Theorems Geometry. Joshua Ruiter. April 8, 2018

Transcription

1 Theorems Geometry Joshua Ruiter Aril 8, 2018 Aendix A: Toology Theorem 0.1. Let f : X Y be a continuous ma between toological saces. If K X is comact, then f(k) Y is comact. 1 Chater 1 Theorem 1.1 (Toological Invariance of Dimension). A nonemty n-dimensional toological manifold cannot be homeomorhic to an m-dimensional manifold unless m = n. Theorem 1.2. Every toological manifold has a countable basis of recomact oen balls. Theorem 1.3. Let M 1,..., M k be toological manifolds of dimensions n 1,..., n k resectively. The roduct sace i M i is a toological manifold of dimension i n i. For ( i ) i M i, we choose charts (U i, φ i ) in each M i with i U i and then the mas φ 1... φ k : i U i R n n k given by are the coordinate charts. ( 1,..., k ) (φ 1 ( 1 ),..., φ k ( k )) Theorem 1.4. Every toological manifold is locally comact. Theorem 1.5. Every toological manifold is aracomact. Theorem 1.6. A toological manifold has countably many comonents, each of which is an oen subset and a connected toological manifold. Theorem 1.7. Every smooth atlas for a manifold M is contained in a unique maximal smooth atlas. Theorem 1.8. Two smooth atlases for a manifold M determine the same smooth structure if and only if their union is a smooth atlas. 1

2 Theorem 1.9. Every smooth manifold has a countable basis of regular coordinate balls. Theorem Any oen subset U of a manifold M is also a manifold, by forming smooth charts for U by taking intersections of U with smooth charts for M. (It has the same dimension rovided U is nonemty.) Theorem Let M 1,..., M k be smooth manifolds of dimension n 1,..., n k resectively. Then the charts defined for the roduct manifold are smoothly comatible. Theorem 1.12 (Smooth Manifold Chart Lemma). Let M be a set and suose there is a collection {U α } of subsets and a collection of mas φ α : U α R n so that 1. φ α is a bijection with oen image in R n 2. φ α (U α U β ) is oen for every α, β 3. If U α U β, the ma φ β φ 1 α is smooth 4. A countable subcollection of U α is a cover for M 5. For, q M, either both are contained in some U α or there are disjoint U α, U β so that U α and q U β. Then M has a unique smooth manifold structure such that (U α, φ α ) are smooth charts. 2 Chater 2 Theorem 2.1. Let M be a smooth manifold and f : M R. Then f is smooth if and only if f is smooth in every smooth chart. Theorem 2.2. Let M be a smooth manifold. Then C (M) is a commutative ring (with ointwise addition and multilication). Theorem 2.3. Smooth mas are continuous. Theorem 2.4. Let M, N be manifolds and F : M N be a ma. equivalent: The following are 1. F is smooth. 2. For every M, there exist smooth charts (U, φ), (V, ψ) with U, F () V such that U F 1 (V ) is oen in M and ψ F φ 1 : φ(u F 1 (V )) ψ(v ) is smooth. 3. F is continuous and there exist smooth atlases {U α, φ α } for M and {V β, ψ β } for N such that ψ β F φ 1 α is smooth for all α, β. 4. For every M, there is a neighborhood U such that F U is smooth. 2

3 Theorem 2.5 (Gluing Lemma for Smooth Mas). Let M, N be smooth manifolds, and let {U α } α A be an oen cover for M. Suose that for each α A we have a smooth ma F α : U α N such that they agee on overlas, that is, F α Uα U β = F β Uα U β for all α, β. Then there is a unique smooth ma F : M N such that F Uα = F α for all α A. Theorem 2.6. Let M, N be smooth manifolds, and let F : M N be a ma. Then F is smooth if and only if every coordinate reresentation of F is smooth. Theorem 2.7. Constant mas are smooth. Theorem 2.8. The identity ma is smooth. Theorem 2.9. Inclusion mas are smooth. Theorem The comosition of smooth mas is smooth. Theorem Let M 1,..., M k be smooth manifolds. Let π i : k j=1 M j M i be the rojection ( 1,..., k ) i. A ma F = (F 1,... F k ) : N k i=1 M i is smooth if and only if each F i = π i F : N M i is smooth. Theorem Let M be a smooth manifold and (U, φ) be a smooth chart. Then φ : U φ(u) is a diffeomorhism. Theorem The comosition of diffeomorhisms is a diffeomorhism. Theorem Diffeomorhism is an equivalence relation on smooth manifolds. Theorem 2.15 (Diffeomorhism Invariance of Dimension). Smooth manifolds can only be diffeomorhic if they have the same dimension, unless one is emty (in which case both are emty). Theorem Let f : M R k. If x M \ su(f), then f(x) = 0. Note that we may have f(y) = 0 for y su(f). Proof. If f(x) 0, then x su(f). Theorem Let M be a smooth manifold and X = {X α } α A an oen cover. Then there exists a smooth artition of unity subordinate to X. Theorem Let M be a smooth manifold and A, U be subsets with A U and A closed and U oen. Then there exists a smooth bum function for A suorted in U. Theorem Let M be a smooth manifold, with A M closed and f : A R k a smooth function. Then for any oen subset U with A U, there exists a smooth function f : M R k such that f A = f and su F U. Theorem Let M be a smooth manifold. Then there exists a smooth exhaustion function for M, that is, there exists a smooth function f : M R such that f 1 ((, a]) is comact for all a R. Theorem Let M be a smooth manifold and K a closed subset of M. Then there is a smooth nonnegative function f : M R such that f 1 (0) = K. 3

