to the standard basis (of R

m _{and R}n_{) the differential dF (x}

0) is given

by the Jacobian matrix

∂Fi

∂xj

!

i≤n;j≤m

If n = 1, i.e. if F is a real valued function, then dF_x₀ _{∈ Hom(R}m_{, R) = (R}m)∗ is al linear form and corresponds via the standart scalar product hr, ri on Rm _{to the gradient}

∇F (x₀) of F in x₀:

dFx0 = h∇F (x0), ri

Warning: partial differentiability ; total differentiability. However, if the partial derivatives exist near x0 and are continuous in x0, then the map is differenatiable in x0.

In particular, the following statements are equivalent:

a) F is partially differentiable on U and the partial derivatives are continuous

b) F is differentiable on U and the differentail is continuous as map U −→ Hom(Rm, Rn) ∼= Rnm : x0 −→ dF (x0).

Such maps are called continuously differentiable or of class C1. More generally, one calls k times continuously differentiable maps of class Ck_{, k ∈ N}₀ ∪ {∞} (C0 _{is just}

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A Ck _{diffeomorphism between open sets V, W ⊂ R}m is a bijective Ck-map F : V −→

W whose inverse F−1 : W −→ V is also a Ck-map (if k = 0: homeomorphism).

A map F : U −→ Rm, U ⊂ Rm open, is called a local Ck-diffeomorphism in x₀, if there are open neighbourhoods V of x0 in U and W of F (x0) in Rm s.t. FV : V −→ W

is a Ck-diffeomorphism.

We observe: F is a local diffeomorphism in x₀ ⇒ dF_x₀ is invertible (as a linear map), because by the chain rule:

dGy0◦ dFx0 = d(G ◦ F

| {z }

id_R_m

)_x₀ = id_Rm

Inverse Function Theorem (Umkehrsatz): Suppose that F : U −→ Rm is Ck≥1_{, U ⊂ R}m open, x₀ ∈ U . If dF_x₀ is invertible, then F is a local Ck - diffeo.

Remark. dF_x₀ invertible ⇔ det∂Fi_∂xj(x₀) 6= 0.

Clever application of the Inverse Function Theorem yields as a corollary the following generalization:

Implicit Function Theorem: Suppose that U ⊆ Rm is open, m ≥ n and F : U −→ Rn is Ck≥1_{. Furthermore let R}m = E₁m−nL

E₂n a direct sum decomposition into linear subspaces with dim E₁ = m − n, (x₀ ∈ E₁, y0 ∈ E2) ∈ U and Z0 := F (x0, y0). If

dF (x0, y0)E2 : E2 −→ R

n _{is invertible, then there exist open neighbourhoods V}

1 of x0

in E1, V2 of y0 in E2 and W of z0 in Rn and for each z ∈ W a Ck-map Gz : V1 −→ V2

s.t. F−1(z) ∩ V₁× V₂

| {z }

solutions of F (x,y)=z near (x0,y0)

= {(x, G_z(x))

| {z }

graph of Gz

/x ∈ V1}. Furthermore, Gz depends Ck

-differentiably on z, i.e. the map V₁_{× W −→ R}m : (x, z) −→ G_z(x) is Ck- differentiable, more precisely a Ck-diffeo onto an open neighbourhood of (x0, y0), with Gz0(x0) = y0.

Remark.

i) Near (x₀, y0) the function/map Gzis implicitly given by the equation F (x, Gz(x)) =

0

ii) Let p ∈ U . There exists a decomposition Rm = E₁L

E2 s.t. dFpE2 is invertible

and dF_p is surjective.

Remark.

Let p0 ∈ U . A decomposition Rm = E1⊕ E2 such that dFp0|E2 is invertible, exists iff

dF_p₀ is surjective.

18.10.2012

Definition 1.1. Let F : Rm ⊃ U → Rn _{with U open be} _Ck_{-differentiable.}

1. A point x ∈ U is called regular if dF_x is surjective, and critical or singular other-wise.

2. A value y ∈ Rn is called regular if all points x ∈ F−1(y) are regular, and critical or singular otherwise.

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Remark.

1. Values which are not attained are trivially regular.

2. Sard’s theorem says that the set of critical values has Lebesgue measure zero in Rn.

Definition 1.2.

1. F is called a submersion, if dF_x is surjective for all x ∈ U , i.e. all points are regular, or equivalently, all values are regular.

2. F is called an immersion if dF_x is injective for all x ∈ U .

As a consequence of the implicit function theorem, for y ∈ Rn, the set of solutions of the equation F (x) = y is in every regular point locally a graph. In particular, it is locally parametrizable by independent coordinates. This leads us to the notion of a submanifold of Euclidean space.

1.2 Submanifolds of Euclidean space

Submanifolds of Rn are subsets which are regular in the sense that they can be made flat locally by a suitable coordinate change, i.e. be transformed into an affine subspace. In particular, they can be locally parametrized by independent coordinates.

Definition 1.3. Let 0 ≤ d ≤ n, k ≥ 1. A subset M ⊂ Rn is called a d-dimensional

differentiable submanifold of class Ck, if for every point x ∈ M there exists a Ck -diffeomorphism φ : U → V from an open neighborhood U of x in Rn onto an open subset V ⊂ Rn, such that

φ(M ∩ U ) = Rd× {0}

∩ V Example.

1. The unit sphere

Sn−1= x ∈ Rn n X i=1 x2_i = 1

is covered by the open (relative to the subspace topology) subsets Sn∩ {x|x_k_{≷ 0}.} These can be made flat, for instance by the map

φ(x) = x∓ 1 −X i6=k x2_i 1/2 ek

2. The following subsets are not submanifolds in R2:

We now give two characterizations of submanifolds. First, we characterize them locally as solution sets.

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Theorem 1.4. The inverse images of regular values of Ck≥1-maps F : Rn ⊃ U → Rm

are Ck-submanifolds of dimension n − m, i.e. codimension m. Proof. Consequence of the implicit function theorem.

Example.

1. The unit sphere Sn−1is a C∞-submanifold since it is the inverse image of 1 under theC∞-map F (x) = kxk2.

2. Consider

det : GL(n, R) → R×

where GL(n, R) is viewed as an open subset of Rn×n. det is smooth, since it is polynomial. We have (exercise)

( d det)E = tr .

Hence d detE is surjective and E is a regular point for the determinant. With

the multiplicativity of det follows that all A ∈ GL(n, R) are regular points for det, because differentiating det(AX) = det(A) det(X) with respect to X in E in direction V yields

( d det)_A(AV ) = det(A)( d det)_E(V ) = det(A) tr(V )

so d det_A = det A · tr(A−1−), which is surjective. So all points and values are regular, i.e. det : GL(n, R) → R× is a surjective submersion. In particular, 1 is a regular value which implies that SL(n, R) = det−1(1) is a smooth submanifold of codimension 1.

3. To show that the orthogonal group

O(n) = {A ∈ GL(n, R)|AAt= E} is a submanifold, consider

F : GL(n, R) → Matsym(n × n, R) ∼= Rn(n+1)/2, X 7→ XXt.

Then dFX(H) = HXt+ XHt and dFX(E) = H + Ht. So E is a regular point

for F , F (E) = E, and thus O(n) = F−1(E) is a C∞-submanifold locally at E, i.e. there exists an open neighborhood U of E in GL(n, R) such that O(n) ∩ U is a C∞-submanifold. Due to the homogeneity, O(n) is in every point A locally a submanifold, because left multiplication L_A: X → AX is a C∞-diffeomorphism of GL(n, R) and preserves O(n). So O(n) is a C∞)-submanifold with dimension

n2− n(n + 1)/2.