4 3 Chater 3 Theorem 3.1. Let M be a smooth manifold and M. Then T M is a vector sace (over R). Proof. Let v, w T M and a R. We need to show that v + w T M and av T M, so we need to show that v + w and av are derivations at. (v + w)(fg) = v(fg) + w(fg) = f()vg + g()vf + f()wg + g()wf Thus v + w is a derivation at. = f()(vg + wg) + g()(vf + wf) = f()(v + w)g + g()(v + w)f (av)(fg) = a(v(fg)) = a(f()vg + g()vf) = f()(avg) + g()(avf) Thus av is a derivation at. = g()(av)f + f()(av)g Theorem 3.2. Let M be a smooth manifold, and let M, v T M, and f, g C (M). If f is a constant function, then vf = 0. If f() = g() = 0, then v(fg) = 0. Proof. First suose that f is the constant function f() = 1. vf = v(ff) = f()vf + f()vf = 2vf = vf = 0 Then by linearity, if g() = c we have vg = v(cf) = c(vf) = c(0) = 0. The other assertion is obvious. Theorem 3.3. Let M, N be smooth manifolds, F : M N be smooth, and M. Then df (v) = F (v) is a derivation at F (). Theorem 3.4. Let M, N, P be smooth manifolds, and F : M N and G : N P be smooth mas, and let M. Then 1. df = F : T M T F () N is linear. 2. d(g F ) = dg F () df : T M T G F () P. In the other notation, (G F ) = G F 3. d(id M ) = (Id M ) = Id TM. 4. If F is a diffeomorhism, then df = F is an isomorhism, and (F ) 1 = (F 1 ). Theorem 3.5. Let M be a smooth manifold, M, and v T M. Let f, g C (M) and suose there is a neighborhood U of such that f(x) = g(x) for x U. Then vf = vg. Theorem 3.6. Let M be a smooth manifold and U M be oen. Let ι : U M be the inclusion ma. Then for U, the differential dι = ι : T U T M is an isomorhism. Theorem 3.7. Let M be a smooth n-manifold. Then for M, T M is an n-dimensional vector sace (over R). 4

5 Theorem 3.8. Let V be a finite-dimensional (real) vector sace with its standard smooth manifold structure. For a V the ma v D v a is a canonical isomorhism from V to T a V, and for any linear ma L : V W, the diagram commutes: V W L = = T a V L T La W Theorem 3.9. Let M 1,... M k be smooth manifolds and let π j : (M 1... M k ) M j be the rojection ( 1,... k ) j. Then is an isomorhism. α : T (M 1... M k ) T 1 M 1... T k M k α(v) = ((π 1 ) (v),..., (π k ) (v)) = (d(π 1 ) (v),..., d(π k ) (v)) Theorem Let R n. The derivations x 1, x 2,..., x n form a basis for T R n. Theorem Let M be a smooth n-manifold and M. Then T M is an n-dimensional (real) vector sace, and for any smooth chart (U, φ) = (U, (x 1,..., x n )) containing, the coordinate vectors x 1, x 2,..., x n form a basis for T M. Therefore, any v T M can be written as v = n i=1 v i x i Furthermore, x j : U R is a smooth function, so ( ) v(x j ) = v i (x j ) = v i xj () = x i vj xi Theorem Let M, N be smooth manifolds and let F : M N be a smooth ma. Let (U, φ) be a chart for M with U and (V, ψ) be a chart for N with F () V. Let F = ψ F φ 1 be the coordinate reresentation of F and = φ() be the coordinate reresentation of. Then ( ) F = F j x i x ( ) i y j F () 5

6 That is, the matrix of the linear ma df = F is the Jacobian matrix of F at, F 1 F ()... 1 () x 1 x n..... F n F ()... n () x 1 x n Theorem 3.13 ( Chain Rule for Coordinate Changes). Let M be a smooth n-manifold and (U, φ) = (U, x i ) and (V, ψ) = (V, y i ) be two smooth charts. The change of basis from { } { } to is given by x i y i x i = n ( ) y j x ( ) i y j where = φ(). For v T M, we can write v in terms of both bases as v = a i x i = b i y i j=1 Then the relationshi between a i and b i is given by n ( ) y b i j = x ( ) i i=1 Theorem Let M be a smooth n-manifold. Then T M is a smooth manifold of dimension 2n. Theorem Let M be a smooth n-manifold such that M can be covered by a single smooth chart. Then T M is diffeomorhic to M R n. Theorem Let M, N be smooth manifolds and F : M N be smooth. Then df : T M T N is smooth. Theorem Let M, N, P be smooth manifolds and F : M N, G : N P be smooth mas. Then 1. d(g F ) = dg df 2. d(id M ) = Id T M 3. If F is a diffeomorhism, then df : T M T N is also a diffeomorhism and (df ) 1 = d(f 1 ). Theorem Let M be a smooth manifold and γ : J M be a smooth curve. Then for a function f C (M), γ (t 0 ) acts on f by ( ) γ d (t 0 )f = dγ f = d dt dt (f γ) = (f γ )(t 0 ) t0 t0 6 a i

7 Theorem Let M be a smooth manifold and γ : J M be a smooth curve, and let t 0 J. Let (U, φ) be a smooth chart containing t 0, with φ() = (x 1 (),... x n ()). Then for t sufficiently close to t 0, we can write γ as γ(t) = (γ 1 (t),... γ n (t)) and then γ (t 0 ) = dγi dt (t 0) x i γ(t0 ) Theorem Let M be a smooth manifold and M, and v T M. Then there exists a curve γ : ( ɛ, ɛ) M with γ(0) = and γ (0) = v for some ɛ > 0. Theorem Let F : M N be a smooth ma and let γ : J M be a smooth curve. For t 0 J, the velocity of F γ : J N at t 0 is (F γ) (t 0 ) = df (γ (t 0 )) Theorem Let F : M N be a smooth ma, and M and v T M. Then df (v) = F (v) = (F γ) (0) for any smooth curve γ : J M satisfying 0 J, γ(0) =, and γ (0) = v. 4 Chater 4 Theorem 4.1. Let F : M N. min(dim M, dim N). For all M, the rank of F at does not exceed Theorem 4.2. Let F : M N be a smooth ma and M. If df = F is surjective, then there is a neighborhood U of such that F U is a submersion. Similarly, if df = F is injective, then there is a neighborhood U of such that F U is an immersion. Theorem 4.3. Let M 1,... M k be smooth manifolds, and let π : M 1... M k M i be the rojection ( 1,..., k ) i. Then π i is a smooth submersion. Theorem 4.4. Let M be a smooth manifold and γ : J M a smooth curve. Then γ is a smooth immersion if and only if γ (t) is never zero. Theorem 4.5. Let M be a smooth manifold and T M the tangent bundle. Then the standard rojection π : T M M given by (, v) is a smooth submersion. Theorem 4.6. A comosition of surjective functions is surjective. Theorem 4.7. A comosition of injective functions is injective. Theorem 4.8. A comosition of smooth submersions is a smooth submersion. Theorem 4.9. A comosition of smooth immersions is a smooth immersion. 7

8 Theorem 4.10 (Proerties of Local Diffeomorhisms). 1. A comosition of local diffeomorhisms is a local diffeomorhism. 2. A finite roduct of local diffeomorhisms is a local diffeomorhism. 3. A bijective local diffeomorhism is a diffeomorhism. 4. A local diffeomorhism is a smooth immersion and a smooth submersion. Conversely, a ma that is a smooth submersion and smooth immersion is a local diffeomorhism. 5. A smooth immersion between manifolds of equal dimension is a local diffeomorhism. Likewise for smooth submersions. Theorem Let F : M N be a smooth ma. Then F is a local diffeomorhism if and only if it is both a smooth immersion and a smooth submersion. Consequently, if dim M = dim N and F is either a smooth sumbersion or a smooth immersion, then F is a local diffeomorhism. Theorem 4.12 (Inverse Function Theorem). Let M, N be smooth manifolds and F : M N a smooth ma. If M such that df = F is invertible, then there are connected neighborhoods U 0 of and V 0 of F () such that F U0 : U 0 V 0 is a diffeomorhism. Theorem 4.13 (Rank Theorem). Let M, N be smooth manifolds of dimension m, n resectively and F : M N a smooth ma with constant rank r. Then for M there exist smooth charts (U, φ) for M with U and (V, ψ) for N with F () V such that F (U) V and F has a coordinate reresentation of the form F (x 1,..., x r, x r+1,... x m ) = (x 1,..., x r, 0,... 0) In articular, if F is a smooth submersion, then and if F is a smooth immersion then F (x 1,... x n, x n+1,..., x m ) = (x 1,... x n ) f(x 1,..., x m ) = (x 1,..., x m, 0,..., 0) Theorem Let F : M N be a smooth ma and suose M is connected. Then the following are equivalent: 1. F has constant rank. 2. For M, there exist smooth charts (U, φ) and (V, ψ) with U, F () V, such that the coordinate reresentation F = ψ F φ 1 is linear. Theorem 4.15 (Global Rank Theorem). Let M, N be smooth manifolds such that F : M N is a smooth ma of constant rank. Then 1. If F is surjective, then it is a smooth submersion. 8