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Definition 1.5. A d-dimensional localCk-parametrization of M near p is a map

V0 F−→ U0 = M ∩ U 3 p

where V0, U0 are open in M and F is a homeomorphism and aCk-immersion.

Remark. In general, injective immersions are no homeomorphisms onto their image

(topo-logical embeddings).

Example. Consider the curve

(−1, ∞) c // R2 t // _1+tt3, t 2 1+t3 .

It is injective and C∞-smooth, but its inverse is not continuous.

For a subset M ⊂ Rn holds: M is a submanifold if and only if there exists a local parametrization. (One direction is trivial, the other is given by the Inverse Function Theorem).

Proposition 1.6. Let M ⊂ Rn. M is a submanifold of Rn if and only if there exists a local parametrization.

Proof. We start with a local parametrization F near p, F (0) = p. We thicken F to a

Ck_{-differentiable map}

V0× (−ε, ε)n−d_{−−−−→ R}F n

such that dF₀ is surjective. Then the Inverse Function Theorem yields that F is locally invertible at 0, i.e. there is a neighborhood V1of 0 such thatF |V1 is aC

k_{-diffeomorphism}

onto F (V1). We have

F (Rd× {0} ∩ V₁) ⊂ M ∩ U₁

To achive equality, we shrink V₁ (and U₁). Since F is a homeomorphism onto its image,

F (Rd× {0} ∩ V1) is open in M , so it can be written in the form M ∩ U2, where U2 is an

open subset of U₁. Define V₂:= (F |_V₁)−1(U₂). Then

F (Rd× {0} ∩ V2) = F (Rd× {0} ∩ V1) ∩ F (V2) = (M ∩ U1) ∩ U2= M ∩ U2.

So (F |V2)

−1 _{makes M flat near p, i.e. M is a submanifold.}

Example. Consider the unit sphere Sn−1= {x ∈ Rn| kxk = 1}.

1. The local parametrization results from Sn−1 being locally a graph. For example,

ϕ : {y ∈ Rn−1|kyk < 1} → Sn−1, y 7→ (y, (1 − kyk2)1/2)

is a parametrization of the upper hemisphere: Its image is the open subset Sn−1∩ {x_n > 0} of Sn−1 and it is inverted by the projection onto the first n − 1 compo-nents.

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2. Another way to parametrize the sphere is the stereographic projection: It is given by the formula σN: Sn−1r {N } → Rn−1, x 7→ 1 1 − xn (x1, . . . , xn−1)

where N = (0, . . . , 0, 1) is the north pole of the spere. Its inverse is given by

y 7→ 1

kyk2_{+ 1}(2y1, . . . , 2yn−1, kyk 2_{− 1).}

Geometrically, it projects a point x ∈ Sn−1r {N } to the unique point y ∈ Rn−1 such that x, N , and (y, 0) lie on one line.

Analogously, one can define the projection

σS: Sn−1r {S} → Rn−1 where S = (0, . . . , 0, −1) is the south pole of the sphere. We summerize:

Theorem 1.7. For M ⊂ Rn are equivalent: 1. M is a d-dimensional submanifold.

2. M is locally the image of a regular value of a Rn−d-valued Ck-map. 3. There exist local d-dimensional Ck_{-parametrizations of M .}

The system of local parametrizations forms the differentiable structure on M , which enables us to do Analysis (differentiate etc.) intrinsically (without reference to the ambient space).

Let M be a d-dimensional submanifold of Rn, p ∈ M , and φ : U → V the map that locally flattens M . Then the restriction of φ to M ∩ U is a homeomorphism onto an open subset V0 _{of R}d. It is called a local chart or local coordinate map. It is the inverse of a local parametrization. We observe that the coordinate changes areCk: Let U₁∩ M and

U2∩ M be two open subsets of M (U1, U2 are open in Rn). The twoCk-diffeomorphisms

φ1, φ2 which flatten M map U1, U2 to open subsets φ1(U1), φ2(U2) ⊂ Rn. If U1∩ U26= ∅,

the coordinate changes are given by restrictions of φ₂◦ φ−1₁ and φ₁◦ φ−1₂ :

φ2◦ φ−11 : Rd× {0} ∩ φ1(U1∩ U2) → Rd× {0} ∩ φ2(U1∩ U2)

φ1◦ φ−12 : Rd× {0} ∩ φ2(U1∩ U2) → Rd× {0} ∩ φ1(U1∩ U2)

These are mutually inverseCk-diffeomorphisms.

This compatibility of local charts makes it possible e.g. to define differentiability of functions f : M → R which are given only on M .

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Definition 1.8. A function f : M → R is called l-times differentiable in x ∈ M , l ≤ k, if for some chart κ around x the function f ◦ κ−1 defined on the open subset Im(κ) is

l-times differentiable in κ(x). This is independent of the chosen chart κ. Indeed, if κ0 is another chart around x, then

f ◦ κ0−1= (f ◦ κ−1) ◦ (κ ◦ κ0−1)

Since the coordinate change κ ◦ κ0−1is aCk-diffeomorphism, f ◦ κ0−1isCk-differentiable if and only if f ◦ κ−1 is. This definition of differentiability is intrinsic since it does not refer to the ambient Euclidean space.

This leads us to the notion of abstract manifolds, i.e. manifolds which are not a priori embedded into Euclidean space).

1.3 Abstract differentiable manifolds

Definition 1.9. A topological space (X, T ) is called d-dimensional locally Euclidean if every point has a local neighborhood which is homeomorphic to an open subset of Rd.

In particular, there exist local parametrizations by independent coordinates. Example.

1. C0_{-submanifolds of R}n as abstract topological spaces are locally Euclidean. 2. The eight and the X (see above) are not locally Euclidean.

25.10.2012

Example of an "abstract" locally euclidean space which does not arise in a natural way by a distinguished embedding into an euclidean space: real projective space:

RPd:= {1 − dim linear subspaces of Rd+1}

To a point Rx represented by x = (x0, x1, . . . xn) ∈ Rd+1− {0} one assigns the

homoge-neous coordinates [x] = [x₀: · · · : x_d] . They are unique up to scaling [x] = [x0_{] ⇔ ∃λ ∈ R}∗with x0 = λx.

One equippes RPdwith the quotient topology with respect to the projection π : Rd+1− {0} −→ RPd

x −→ Rx = [x].

Open sets in RPdcorrespond to open double cones in Rd+1. Note that π is an open map. Restricted to the unit sphere Sd _{⊂ R}d+1− {0}, the projection corresponds to dividing out the antipodal involution x −→ −x.

Sd_{←−−−−→ R}inclusion d+1− {0}_{−→ RP}π d Sd π/S d −−−→ 2:1 RP d Sd x→{x,−x}−−−−−−→ 2:1 S d_{/ ± 1}_{−−−−−−−−−−→}{x,−x}→Rx=[x] 1:1 RP d

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Also π|_Sd is open, because for small r > 0 and z ∈ Rd+1 with kzk = p (1 + x2_{) holds} π|_Sd(B(z, r) ∩ Sd) = π( B(z, r) | {z } open in Rd+1_−{0}

) _{is open in RP}d, because π open.

π|_Sd is continuous, open, locally injective (e.g. injective on open hemispheres) ⇒ π|_Sd

is local homeo Sdloc.eucl.⇒ _RPd _{is local euclidean.}

A homeomorphism U _{−→ κ(U ) of an open subset U ⊆ X onto an open subset κ(U ) ⊆ R}κ d

is called a local chart or set of local coordinates. U is called the domain of the chart. If (Ui, κi), i = 1, 2 are two charts, then the coordinate change κ1(U1 ∩ U2)

κ2◦κ−11

−−−−→

κ2(U1∩ U2) is a homeo between (possibly empty) open subsets of Rd.