9 2. If F is injective, then it is a smooth immersion. 3. If F is bijective, then it is a diffeomorhism. Theorem The comosition of smooth embeddings is a smooth embedding. Theorem If M is a smooth manifold and U M is an oen submanifold, then the inclusion ma is a smooth embedding. Theorem Let M 1,..., M k be smooth manifolds and i M i be oints. Then the ma ι j : M j i M i given by is a smooth embedding. ι j (q) = ( 1,..., j 1, q, j+1,..., k ) Theorem Let M, N be smooth manifolds and F : M N be an injective smooth immersion. If any of the following holds, then F is a smooth embedding. 1. F is an oen or closed ma. 2. F is a roer ma. 3. M is comact. 4. M has emty boundary and dim M = dim N. Theorem 4.20 (Local Embedding Theorem). Let M, N be smooth manifolds and F : M N be a smooth ma. Then F is a smooth immersion if and only if every M has a neighborhood U M such that F U : U N is a smooth embedding. Theorem 4.21 (Local Section Theorem). Let M, N be smooth manifolds and π : M N a smooth ma. Then π is a smooth submersion if and only if every M is in the image of a local section of π. Theorem Let M, N be smooth manifolds and π : M N a smooth submersion. Then π is an oen ma. If π is surjective, then it is a quotient ma. 5 Chater 5 Theorem 5.1. Let M be a smooth manifold. The embedded submanifolds of codimension zero in M are exactly the oen submanifolds. Theorem 5.2. Let M, N be smooth manifolds, and F : N M be a smooth embedding. Let S = F (N). Then with the subsace toology (from M), S is a toological manifold, and it has a unique smooth structure making it into an embedded submanifold of M with the roerty that F is a diffeomorhism between N and F (N). Theorem 5.3. Let M, N be smooth manifolds and let N. Then the slice M {} is an embedded submanifolds of M N, and it is diffeomorhic to M. 9

10 Theorem 5.4. Let M, N be smooth manifolds with dimension m, n resectively. Let U M be oen and f : U N be a smooth ma. Let Γ(f) M N be the grah of f. Then Γ(f) is an embedded m-dimensional submanifold of M N. Theorem 5.5. Let M be a smooth manifold with or without boundary and S M an embedded submanifold. Then S is roerly embedded if and only if it is a closed subset of M. Theorem 5.6. Every comact embedded submanifold is roerly embedded. Theorem 5.7. Let M, N be smooth manifolds, where N may have boundary, and let f : M N be a smooth ma. Then Γ(f) is a roerly embedded submanifold of M N. Theorem 5.8 (Local Slice Criterion for Embedded Submanifolds). Let M be a smooth n- manifold. If S M is an embedded k-dimensional submanifold, then S satisfies the local k-slice condition. Conversely, if S M is a subset that satisfies the local k-slice condition, then S is a toological manifold with the subsace toology and has a smooth structure making it a k-dimensional embedded submanifold of M. Theorem 5.9. Let M be a smooth n-manifold with boundary, and give M the subsace toology. Then M is a toological (n 1)-dimensional submanifold (without boundary), and it has a unique smooth structure such that it is a roerly embedded submanifold of M. Theorem 5.10 (Constant-Rank Level Set Theorem). Let M, N be smooth manifolds and let φ : M N be a smooth ma with constant rank r. Then every level set of φ is a roerly embedded submanifold of codimension r in M. Theorem 5.11 (Submersion Level Set Theorem). Let M, N be smooth manifolds and φ : M N be a smooth submersion. Then every level set of φ is a roerly embedded submanifold of M, with codimension equal to the dimension of N. Theorem Let M, N be smooth manifolds and φ : M N a smooth ma. Suose that dim M < dim N. Then every oint in M is a critical oint of φ. Theorem Let M, N be smooth manifolds and φ : M N a smooth submersion. Then every oint in M is a regular oint. Theorem Let M, N be smooth manifolds and φ : M N a smooth ma. The sets of regular oints of φ is an oen subset of M. Theorem 5.15 (Regular Level Set Theorem). Every regular level set of a smooth ma between smooth manifolds is a roerly embedded submanifold with codimension equal to the dimension of the codomain. Theorem Let S be a subset of a smooth m-manifold M. Then S is an embedded k-submanifold of M if and only if every oint of S has a neighborhood U M such that U S is a level set of a smooth submersion φ : U R m k. Theorem Every embedded submanifold is also an immersed submanifold. 10