An atlas is a family (A) of charts (U_i, κi), i ∈ I (index set), which cover X, X = S i∈I

Ui.

Two charts are Ck- compatible if the coordinate changes in boths directions are Ck -differentiable and hence Ck-diffeomorphisms. An atlas is called Ck-differentiable if any two of its charts are Ck-compatible. A Ck-differentiable atlas is contained in a unique maximal Ck-differentiable atlas which arises by adding all Ck-compatible charts. Definition 1.10. A Ck- differentiable structure, 1 ≤ k ≤ ∞, on a loc. eucl. space is a max Ck- differentiable atlas.

Example:

1) If M ⊂ Rn is a Ck-submanifold, then the local parametrizations, 1 ≤ l ≤ k, form a

Ck-differentiable structure on M, the natural one induced by Rn.

2) Sn−1 _{= {kxk = 1} ⊂ R}n: The natural C∞_{-differentiable structure induced by R}n is generated by an atlas consisting only of two charts, namely the sterogrphic projections

κ±: Sn−1− {±en} −→ Rn−1, x −→

1 1 ∓ xn

(x1, . . . xn−1).

We compute the coordinate change:

y = 1 1 + xn (x₁, . . . xn−1) κ−1₋ −−→ x κ+ −−→ 1 1 − xn (x₁, . . . xn−1) = 1 kyk2y kyk2₌ 1 − x2n (1 + xn)2 = 1 − xn 1 + xn

3) RPd. Consider for i = 0, . . . , d the bijection

Ui := {[x] ∈ RPd/xi 6= 0} κi −→ Rd [x₀: · · · : x_d] −→ _x 0 xi , . . .xˆi xi ,xd xi

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Rd+1− {0}−→ Yf (universal property of quot. topology) Rd+1− {0}−→ RPπ d RPd ¯ f −→ Y ¯ f continuous ⇔ f continuous κi is continuous because κi◦ π|π−1_(U i) : x −→ _x 0 xi, . . . ˆ xi xi, xd xi

is continuous (we use here the universal property of the quotient topology).

The coordinate changes are rational functions defined on hyperplane complements and therfore smooth (= C∞), for instance

{y ∈ Rd/y16= 0} κ−1₀ −−→ U0∩ U1 κ1 −→ {y ∈ Rd/y1 6= 0} (y1, . . . yd) −→ [1 : y1: · · · : yd] −→ ₁ y1 ,y2 y1 , . . .yd y1

We see that the atlas is C∞-differentiable and defines a C∞-differentiable (smooth) structure on RPd.

The differentiable structure enables us to transfer local analytic concepts from eucl. spaces to local eucl. spaces.

Let Xd F−→ Yd0 _{be a continuous map equipped with C}k≥1_{-differentiable structures. For}

x ∈ X and charts X ⊃ U −→ κ(U ) of X around x and Y ⊃ Uκ 0 κ₋_{→ (U}0 0_{) of Y around F (x),}

we call κ(U ∩ F−1(U0) | {z } open in X ) | {z } open nbh. of κ(x) in Rd κ0◦F ◦κ−1 −−−−−−→ κ0(U0_{) ⊂ R}d0

Definition 1.11. F is called l times resp. Cl_{-differentiable in x, if the local coordinate}

representations κ0◦ F ◦ κ−1 of F near x are l times, resp. Cl-differentiable in κ(x).

30.10.2012

Definition 1.12. Let X, Y be Ck_{–manifolds. A function F : X → Y is called l times}

differentiable (Cl-differentiable) in x (1 ≤ l ≤ k) if there exist local coordinates κ around x and κ0 around F (x) such that κ0◦ F ◦ κ−1 _{is l times differentiable at κ(x).}

Remark.

1. A function G : Rd ⊃ V → Rd0 _{is l times differentiable in y ∈ V if G is} _Cl−1

-differentiable on an open neighbourhood of y in V and the (l − 1)-st differential is differentiable in y.

2. Differentiablility does not depend on the choice of local coordinates: Let κ1, κ2 be

two coordinate systems near x ∈ X and κ0₁, κ0₂ coordinate systems near F (x) ∈ Y , then

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and (κ0₂ ◦ κ0−1₁ ) as well as (κ1 ◦ κ−12 ) are Ck-diffeomorphisms, so (κ02 ◦ κ −1 2 ) is

Cl_{-differentiable if and only if (κ}0

1◦ F ◦ κ −1

1 ) isCl-differentiable.

3. If F : X → Rd0 is a continuous function, then we regard Rd0 as a locally Euclidean space equipped with the natural differentiable structure generated by the atlas {id

Rd0}, and define differentiability using the local coordinate representations F ◦

κ−1.

We denote the space of all Cl-maps X → Y withCl(X, Y ). In particular Cl(X) := Cl_{(X, R). The composition of C}l_{-maps is} _Cl_.

Definition 1.13. A homeomorphism F : X → Y of Ck–manifolds X, Y is called a Cl -diffeomorphism, 1 ≤ l ≤ k, if F and F−1 areCl-differentiable. A function F : X → Y is a local Cl-diffeomorphism if every point in X has an open neighbourhood U such that F |U: U → F (U ) is aCl-diffeomorphism onto an open subset F (U ) ⊂ Y .

Example.

1. The natural 2:1 covering Sd_{→ RP}dis a local C∞-diffeomorphism.

2. For any A ∈ O(d + 1), A : Sd → Sd _{is a} _C∞_{-diffeomorphism, and for all A ∈}

GL(d + 1, R), A : RPd→ RPd _{is a}_C∞_{-diffeomorphism.}

Definition 1.14.

1. A topological manifold (C0-manifold) is a locally Euclidean Hausdorff space whose

topology admits a countable basis (is 2nd countable).

2. ACk-differentiable manifold is a topological manifold together with aCk -differen-tiable structure.

Remark. One asks the Hausdorff property to be able to seperate points by continuous

functions, and the 2nd axiom of countablility implies the existence of a partition of unity. Example.

1. The Euclidean space Rn is locally Euclidean, Hausdorff and 2nd countable. Its C∞_{-differentiable structure is generated by the atlas {id}

Rn}.

2. TheCk_{-submanifolds of R}nare in a natural way Ck-differentiable manifolds. 3. RPd is a C∞-differentiable manifold, because the Hausdorff and 2nd countability

properties carry over from Rd+1.

Definition 1.15. Let M be an m-dimensional Ck≥0-differentiable manifold. A subset

N ⊂ M is an n-dimensionalCl≤k-submanifold, if around any point x ∈ N there exists a

Cl_{-differentiable chart (U, κ) such that κ(N ∩U ) = R}n_×{0}m−n_{∩κ(U ). Then N is (with}

respect to the relative topology inherited from M ) n-dimensional locally Euclidean. The Hausdorff and 2nd countability properties carry over from M to N . The restrictions of the charts generate the natural differentiable structure of N . So submanifolds are manifolds in a natural way.

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Example.

1. Consider (R × {0} ∪ R × {1})/ ∼ where (x, 0) ∼ (x, 1) for all x ∈ R r {0}. This space is 2nd countable and locally Euclidean, but not Hausdorff.

2. The long line: Let (W, <) be a well-ordered set and L = W × [0, 1) and give it the lexicographical order, i.e.

(w, t) < (w0, t0) ⇐⇒ w < w0∨ (w = w0∧ t < t0).

This order induces an order topology (generated by the the open intervals). We call x ∈ W good, if L_(x,0) := {(w, t) < (x, 0)} is empty or homeomorphic to [0, 1). We observe the following:

a) A limit of an increasing sequence of good elements of W is also good.

b) if W_x = {w | w < x} is countable, x is good. To prove this, assume x is bad. Without loss of generality all w ∈ Wx are good. (If not, replace x by the

smallest bad w in Wx). There exists a sequence (xn) ⊂ Wx such that xn% x,

so x is good.

c) If W_x is uncountable, x is bad, because L_(x,0) contains uncountably many disjoint open subsets.

To construct the long line, choose W uncountable such that all initial segments

Wx are countable. Then for L+ = L r min L holds:

a) L+ is 1-dimensional locally Euclidean, since it is covered by L_(x,0)∼= [0, 1). b) L+ is path connected.

c) The topology of L+ has no countable basis, because there exists an uncount-able family of disjoint open sets.

06.11.2012

Remark. One can equip a set directly with a structure as differentiable manifold by

providing a suitable atlas. The topology will result implicitly. We start with the following data: A set M and an atlas consisting of charts

κi: Ui → Vi i ∈ I

which are bijections from U_i ⊂ M to open subsets V_i _{⊂ R}d_{, such that the coordinate}

changes κj◦ κ−1i are defined on open subsets andCk-differentiable.

The topology arises as follows. We define the neighbourhoods of a point p on the subset

W ⊂ M such that for the charts κiwith pinUithe subset κi(W ∩ Ui) is a neighbourhood

of κi(p) in Vi. This is independent of the chart κi! The axioms for neighbourhoods are

satisfied (finite intersections and larger subsets are again neighbourhoods).

The open subsets are then those which are neighbourhoods of all their points. We obtain:

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With this topology on M the charts κi become homeomorphisms of open subsets and

M become a d-dimensional locally Euclidean space. If the atlas is countable (or has a

countable subatlas), the topology of M has a countable basis. The Hausdorff property does in general not follow for free and has to be verified in the concrete case at hand. The given atlas yields a Ck-differentiable structure on M .

Example (RPd revisited). On the set RPd of 1-dimensional subspaces of Rd+1 we consider the same atlas as before

κi: Ui= {[x] | xi 6= 0} → Rd, [x] 7→ _x 0 xi , . . . ,cxi xi , . . . ,xd xi

The coordinate changes are defined on hyperplane complements and smooth. The Haus-dorff property is clear. Hence we obtain on RPd a structure as smooth manifold. Example. The Grassmannian Grk(Rn) is the space of all k-dimensional linear subspaces

Lk ⊂ Rn_{, 1 ≤ k ≤ n − 1. We will put on it a structure as a smooth manifold by giving}

a suitable atlas. Every direct sum decomposition Rn= Vk⊕ Wn−k _{yields a chart}

κV,W: UV,W = {Lk⊂ Rn| L t W } → Hom(V, W ), graph(F ) 7→ F

We investigate the domains of definition and smoothness of the coordinate changes. The components with respect to two decompositions V ⊕ W = Rn= V0⊕ W0 transform into each other linearly, i.e.

v + w = v0+ w0⇒

(

v0 = Av + Bw

w0 = Cv + Dw where A, B, C, D are linear maps such that

A B

C D

!

is a block decomposition of id_Rn with respect to the two direct sum decompositions.

Let L ∈ UV,W ∩ UV0_,W0, i.e. L t W, W0, and let graph(F ) = L = graph(F0) with

F ∈ Hom(V, W ) and F0 ∈ Hom(V0_{, W}0_{). We decompose the vectors in L and obtain}

v + F v = v0+ F v0 ⇒

(

v0= Av + BF v = (A + BF )v

F0v0 = Cv + DF v = (C + DF )v where A + BF ∈ Hom(V, V0) is invertible, because L_{t W, W}0, so

F0= (C + DF )(A + BF )−1

Hence the coordinate changes are defined on open subsets and are smooth.

There exist finite subatlasses (e.g. corresponding to decompositions by coordinate subspaces), and hence a countable basis for the topology of Gr_k_(Rn). The Hausdorff property is easy to check (exercise). We obtain on Gr_k_(Rn) a stucture as a smooth manifold.

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Remark.

1. The group GL(n, R) operates on Grk(Rn) by diffeomorphisms, because it preserves

our atlas.

2. The projective space is a special case of the Grassmannian, Gr₁_(Rn_{) = RP}n−1, and Grn−k(Rn) is diffeomorphic to Grk(Rn).

1.4 The tangent bundle of a differentiable manifold

1.4.1 Tangent spaces to submanifolds of Euclidean space

Let first A ⊂ Rn be an arbitrary subset. A vector v ∈ Rn is a tangent vector to A at

p ∈ A if it is the velocity vector of a curve in A through p, i.e. there is aC1-differentiable curve

c : (−ε, ε) → A

such that c(0) = p and ˙c(0) = v.

The set T_pA of all tangent vectors to A in p may be called the tangent (double) cone

to A in p, since v ∈ TpA ⇒ Rv ⊂ TpA because d dt t=0 c(λt) = λ ˙c(0)

The tangent cone is preserved by (differentials of) diffeomorphisms: If A1, A2 ⊂ Rn

are subsets and if ψU₁ → U₂is a diffeomorphism of open subsets such that ψ(A₁∩ U₁) =

A2∩ U2, then for every point p1 ∈ A1∩ U1 and p2= ψ(p1) ∈ A2∩ U2 holds

dψ_p₁(T_p₁A1) = Tp2A2.

Indeed, if c1: (−ε, ε) → A1∩U1is a differentiable curve then c2 := ψ◦c1is a differentiable

curve in A2∩ U2 and the chain rule implies

dψ ˙c₁(0)

= (ψ ◦ c₁)0(0) = ˙c₂(0). For linear subspaces L ⊂ Rn holds

TxL = L ∀x ∈ L

It follows for submanifolds: The tangent cones T_pM to a d-dimensionalCk≥1-differentiable submanifold M ⊂ Rn are d-dimensional linear subspaces and are called the tangent

spaces to M at p ∈ M .

08.11.2012

The property of T_pM of being a linear subspace can also be seen as follows: If V0 F−→

M ∩ U is a local Ck-parametrization near p, F (0) = p, and if U −_{→ R}s n−d _{is a C}k

-submersion s.t. M ∩ U = s−1(0), then we have imdF₀ | {z } dim =d ⊆ T_pM ⊆ ker dSp |{z} surj. | {z } dim=d

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⇒ equality holds: TpM is d-dim linear subspace.

Towards an intrinsic definition of the tangent space: Two curves (−_i, i)

ci

−−→

C1 M represent the same tangent vector if they agree up to first

order in 0, i.e.

c1(0) = c2(0) ∧ ˙c1(0) = ˙c2(0)

This equivalence relation on differentiable curves in M can be expressed in terms of local coordinate charts and one thereby obtains an intrinsic definition of tangent spaces which carries over to. . .

1.4.2 Tangent vectors and differentials

Let Mm be a Ck≥1 manifold. We say that two C1-curves (−i, i) ci

−→ M agree up to first order in 0, if c₁(0) = c₂(0) := p and if for a chart around p holds

(κ ◦ c₁)˙(0) = (κ ◦ c₂)˙(0). This is independent of the chart:

(κ0◦ ci)˙(0) | {z } coo. change (κ0_◦κ−1_)◦(κ◦c i) = |{z} chain rule d(κ0◦ κ−1)_κ(p) | {z } invertible (κ ◦ ci)˙(0)

An equivalence class of curves agreeing up to first ofder is called a 1-jet (of curves) (here in p).

Definition 1.16. A tangent vector to M is a 1-jet.

The set of all tangent vectors to M in a point p ∈ M is called the tangent space TpM .