11 Theorem Let M be a smooth manifold with or without boundary, and N be a smooth manifold, and F : M N an injective smooth immersion. Let S = F (N). Then S has a unique toology and smooth structure such that it is a smooth immersed submanifold of M and such that F : N S is a diffeomorhism. Theorem Let M be a smooth manifold with or without boundary and S M an immersed submanifold. If any of the following holds, then S is an embedded submanifold. 1. S has codimension zero in M. 2. The inclusion ma S M is roer. 3. S is comact. Theorem 5.20 (Immersed Submanifolds are Locally Embedded). Let M be a smooth manifold with or without boundary and S M an immersed submanifold. Then for S, there exists a neighborhood U of with U S such that U is an embedded submanifold of M. Theorem Let M, N be smooth manifolds with or without boundary, and F : M N a smooth ma, and S M an immersed (or embedded) submanifold. Then F S : S N is smooth. Theorem Let M be a smooth manifold without boundary, and S M an immersed submanifold, and F : N M a smooth ma such that f(n) S. If F is continuous as a ma from N S, then F : N S is smooth. Theorem Let M be a smooth manifold and S M an embedded submanifold. Then every smooth ma F : N M whose image is contained in S is also smooth as a ma from N to S. Theorem Let M be a smooth manifold and S M an immersed submanifold, and f C (S). Then 1. If S is embedded, then there exists a neighborhood U of S in M and a smooth function f C (U) such that f S = f. 2. If S is roerly embedded, then the neighborhood above can be taken to be all of M. Theorem Let M be a smooth manifold with or without bondary, and S M an immersed (or embedded) submanifold, and S. Then for v T M, we have v T S if and only if there is a smooth curve γ : J M whose image is contained in S, where γ is also smooth as a ma into S, such that 0 J, γ(0) =, and γ (0) = v. Theorem Let M be a smooth manifold and S M an embedded submanifold, and S. As a subsace of T M, the tangent sace T S is recisely T S = {v T M : vf = 0 whenever f C (M) and f s = 0} Theorem Let M be a smooth n-manifold with boundary, and let M and (x i ) be smooth boundary coordinates on a neighborhood of. Write v T M as v i. Then x i v is inward ointing v n > 0 v is outward ointing v n < 0 v T ( M) v n = 0 11

12 6 Chater 6 Theorem 6.1. Let A R n with m(a) = 0 and F : A R n be a smooth ma. Then m(f (A)) = 0. Theorem 6.2. Let F : M N be a smooth ma between smooth manifolds. If A M has measure zero, then F (A) N has measure zero. Theorem 6.3 (Sard s Theorem). Let F : M N be a smooth ma between smooth manifolds. The set of critical values of F has measure zero. Theorem 6.4 (Corollary to Sard s Theorem). Let F : M N be a smooth ma between smooth manifolds. If dim M < dim N, then F (M) has measure zero in N. 7 Chater 7 Theorem 7.1. If G is a smooth manifold with a grou structure such that the ma (g, h) gh 1 is smooth, then G is a Lie grou. Theorem 7.2. Every Lie grou homomorhism has constant rank. Theorem 7.3. A Lie grou homomorhism is a Lie grou isomorhism if and only if it is bijective. Theorem 7.4. Let G be a Lie grou and W G an oen neighborhood of the identity. Then W generates an oen subgrou of G. If W is connected, it generates a connected oen subgrou of G. If G is connected, then W generates G. Theorem 7.5. Let G be a Lie grou and let G 0 be the connected comonent containing the identity. Then G 0 is a normal subgrou of G and it is the only connected oen subgrou. Every connected comonent of G is diffeomorhic to G 0. Theorem 7.6. Let F : G H be a Lie grou homomorhism. Then ker F is a roerly embedded Lie subgrou of G, with codimension equal to rank F. Theorem 7.7. Let F : G H be an injective Lie grou homomorhism. Then im F has a unique smooth manifold structure such that it is a Lie subgrou of H and F : G im F is a Lie grou isomorhism. Theorem 7.8 (Equivariant Rank Theorem). Let G be a Lie grou and let M, N be smooth manifolds such that G acts transitively (and smoothly) on M and N. If F : M N is a smooth equivariant ma, then F has constant rank. 8 Chater 8 Theorem 8.1 (Smoothness Criterion for Vector Fields). Let M be a smooth manifold with or without boundary and X : M T M a rough vector field. If (U, x i ) is a smooth chart on M, then the restriction of X to U is smooth if and only if its comonent functions with resect to U are smooth. 12