The set T M = S

p∈M

TpM of all tangent vectors is called the tangent bundle.

Remark: Tangent spaces to different points are disjoint (think of them as vertical to the manifold).

A differentiable map Mm F−→ Nn _{of differentialbe mainfolds induces the map of tangent}

bundles T M −dF−→ T N, [c] −→ [F ◦ c], the differential of F. It is well defined because for charts κ around c(0) and κ0 around F (c(0)) holds:

κ0◦ (F ◦ c) = (κ0◦ F ◦ κ−1)

| {z }

loc. coo. rep. of F

◦(κ ◦ c)

⇒ chain rule: (κ0◦ (F ◦ c))˙_{(0) = d(κ}0_{◦ F ◦ κ}−1₎

κ(c(0))

| {z }

old differential (Ana II)

(κ ◦ c)˙(0).

Hence [F ◦ c] depends only on [c].

The differential maps tangent spaces into tangent spaces, dF (TpM ) ⊆ TF (p)N and we

write dF_p := dF |_TpM. The chain rule holds: If M −F→ N −G→ W are diff. manifolds, then

d(G ◦ F ) = dG ◦ dF because

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Remark: The differentiable structure as defined above is consistent with the “old” differential because for a diff. map

Rm⊃ U open−F→ Rn, (F ◦ c)˙(0) = dFc(0)· ˙c(0)

holds according to the old chain rule.

1.4.3 The linear structure on tangent spaces

For open subsets U ⊂ Rn, we have the natural identifications

TpU ∼ = −−−−−→ c−→ ˙c(0) R n_,

and therefore linear structures on the tangent spaces. The differentials of differentiable maps U −→ Rm are linear. This carries over (via charts) to abstract manifolds.

Let M_m be a differentiable manifold. A chart (U, κ) around p ∈ M yields the identifica-tion TpM dκp −−→ 1:1 Tκ(p)U ∼= R m

And thus a linear structure on TpM . It does not depend on the chart because according

to the chain rule

dκ0_p◦ (dκ_p)−1 | {z } (dκ−1₎ κp = d(κ0◦ κ−1)_κ(p) | {z }

old differential, hence linear

is an isomorphism of vector spaces. It is clear that the differentials act linearly on tangent spaces i.e. that the maps

dFp : TpM −→ TF (p)N

are linear.

Local coordinates distinguish bases of tangent spaces: Let (U,x) be a local chart around p. W.r.t. the identification (∗) T_p_{M −→ R}m, [c] −→ (x ◦ c)˙ (0) the standard basis vector ei∈ Rm corresponds to the tangent vector [ci] with ci(t) = x−1(x(p) + tei) near t = 0.

For reasons which become clear later (when we interpret tangent vectors as directoral derivatives) we use the notation

∂ ∂xi

|_p := [c_i]

Definition 1.17. {_∂xi∂ |_p} is called the standard basis of T_pM w.r.t. local coordinates

x. An arbitrary tangent vector [c] ∈ TpM corresponds via (∗) to the vector (x ◦ c)˙ (0) =

P

i(xi◦ c)˙(0)ei, and we obtain the representation [c] = m

P

i=1

(xi◦ c)˙(0)_∂xi∂_|_p

Matrix representation of the differential dF : T M −→ T N w.r.t. the standard basis:

Mm x F _// Nn y VM ⊂ Rm ˜ F =y◦F ◦x−1 //VN ⊂ R n [c] (∗) dFp _// [F ◦ c] (∗) (x ◦ c)˙₍₀₎ d ˜Fx(p) //(y ◦ F ◦ c)˙₍₀₎

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W.r.t the standard bases {ej} resp. {ei} of Rm and Rn, d ˜Fx(p) is given by the Jakobi

matrix _∂xj∂ ˜Fi(x(p))

i=1,...n. These correspond via the differentials of local charts x resp.

y to the associated standard bases {_∂xj∂ |p} of TpM , resp. {_∂xi∂ |F (p)} of TF (p)N .

Therefore dFp ∂ ∂xj |p = n X i=1 ∂ ˜Fi ∂xj (x(p)) ∂ ∂yi |_{F (p)}

Transformation of standard bases in case of coordinate change: We put M = N and

F = id M . We write the coordinate change suggestively as ˜x(x). Then ∂ ∂xj |_p= n X i=1 ∂ ˜xi ∂xj (x(p)) ∂ ∂ ˜xi |_p, respectively, ∂ ∂ ˜xi |_p = m X j=1 ∂xj ∂ ˜xi (˜x(p)) ∂ ∂ ˜xj |_p, 13.11.2012

1.4.4 The differentiable structure on the tangent bundle LetMm be a Ck–manifold. The natural projection

T M = a

p∈M

TpM → M

is given by the footpoint projection, i.e. π−1(p) = T_pM . A chart (U, x) of M induces

local coordinates: T x : T U = a p∈U TpU → x(U ) × Rm ⊂ R2m, X i vi ∂ ∂xi p 7→ x(p),X i viei !

which we will use as a chart for T M . If (U ,e x) is another chart on M , then the coordinate_e

change is given by x(U ∩U ) × Re m →x(U ∩_e U ) × Re m, (x(p), v) 7→   ex(p), X i X j vj ∂xei ∂xj (x(p))_eei  .

We see that coordinate changes are defined an open sets and areCk−1–differentiable. It follows (compare the technical remark above) that the atlas consisting of the charts T x yields on the set T M a natural topology as 2m–dimensional locally Euclidean space and aCk−1–differentiable structure. The topology on T M has a countable basis since there are countable subatlases. The Hausdorff property is clear. We conclude

Theorem 1.18. If M is an m–dimensional Ck≥1–manifold, then T M carries a natural induced structure as a 2m–dimensional Ck−1–manifold.

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Remark. The projection π : T M → M is Ck−1–differentiable, which is the maximum possible degree of differentiability.

If F : M → N is a Cl–map of Ck–manifolds with 1 ≤ l ≤ k, then a local coordinate representation shows that its differential dF : T M → T N isCl−1–differentiable.

dFp ∂ ∂xj p ! =X i ∂fF_i ∂xj (x(p)) ∂ ∂yi F (p) e F = y ◦ F ◦ x−1

because the entries of the Jacobian matrix dependCl−1_{–differentiably on the point.}

1.4.5 Tangent vectors as derivations

One can regard tangent vectors (analytically) as differential operators. This will be useful for us from a technical point of view. To a tangent vector [c] ∈ TpM we can assign

its directional derivative

∂[c]f := df [c] = [f ◦ c] = (f ◦ c)0(0)

where we canoncially identify T_{(f ◦c)(0)}R with R. The differential operator ∂[c] has the

following properties: 1. It is R–linear:

∂[c](λ1f1+ λ2f2) = λ1∂[c]f1+ λ2∂[c]f2

2. It is a derivation, i.e. it satisfies the product rule

∂_[c](f g) = (∂_[c]f )g(p) + f (p)(∂_[c]f )

Directional derivatives are local operatores, i.e. ∂_[c]f depends only on the values of f

near p. It is therefore natural to pass to germs of functions. Let M be aCk≥1-manifold and p ∈ M . We call two functions defined on neighbourhoods U₁ resp. U₂of p equivalent, if they agree on a neighbourhood W ⊂ U₁∩ U₂ of p. An equivalence class is called a

germ (of a function) in p, andCk–differentiable, if the functions representing it areCk– differentiable in a neighbourhood of p. We denote the germ of f in p by [f ]p. The set

Ck_{(M )}

p of Ck–germs in p carries a natural structure as an R–algebra.