13 Theorem 8.2 (Extension Lemma for Vector Fields). Let M be a smooth manifold with or without boundary and let A M be a closed subset. Suose X is a smooth vector field on A. If U is an oen subset containing A, then there is a smooth global vector field X on M such that X A = X and su X U. Theorem 8.3. Let M be a smooth manifold with or without boundary and let X : M T M be a rough vector field. The following are equivalent: 1. X is smooth. 2. For every f C (M), the function Xf is smooth. 3. For every oen subset U M and every f C (M), the function Xf is smooth on U. Theorem 8.4. A ma D : C (M) C (M) is a derivation if and only if it is of the form Df = Xf for a smooth vector field X X(M). Theorem 8.5. Let F : M N be a smooth ma and X X(M) and Y X(N). Then X and Y are F -related if and only if for every smooth real-valued function f defined on an oen subset of N we have X(f F ) = (Y f) F Theorem 8.6. Let F : M N be a diffeomorhism. Then for every X X(M), there is a unique smooth vector field on N that is F -related to X. Theorem 8.7 (Coordinate Formula for Lie Bracket of Vector Fields). Let X, Y X(M), and let (U, x i be a smooth chart for M. We can write X = X i and Y = Y i on U. x i x i Then the coordinate exression for [X, Y ] is given by [X, Y ] = (X i Y j x i Y i Xj x i ) x j = (XY j Y X j ) x j Theorem 8.8. Let M be a smooth manifold and (U, x i ) be local coordinates. Then [ ] x, = 0 i x j Theorem 8.9 (Proerties of the Lie Bracket). The Lie bracket of vector fields is bilinear, antisymmetric, and satisfies the Jacobi identity. For f, g C (M), [fx, gy ] = fg[x, Y ] + (fxg)y (gy f)x Theorem Let F : M N be a smooth ma, and let X 1, X 2 X(M) and Y 1, Y 2 X(N) such that X 1 is F -related to Y 1 and X 2 is F -related to Y 2. Then [X 1, X 2 ] is F -related to [Y 1, Y 2 ]. Theorem Let F : M N be a diffeomorhism and X 1, X 2 X(M). Then F [X 1, X 2 ] = [F X 1, F X 2 ] 13

14 Theorem Let G be a Lie grou. The set of left-invariant smooth vector fields on G is a Lie algebra under the above bracket. That is, it is a vector sace and closed under brackets. Theorem Let G be a Lie grou. Define ɛ : Lie(G) T e G by ɛ(x) = X e. Then ɛ is a vector sace isomorhism. Thus dim Lie(G) = dim G. Theorem Every Lie grou is arallelizable. 9 Chater 9 Theorem 9.1. Let V be a smooth vector field on a smooth manifold M. For M, there exists ɛ > 0 and a smooth curve γ : ( ɛ, ɛ) M so that γ is an integral curve of V, with γ(0) =. Theorem 9.2. Let F : M N be a smooth ma and X X(M) and Y X(N). Then X and Y are F -related if and only if F mas integral curves of X to integral curves of Y. That is, if γ is an integral curve of X, then F γ is an integral curve of Y. Theorem 9.3. Let θ : R M M be a smooth global flow on M. The infinitesimal generator of θ is a smooth vector field on M, and each curve θ is an integral curve of V. Theorem 9.4. Let θ : D M be a smooth flow. The infinitesimal generator of θ is a smooth vector field and the curves θ are integral curves of V. Theorem 9.5 (Fundamental Theorem of Flows). Let V be a smooth vector field on M. There is a unique smooth maximal flow θ : D M whose infinitesimal generator is V. For this flow, θ is a maximal integral curve of V. Theorem 9.6. Let F : M N be a smooth ma and X X(M) and Y X(N). Let θ be the flow of X and η the flow of Y. If X and Y are F -related, then the diagram commutes. M t θ t M t F N t η t F N t Theorem 9.7 (Diffeomorhism Invariance of Flows). Let F : M N be a diffeomorhism. If X X(M) and θ is the flow of X, then the flow of F X is η t = F θ t F 1. Theorem 9.8. Let V be a smooth vector field on M. Suose there exists ɛ > 0 such that for every M, the domain of θ contains ( ɛ, ɛ). Then V is comlete. Theorem 9.9. Every comactly suorted smooth vector field is comlete. Theorem On a comact manifold, every smooth vector field is comlete. Theorem Every left-invariant vector field on a Lie grou is comlete. 14

15 Theorem Let V be a smooth vector field on M and let θ : D M be the flow of V. If M is a regular oint, then θ : D M is a smooth immersion. Theorem Let M be a smooth manifold and V, W X(M). The following are equivalent: V and W commute, V is invariant under the flow of W, and W is invariant under the flow of V. Theorem Let M be a smooth manifold, and let V X(M). A smooth covariant tensor field A is invariant under the flow of V if and only if L V A = Chater 10 Theorem Let E be a smooth vector bundle over M. The rojection ma π : E M is a smooth submersion. Theorem The tangent bundle is a vector bundle. Theorem Let π : E M be a smooth vector bundle of rank k over M. Suose φ : π 1 (U) U R k and ψ : π 1 (V ) V R k are local trivializations such that U V. Then there is a smooth function τ : U V GL(k, R) such that Such a ma τ is called a transition function. φ ψ 1 (, v) = (, τ()v) Theorem A vector field is a global section of the tangent bundle. 11 Chater 11 Theorem Let V be a finite-dimensional vector sace. If (E 1,..., E n ) is a basis of V, then define ɛ i : V R by ɛ i (E j ) = δ j i. Then (ɛ1,..., ɛ n ) is a basis for V, called the dual basis. Theorem The assignment that sends a vector sace to its dual sace and a linear ma to its dual ma is a contravariant functor from the category of real vector saces to itself. Theorem Let V be a finite-dimensional vector sace and define ξ : V V by defining ξ(v) : V R to be the ma ξ(v)(ω) = ω(v). Then ξ is a vector sace isomorhism. ( ) Theorem If (U, x i ) is a smooth chart for a manifold, then is a basis for T x i M, and its dual basis is (dx i ). Theorem Let F C (M). comonent of M. Then df = 0 if and only if f is constant on each Theorem 11.6 (Pullback Commutes with d). Let F : M N be a smooth ma, and let u C (M) and ω be a 1-form on N. Then F (du) = d(u F ) 15

16 Theorem Let F : M N be a smooth ma and let ω be a 1-form on N. Let (U, x i ) a smooth chart on M and (V, y j ) be a smooth chart on N with F (U) V. Then we can write ω as ω = ω j dy j, and 12 Chater Chater Chater 14 F ω = (ω j F )d(y j F ) = (ω j F )df j Theorem 14.1 (Proerties of Elementary k-covectors). Let (E i ) be a basis for V and (ɛ i ) the dual basis, and I a multi-index. If I has a reeated index, ɛ I = 0. If J = I σ for some σ S k, then ɛ I = (sgn σ)ɛ J. Also, ɛ I (E j1,..., E jk ) = δ I J Theorem Let V be an n-dimensional vector sace. If (ɛ i ) is a basis for V, then the collection {ɛ I : I is an increasing multi-index of length k} is a basis for Λ k (V ). Theorem Let I, J be multi-indices. Then ɛ I ɛ J = ɛ IJ. Theorem 14.4 (Proerties of Wedge Product). The wedge roduct is bilinear, associative, and anti-commutative. The anticommutative roerty says that where k, l are the resective degrees of ω, η. ω η = ( 1) kl (η ω) Theorem Let V be an n-dimensional vector sace. If (ɛ i ) is a basis for V, then As a result, ɛ i 1... ɛ i k = ɛ I ω 1... ω k (v 1,..., v k ) = det(ω j (v i )) Theorem 14.6 (Proerties of Interior Multilication). Let V be a finite-dimensional vector sace and v V. Then i v i v = 0, and for ω Λ k (V ) and η Λ l (V ) we have More generally, i v (ω η) = (i v ω) η + ( 1) k ω (i v η) = (v ω) η + ( 1) k ω (v η) v (ω 1... ω k ) = k ( 1) (i 1) ω i (v)ω 1... ω i... ω k i=1 16