Definition 1.19. A derivation on M in p is an R–linear functional D : Ck(M )_p _{→ R} which satisfies the product rule:

D(f g) = (Df )g(p) + f (p)(Dg)

The derivations in p form an R–vector space Dp(M ). It contains the tangent space, i.e.

there is the natural linear embedding

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If (U, x) are local coordinates at p, thento the tangent vector _∂∂ xi|p is assigned the derivation f 7→ d dt t=0(f ◦ x −1 )(x(p) + tei) = ∂(f ◦ x−1) ∂xi (x(p))

which is the i–th partial derivative wrt. the coordinates x. This motivates our notation

∂ ∂xi

_p. From now on we identify the tangent vector v with the corresponding derivation ∂v.

Let us discuss the surjectivity of (∗). We denote by I_p ⊂Ck_{(M )}

p the maximal ideal of

the germs vanishing in p, i.e. the krenel of the evaluation map (algebra homomorphism!) Ck_{(M )}

p→ R. We observe that every derivation D : Ck(M )p→ R vanishes on germs of

constant functions, since

D(1) = D(1) · 1 + 1 · D(1) − D(1) = D(1 · 1) − D(1) = D(1) − D(1) = 0

and as well on the idal I_p2 (the linear combinations of products f₁f2 for f1f2 ∈ Ip),

because

D(f1f2) = D(f1)f2(p) + f1(p)D(f2) = D(f1) · 0 + 0 · D(f2) = 0 f1, f2 ∈ Ip

Vice versa, every linear functional D : Ck(M )_p_{→ R, which vanishes on R·1∪I}_p2 ⊂Ck_{(M )}

is a derivation:

D((c1· 1 + f1)(c2· 1 + f2)) = D(c1c2· 1 + c2f1+ c1f2+ f1f2)

= c2D(f1) + c1D(f2)

= D(c₁˙1 + f₁)(c₂· 1 + f₂)(p) + (c₁˙1 + f₁)(p) · D(c₂· 1 + f₂) Hence derivations correspond to linear forms on the vector space Ip/Ip2:

Dp(M ) ∼= (Ip/Ip2)∗

Proposition 1.20. If k = ∞, then the embedding (∗) is a linear isomorphism.

Proof. Let (U, κ) be a chart around p with κ(p) = 0 and [f ] ∈ I_p. Then we obtain for x near 0: (f ◦ κ−1)(x) = Z 1 0 _d dt(f ◦ κ −1 )(tx) dt =X i xi Z 1 0 ∂(f ◦ κ−1) ∂xi (tx) dt | {z } =:gi(x) so f = dim M X i=1 (x_i◦ κ)(g_i◦ κ) ⇒ [f ] = dim M X i=1 [x_i◦ κ]g_i(0) mod I_p2.

Therefore Ip/Ip2 is generated by the [xi◦ κ] + Ip2 and (∗) is also surjective by dimension

reasons.

Remark. If 1 ≤ k < ∞, then dimD_p(M ) = dim I_p/I_p2 = ∞, because the germs in I_p2 are even (k + 1)–times differentiable.

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1.4.6 Submersions and Immersions

After having defined the differential, we can generalize these notions to the setting of manifolds.

Let Mm F−→ Nn _{be a C}k≥1_{-map of C}k_-manifolds.

Definition 1.21. F is called a submersion if its differential dFp is surjective at all points

p ∈ M .

We know (compare our discussion of submanifolds and the Inverse Function Theorem): Proposition 1.22. The inverse images F−1(y) of values of submersions are

submani-folds.

Definition 1.23. i) F is called an immersion, if dFp is injective for all p ∈ M .

ii) F is called embedding, if its image F (M ) ⊂ N is a submanifold and M−F→ F (M ) is a diffeomorphism.

immersions are locally embeddings (compare the earlier discussion of local parametriza-tions of submanifolds)

Proposition 1.24. Every point p ∈ M has an open neighbourhood U s.t. F |U is an

embedding

Proof. We thicken F locally to make it a diffeomorphism (working in a single chart).

For a sufficiently small open nbh. U of p (contained in one chart) and > 0 exists an extension of F |U to U × (−, )(n−m) F |

U

−−→

Ck N s.t. (dF |U)(p,0) is invertible.

According to Inverse Function theorem we can achieve by shrinking U and , that F |_U is a diffeomorphism. then F |_U is an embedding.

injective immersions are in general no embeddings!

Proposition 1.25. An immersion is an embedding if and only if it is a homeomorphism

onto its image.

Proof. ” ⇐ ”: Let p ∈ M . by the previous proposition, there exists an open

neighbour-hood U of p s.t. F (U ) ⊂ N is a submanifold. By assumption, F (U ) is open in F (M ), hence ∃O ∈ N open s.t. F (U ) = F (M ) ∩ O.

It follows that F (M ) is a submanifold near F (p). Thus F (M ) ⊂ N is a submanifold. Then

M −−−−−−−−→F

homeo+imm F (M )_| _{z _}

(sub)manif old

is a local diffeomorphism. Since it is also a homeomorphism, it is also a global manifold.

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Proof. Bijective continuous maps from compact spaces to Hausdorff spaces are

homeo-morphisms.

We call the images of injective immersions M → N immersed submanifolds of N (Relaxing our notion of (embedded) submanifolds.)

1.5 Vector fields, flows, Lie brackets

(everything C∞)

Definition 1.27. A (smooth) vector field on M is a smooth map X : M → T M with

π ◦ M = idM i.e. X(p) ∈ TpM ∀p ∈ M .

Geometric intuition: In every point, a direction is chosen.

in local coordinates (U,x) the vector field can be expressed in terms of the cononical basis:

X(p) =Xai(p)

∂ ∂xi

|_p.

The smoothness of X on U is equivalent to the smoothness of the coefficient functions

ai.

Analytically, we can regard vector fields as certain differential operators on functions. To the vectorfield X corresponds the operator, also denoted by X:

X : C∞(M ) −→ C∞(M )

f −→ Xf : p −→ X(p)

| {z }

derivation

f.

"derivative in direction of the vector field X" Properties:

i) R- linear

ii) product rule: X(f g) = (Xf )g + f (Xg)

Vice versa, to an R -linear operator X satisfying the product rule corresponds a vector field. Namely, X determines in a point p ∈ M the derivation X(p) ∈ TpM given by

f −−−→ (X(f ))(p) for f ∈ CX(p) ∞_{(M ).}

We use:

Lemma 1.28. This operator is local, i.e. f := 0 near p ⇒ (Xf )(p) = 0

Proof. —Unvollständig——— ⇒ 0 = X(ϕf ) = (Xϕ) f |{z} :=0 + ϕ |{z} :=1 near p (Xf ) ⇒ Xf := 0 near p, i.p. (Xf)(p) = 0.

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The smoothness of the vector field p → X(p) determined by the operator X can be seen near a point p by choosing local coordinates:

X(q) = m X i=1 (Xhi)(q) ∂ ∂xi |q for q near p,

where the hi are auxiliary functions which coincide near p with the i.th coordinate

function.

The space Γ(T M ) of smooth vector fields on M is a module over the ring C∞(M ), where multiplication is defined pointwise:

(ϕX)_(p) = ϕ(p)X(p) for ϕ ∈ C∞(m), X ∈ Γ(T M ). In terms of derivations:

(ϕX)f = ϕ(Xf ), ϕ, f ∈ C∞(M ), X ∈ Γ(T M ).

An integral curve or trajectory of the vector field X is a differential curve c: I → M with ˙c = X ◦ c (∗) where ˙c = [τ → c(t + τ )] = dc [τ → t + τ ] | {z } ∂ ∂τ|t = dc ∂ ∂τ|t∈ Tc(t)M.

We rewrite (∗) in local coordinates:

X(p) =Xai(p) ∂ ∂xi |_p x ◦ c = (c1, . . . cm); ˙c(t) = X ˙ci(t) ∂ ∂xi |c(t)

(∗) becomes a system of first order ODEs:

˙c_i= a_i◦ c = (a_i◦ x−1)

| {z }

:=˜ai

◦(x ◦ c)

⇒ ˙ci(t) = ˜ai c1(t), . . . cm(t)first order smooth coefficients.

The local theory of ODEs yields that the initial value problem ˙c = X ◦ c, c(t₀) = p₀ has unique local solutions, which depend smoothly on the initial conditions.

In other words: To (p0, t0) ∈ M × R exists an open nbh U0 of p0, > 0 and a smooth

map Φ : U₀× (t₀− ; t₀+ ) −→ M with (_∂Φ ∂t = X ◦ Φ Φ(r, t0) = idU0 , where ∂Φ ∂t := dΦ ∂ ∂t.

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The local flow Φ(p, r) is the integral curve with t0 → p.

Globally one obtains: Due to the uniquenes of local solutions, there is for p ∈ M an unique maximal integral curve

cp: ( α(p) | {z } ∈[−∞,0) , ω(p) | {z } ∈(0,+∞] ) −→ M with cp(0) = p. 20.11.2012

The function α : M → [−∞, 0) is upper semicontinuous and ω : M → (0, ∞] is lower semicontinuous, i.e. for p ∈ M and α(p) < a < 0 < b < ω(p) there exists a neighbour-hood U of p such that cq is defined on an interval containing [a, b] for all q ∈ U . In other

words, the set

DX _{:= {(p, t) ∈ M × R | α(p) < t < ω(p)}}

is open in M × R. It is the natural domain of definition for the global flow

φX: DX → M, (p, t) 7→ c_p(t)

of X. For a vector field X ∈ Γ(T M ) and p ∈ M , there is a maximal integral curve

cp: (α(p), ω(p)) → M with cp(0) = p. The set DX is open in M × R. The smoothness

of the lcoal flows implies the uniqueness of φX. The uniqueness of the solutions to the ODE implies the group property of the flow φX:

φX(p, t1+ t2) = φX(φX(p, t1), t2)

This holds whereever it is defined.

Definition 1.29. A vector field X ∈ Γ(T M ) is called complete if DX = M × R. That is, all integral curves are defined for all R.

In this case the maps φ_t := φX_{(_, t) : M → M are well–defined and are}

diffeomor-phisms. We have the group property

φt1+t2 = φt1◦ φt2

So the map R → Diff(M ), t 7→ φt is a group homomorphism Such a homomorphism is

called a 1–parameter–subgroup of diffeomorphisms.

Example. In a compact manifold all vector fields are complete. Example.

1. If there is a ε0> 0 such that all integral curves cp of the vector field X are defined

(at least) in (−ε₀, ε0), then X is complete.

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1.5.1 The Lie bracket

Let X, Y ∈ Γ(T M ) be vector fields. We think of a vector field as a derivation in order du construct the operator:

[X, Y ] :C∞(M ) →C∞(M ), f 7→ X(Y f ) − Y (Xf )

This is a differential operator of order ≤ 2, but because we are taking skew-symmetrization, the second order part disappears and we actually obtain an operator of first order. We verify that [X, Y ] is again a derivation; in particular, it is a vector field.

• R–linearity is clear.

• The product rule can be seen by calculation: [X, Y ](f g) = X(Y (f g)) − Y (X(f g))

= (XY f )g + (Y f )(Xg) + (Xf )(Y g) + f (XY g) − (Y Xf )g − (Xf )(Y g) − (Y f )(Xg) − f (Y Xg)

= (XY f − Y Xf )g + f (XY g − Y Xg) = g[X, Y ]f + f [X, Y ]g Definition 1.30. The map [_, _] : Γ(T M ) × Γ(T M ) → Γ(T M ) is called the Lie bracket.

We can write the Lie bracket in local coordinates: Let X = Xi∂i, Y = Yi∂i and

[X, Y ] = Zi∂i, then [X, Y ] = Xi∂i(Yj∂j) − Yi∂i(Xj∂j) = Xi(∂_iYj)∂_j+ XiYj∂i∂j− Yi(∂iXj)∂j− YiXj∂i∂j = Xi(∂_iYj)∂_j− Yi(∂_iXj)∂_j so Zj =X i Xi∂Y j ∂xi − Yi∂X j ∂xi !

Remark. Our definition of the Lie bracket only works for C∞ vector fields, since we are considering vector fields as derivations. We can take the description in local coordinates as a definition of the Lie bracket forCk≥1vector fields. It is well defined and independent of the local coordinate chart that we chose because this is true in the C∞ case. The formula also shows that the Lie bracket of two Ck vector fields (i.e. Xi, Yi are Ck functions) is aCk−1 vector field.

Proposition 1.31 (Algebraic properties of the Lie bracket). The Lie bracket

1. is R–linear.

2. is skew–symmetric: [X, Y ] = −[Y, X]. 3. fulfills the Jacobi identity:

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Proof. R–linearity and skew–symmetry are clear. To prove the Jacobi identity, observe

that

[[X, Y ], Z]f = [X, Y ]Zf − Z[X, Y ]f = XY Zf − Y XZf − ZXY f + ZY Xf [[Y, Z], X]f = [Y, Z]Xf − X[Y, Z]f = Y ZXf − ZY Xf − XY Zf + XZY f [[Z, X], Y ]f = [Z, X]Y f − Y [Z, X]f = ZXY f − XZY f − Y ZXf + Y XZf sums up to 0.

Definition 1.32. An R–vector space V together with a multiplication [_, _] : V × V →

V with the properties 1-3 is called a Lie algebra over R.

The geometric meaning of the Lie bracket will become clear after we discuss the notion of a Lie derivative in the next section.

If ϕ, ψ ∈C∞(M ), then [ϕX, ψY ] = ϕψ[X, Y ] + ϕ(Xψ)Y − ψ(Y ϕ)X.

1.5.2 The Lie derivative of vector fields

We cannot define a directional derivative ∂vY of a vector field Y ∈ Γ(T M ) in the

direction of a vector v ∈ T_pM without some additional structure (a connection, this will

be defined later) because the vectors Y (q) ∈ T_qM live in different tanget spaces.

But with a flow φX of a vector field X ∈ Γ(T M ) we are able to idetify the tangent spaces along a trajectory of the the flow and in this way to derivate a vector field Y in the direction of the flow of X.

Let X, Y ∈ Γ(T M ). We define the derivative of Y in the direction of X, called Lie

derivative, by: LXY (p) = d dt t=0 dφX_−t(Y ◦ φX_t (p)) ∈ T_pM

The fact that this is smooth can be seen e.g. in local coordinates. Therefore the vetor field LXY is smooth, i.e. LXY ∈ Γ(T M ).

Proposition 1.33. LXY = [X, Y ].

Lemma 1.34. If X(p) 6= 0, then we can find local coordinates near p ∈ M such that

X = ∂1.

Proof. We choose a smooth map q : B_2δ_{(0) ⊂ R}n−1 → M such that q(0) = p. It has an injective differential dq0 at 0 and X(p) 6∈ Im( dq0). For ε > 0 small enough we consider

the smooth function

H : (−ε, ε) × Bδ(0) → M, (x1, . . . , xn) 7→ φX(q(x2, . . . , xn), x1)

and see that dH₀ is invertible ( dH₀ = (X(p), dq₀)). Hence H is a local diffeomorphism, i.e. for ε, δ > 0 small enough H is a diffeomorphism into an open neighbourhood of p. The inverse H−1 is a local chart with the desired property.