17 Theorem Let F : M N be a smooth ma. Then F (ω η) = F ω F η. In a smooth chart (V, y i ) for N, we have ( ) F ω I dy i 1... dy i k = (ω I F )d(y i 1 F )... d(y i k F ) I I Theorem 14.8 (Pullback for To Degree Forms). Let F : M N be smooth. Let (U, x i ) be a chart for M and (V, y i ) be a chart for N, and f C (V ). Then on U F 1 (V ), we have F (udy 1... dy n ) = (u F )(det J(F ))dx 1... dx n (Note that J(F ) is the Jacobian of F in these coordinates.) Theorem If (U, x i ) and (V, y j ) are overlaing charts on M, then on U V we have ( ) dy dy 1... dy n j = det dx 1... dx n dx i Theorem The exterior derivative exists and is unique. Theorem Let F : M N be a smooth ma. For each k, the ma F : Ω k (N) Ω k (M) commutes with d. That is, F (dω) = d(f (ω) Theorem Every exact form is closed. Theorem Let M be a smooth manifold and V X (M). A smooth covariant tensor field A is invariant under the flow of V if and only if L V A = 0. Theorem (Cartan s Magic Formula). Let M be a smooth manifold and V a smooth vector field on M, and ω a differential form on M. Then 15 Chater 15 L V ω = V (dω) + d(v ω) Theorem Let M be a smooth n-manifold. Any nonvanishing n-form on M determines a unique orientation on M. Conversely, if M is oriented, there is a smooth nonvanish n-form on M that determines that orientation. Theorem Let M be an oriented smooth manifold and D M a smooth codimesion-0 (immersed) submanifold. The orientation on M restricts to an orientation on D. If ω is an orientation form for M, then ι ω is an orientation form for D. (ι : D M is the inclusion ma.) Theorem Let F : M N be a local diffeomorhism and N be oriented. Then M has a unique orientation such that F is orientation reserving. Theorem Every arallelizable manifold is orientable. Theorem Let M be an oriented smooth n-manifold with boundary. The M is orientable, and all outward-ointing vector fields along M determine the same orientation on M. 17

18 16 Chater 16 Theorem Let U and V be oen sets in R n and let G : U V be an orientationreserving diffeomorhism. If ω is a comactly suorted n-form on V, then G ω = ω If G is orientation reversing instead of orientation reserving, then G ω = ω U U Theorem 16.2 (Proerties of Integrals). Let M, N be non-emty oriented smooth n-manifold and ω, η comactly suorted n-forms on M, and let a, b R. Denote M with oosite orientation by M. Then aω + bη = a M M = M ω is ositively oriented = V V M ω + b η M M ω ω > 0 Theorem If F : M N is an orientation reserving diffeomorhism, then ω = F ω If F is orientation reversing instead, then ω = M M Theorem Let M be an oriented smooth n-manifold and ω a comactly suorted n- form on M. Suose D 1,..., D k are oen domains of integration in R n and we have smooth mas F i : D i M satisfying 1. F i restricts to an orientation reserving diffeomorhism from D i to an oen subset W i M. 2. W i W j = for i j. 3. su ω W 1... W k N N F ω Then M ω = k i=1 D i F ω 18

19 The above theorem says, in an overly comlicated way, that if we arametrize our manifold M by finitely many charts that don t overla, then we can integrate ω over M by integrating the ullback of ω on each chart. All the technicalities of the theorem say that you don t actually need to arametrize the whole manifold - you just need to arametrize the art where ω is nonzero, and you can throw away sets of measure zero, because they don t affect integration. Theorem 16.5 (Stokes s Theorem). Let M be an oriented smooth n-manifold with boundary, and let ω be a comactly suorted smooth (n 1) form on M. Then dω = ω M The ω on the RHS actually means ι ω where ι : M M is the inclusion. If M =, then the RHS is zero. M 17 Chater 17 Theorem Diffeomorhic smooth manifolds have isomorhic de Rhan cohomology grous. Theorem Let M be a smooth manifold, written as a disjoint union of its connected comonents, M = i M i. Then H n (M) = i Hn (M i ). The above theorem means that it is trivial to comute the de Rham cohomology of a disjoint union if we know the cohomologies of the ieces. Theorem Let M be connected. Then H 0 (M) = R. Theorem Let M and N are homotoy equivalent manifolds, then H n (M) = H n (N) for all n. Theorem If M is a contractible smooth manifold, then H n (M) = 0 for n 1. Theorem 17.6 (Poincare Lemma). If U is a star-shaed oen subset of R n, then H n (U) = 0 for 1. Theorem The de Rham cohomology of R k is given by { H n (R k R n = 1 ) = 0 n 1 Theorem Let M be a connected smooth manifold. Then φ : H 1 (M) Hom(π 1 (M), R) is well-defined and injective. Theorem If M is a connected smooth manifold with finite fundamental grou, then H 1 (M) = 0. 19

20 Theorem If M is a comact connected oriented smooth n-manifold, then H n (M) = R. Theorem Let f : M N be a smooth ma between smooth manifolds. If dim M < dim N, then f is not surjective. Proof. Every oint in M is critical, since rank df < dim M < dim T f() N. Thus f(m) has measure zero by Sard s Theorem. Hence f(m) N. 20