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Proof of proposition. We have to show that LXY (p) = [X, Y ](p) for every p ∈ M . First

assume X(p) 6= 0. We work in local coordinates. By the lemma we can assume X = ∂₁. Let ψ : M ⊃ U → Rnbe a corresponding coordinate chart. We identify U with its image

ψ(U ) and work in ψ(U ) to simplify the notation. We have φX_t (x) = x + te1, so dφXt =

id_Rn. From Y ◦ φX_t (x) = Y (x + te₁) it follows that dφX_−t(Y ◦ φX_t (x)) = Y (x + te₁) ∈ Rn.

Write Y in local coordinates, Y =P

Yj∂j|p, then LXY (p) = d dt t=0 X j Xj(x + te1) ∂ ∂xj p =X j ∂Xj ∂x1 ∂ ∂xj p =   ∂ ∂x1 ,X j Xj ∂ ∂xj  (p)

which is [X, Y ](p), so L_XY (p) = [X, Y ](p). Now let X(p) = 0. Consider the set B = {q ∈ M | X(q) 6= 0} ⊂ M.

We have shown that L_XY (q) = [X, Y ](q) for all q ∈ B. By continuity it also hold in B.

It remains to show the proposition for the case X = 0 in a neighbourhood of p. In this case the equality LXY (p) = 0 = [X, Y ](p) is trivial.

Another geometric interpretation of the Lie bracket is that it measures the noncom-mutativity of flows:

Proposition 1.35. For vector fields X, Y ∈ Γ(T M ) the following assertions are

equiv-alent:

1. [X, Y ] = 0.

2. φX and φY commute for small times in the following sense: For p ∈ M there exists an open neighbourhood U of p and ε > 0 such that φX_t φY_s = φY_sφX_t in U for all s, t ∈ [−ε, ε].

Remark. If X, Y are complete vector fields and the flows commute for small times, then

they commute φX_s φY_t = φY_t φX_s for all s, t ∈ R.

Proof of proposition. So show ‘⇒’, let q ∈ U ⊂ M and t0∈ (−ε, ε). Then we have

d dt t=t0 dφX_−t(Y ◦ φX_t (q)) = dφX_−t 0 d dh h=0 dφX_−h(Y ◦ φX_h(φX_t 0(q))) = dφX_−t 0LXY (φ X t0(q)) = 0

Hence dφX_−t(Y ◦ φX_t (q)) = Y (q) for all t ∈ (−ε, ε). We claim that φX_t sends trajectories of Y in trajectories of Y . This holds since φX_t ◦ cY

q(0) = φXt (q) and

(φX_t ◦ cY_q)•(s) = dφX_t ( ˙cY_q(s)) = dφ_tX(Y (cY_q(s)) = Y (φX_t (cY_q(s))) = Y (φX_t ◦ cY_q(s)) , so φX_t ◦ cY

q = cYφX t (q)

is a trajectory of Y through φY_t (q). That is

φX_t ◦ φY_s(q) = c_φX

t (q)(s) = φ Y

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Conversely, if for (p, s, t) in a neighbourhood of (q, 0, 0) ∈ M × R × R holds φYs ◦ φXt (p) =

φX_t ◦ φY

s(p), then φX−t(cφX

t (p)(s)) = φ Y

s(p) = cYp(s) and derivation with respect to s at

s = 0 gives

dφX_−t(Y ◦ φX_t (p)) = Y (p) which again derived wrt. t at t = 0 yields

[X, Y ](p) = L_XY (p) = 0. 1.6 Distributions and foliations

1.6.1 Foliations

An immersed submanifold of a manifold M is the image of an injective immersion ι : N →

M of a manifold N . It relaxes the notions of an (embedded) submanifold. Recall: ι(N )

is an embedded submanifold if and only if ι is a homeomorphism into its image.

Definition 1.36. A k–dimensional foliation of a manifold Mm is a partition of M in disjoint k–dimensional submanifolds (called the leaves of the foliation). This partition is locally trivial in the following sense: At every point x ∈ M there is a local chart of the form κ : U → Dk× Dm−k_{, where D}k _{is a k–dimensional disk, such that the intersection}

of each leaf with U is a (necessarily countable) union of ‘layers’ of the form κ−1(Dk× pt). In other words: The partition must be locally equivalent (via local diffeomorphism) to the model foliation of Rm = Rk× Rm−k _{by the leaves R}k_{× pt.}

27.11.2012

One can regard foliations as geometric structures on manifolds givenby atlases with certain restrictions on the coordinate changes: A k–dimensional foliation on M (with connected leaves) corresponds to a maximal subatlas of the differentiable structure whose coordinate changes are of the form

Rk× Rm−k 3 (x, y) 7→ (f (x, y), g(y))

Starting from such an atlas, one recovers the (connected) leaves as equivalence classes of ‘horizontally connectable’ points: x ∼ y if and only if there exists a curve c from x to

y such that wrt. the foliation charts κ the Rm−k–component of κ ◦ c is locally constant. Equivalently, one can recover the leaves as path components of the finer topology on

M generated by the subsets κ−1(U ×pt) with U ⊂ Dkopen. (‘transverse discretization’). Example.

0. Products M = M₁× M₂ is a foliation by the leaves M₁× pt.

1. Submersions f : Mm → Nn_{. The level sets f}−1_{(pt) form an (m − n)–dimensional}

foliation of M as a consequence of the implicit function theorem. The leaves are embedded submanifolds.

2. Let X be a vector field on M without zeros. The (traces of the) trajectories form a 1–dimensional foliation. The leaves are in general not embedded.

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Definition 1.37. A foliation is called a fiber bundle if

1. All leaves (also called fibers) are diffeomorphic to a fixed manifold (‘model fiber’)

F ,

2. The foliation is locally trivial in transverse direction: Every leaf has a saturated (i.e. a union of leaves) open neighbourhood U on which the foliation is a product foliation, i.e. there exist bundle charts κUm → Fk_{× S}m−k _{such that the subsets}

κ−1(F × pt) are leaves.

The space of leaves B carries a natural strcuture as a smooth manifold such that the natural projection π : M → B becomes a submersion. (In particular, B carries the quotient topology wrt. π). Namely, the bundle charts κ induce chartsκ : π(U ) → S and

the coordinate changesκ0◦ κ−1are defined on open subsets and are smooth. B is clearly Hausdorff and therefore a smooth manifold.

We call F the fiber, M the total space and B the base space and write the fiber bundle like this: F //M π B Example.

0. The product foliation π : F × B → B is a fiber bundle.

1. The tangent bundle π : T M → M of an m–dimensional manifold M is a bun-dle with fiber Rm. Here, the fiber also carries an algebraic structure as an m– dimensional R–vector spaces, varying smoothly. So the tangent bundle is a vector

bundle.

2. The Hopf fibration is the fiber bundle

C ⊃ S1 //S2n−1 ⊂ Cn

π

CPn−1

The fibers are the trajectories of the vector field z 7→ iz. We have the following bundle charts: Ui = {z ∈ S2n−1| zi 6= 0} → S1× Cn−1 z 7→ _z i |zi| ,z1 zi , . . . ,czi zi , . . . ,zn zi 1.6.2 Distributions

Definition 1.38. A k–dimensional distribution on a manifold M is a family D = (Dp)p∈M of k–dimensional vector subspaces Dp ⊂ TpM which depend smoothly on p

in the sense: For every p ∈ M exist smooth vector field X₁, . . . , Xk near p such that

Differential Geometry

Vorlesung aus dem Wintersemester 2012/13