Geometry of manifolds, lecture 9 M. Verbitsky Geometry of manifolds Lecture 9: Serre-Swan theorem Misha Verbitsky Math in Moscow and HSE April 15, 2013 1
Geometry of manifolds, lecture 9 M. Verbitsky Locally trivial fibrations DEFINITION: A smooth map f : X − → Y is called a locally trivial fi- bration if each point y ∈ Y has a neighbourhood U ∋ y such that f − 1 ( U ) is diffeomorphic to U × F , and the map f : f − 1 ( U ) = U × F − → U is a projection. In such situation, F is called the fiber of a locally trivial fibration. DEFINITION: A trivial fibration is a map X × Y − → Y . DEFINITION: A total space of a vector bundle on Y is a locally trivial → Y with fiber R n , with each fiber V := f − 1 ( y ) equipped fibration f : X − with a structure of a vector space, smoothly depending on y ∈ Y . DEFINITION: A vector bundle is a locally free sheaf of C ∞ M -modules. REMARK: Let π : B − → M be a total space of a vector bundle, U ⊂ M open π subset, and B ( U ) the space of all smooth sections of π − 1 ( U ) − → U . Then B is a locally free sheaf of C ∞ M -modules . REMARK: This construction is an “equivalence of categories”; see below for a definition. 2
Geometry of manifolds, lecture 9 M. Verbitsky Categories DEFINITION: A category C is a collection of data called “objects” and “morphisms between objects” which satisfies the axioms below. DATA. Objects: The set O b( C ) of objects of C . Morphisms: For each X, Y ∈ O b( C ), one has a set M or( X, Y ) of mor- phisms from X to Y . Composition of morphisms: For each ϕ ∈ M or( X, Y ) , ψ ∈ M or( Y, Z ) there exists the composition ϕ ◦ ψ ∈ M or( X, Z ) Identity morphism: For each A ∈ O b( C ) there exists a morphism Id A ∈ M or( A, A ). AXIOMS. Associativity of composition: ϕ 1 ◦ ( ϕ 2 ◦ ϕ 3 ) = ( ϕ 1 ◦ ϕ 2 ) ◦ ϕ 3 . Properties of identity morphism: For each ϕ ∈ M or( X, Y ), one has Id x ◦ ϕ = ϕ = ϕ ◦ Id Y . 3
Geometry of manifolds, lecture 9 M. Verbitsky Functors and equivalence of categories DEFINITION: Let C 1 , C 2 be categories. A covariant functor from C 1 to C 2 is the following collection of data. (i) A map F : O b( C 1 ) − → O b( C 2 ). (ii) A map F : M or( X, Y ) − → M or( F ( X ) , F ( Y )), defined for each X, Y ∈ O b( C 1 ). These data define a functor from C 1 to C 2 , if F ( ϕ ) ◦ F ( ψ ) = F ( ϕ ◦ ψ ), and F (Id X ) = Id F ( X ) . DEFINITION: Two functors F, G : C 1 − → C 2 are called equivalent if for each X ∈ O b( C 1 ) there exists an isomorphism Ψ X : F ( X ) − → G ( X ), such that for each ϕ ∈ M or( X, Y ) one has F ( ϕ ) ◦ Ψ Y = Ψ X ◦ G ( ϕ ) . DEFINITION: A functor F : C 1 − → C 2 is called equivalence of categories if there exist functors G, G ′ : C 2 − → C 1 such that F ◦ G is equivalent to an identity functor on C 1 , and G ′ ◦ F is equivalent to identity functor on C 2 . EXAMPLE: Let C be a category of finite-dimensional vector spaces ovet R with a fixed basis (morphisms are linear maps), and C ′ a category with O b( C ′ ) = {∅ , R , R 2 , R 3 , ... } , and morphisms also linear maps. Prove that the inclusion map C ′ − → C is an equivalence of categories, but not an isomor- phism . 4
Geometry of manifolds, lecture 9 M. Verbitsky Total space of a vector bundle from its sheaf of sections DEFINITION: Category of vector bundles C b is a category where objects are locally free C ∞ M -sheaves, and morphisms are morphisms of C ∞ M -sheaves such that all kernels and cokernels are locally free. EXERCISE: Prove that it is a category. DEFINITION: Category of total spaces of vector bundles C t is a category where objects are total spaces of vector bundles, and morphisms of total spaces over M are maps B 1 − → B 2 compatible with projection to M , the multiplicative structure, and of constant rank at each fiber. EXERCISE: Prove that it is a category. THEOREM: Let π : B − → M be a total space of a vector bundle, U ⊂ M π open subset, and B ( U ) the space of all smooth sections of π − 1 ( U ) − → U . Then this map defines an equivalence of categories C b − ˜ → C t . REMARK: The proof was given in the last lecture, using different lan- guage. EXERCISE: Produce a proof of this theorem. 5
Geometry of manifolds, lecture 9 M. Verbitsky Tensor product DEFINITION: Let V, V ′ be R -modules, W a free abelian group generated by v ⊗ v ′ , with v ∈ V, v ′ ∈ V ′ , and W 1 ⊂ W a subgroup generated by combinations rv ⊗ v ′ − v ⊗ rv ′ , ( v 1 + v 2 ) ⊗ v ′ − v 1 ⊗ v ′ − v 2 ⊗ v ′ and v ⊗ ( v ′ 1 + v ′ 2 ) − v ⊗ v ′ 1 − v ⊗ v ′ 2 . Define the tensor product V ⊗ R V ′ as a quotient group W/W 1 . EXERCISE: Show that r · v ⊗ v ′ �→ ( rv ) ⊗ v ′ defines an R -module structure on V ⊗ R V ′ . REMARK: Let F be a sheaf of rings, and B 1 and B 2 be sheaves of locally free ( M, F )-modules. Then U − → B 1 ( U ) ⊗ F ( U ) B 2 ( U ) is also a locally free sheaf of modules. DEFINITION: Tensor product of vector bundles is a tensor product of the corresponding sheaves of modules. 6
Geometry of manifolds, lecture 9 M. Verbitsky Dual bundle and bilinear forms DEFINITION: Let V be an R -module. A dual R -module V ∗ is Hom R ( V, R ) with the R -module structure defined as follows: r · h ( . . . ) �→ rh ( . . . ). CLAIM: Let B be a vector bundle, that is, a locally free sheaf of C ∞ M - π Define B ∗ ( U ) as a space of modules, and Tot B − → M its total space. smooth functions on π − 1 ( U ) linear in the fibers of π . Then B ∗ ( U ) is a locally free sheaf over C ∞ ( M ) . DEFINITION: This sheaf is called the dual vector bundle , denoted by B ∗ . Its fibers are dual to the fibers of B . DEFINITION: Bilinear form on a bundle B is a section of ( B ⊗ B ) ∗ . A symmetric bilinear form on a real bundle B is called positive definite if it gives a positive definite form on all fibers of B . Symmetric positive definite form is also called a metric . A skew-symmetric bilinear form on B is called non-degenerate if it is non-degenerate on all fibers of B . 7
Geometry of manifolds, lecture 9 M. Verbitsky Subbundles DEFINITION: A subbundle B 1 ⊂ B is a subsheaf of modules which is also a vector bundle, and such that the quotient B / B 1 is also a vector bundle. DEFINITION: Direct sum ⊕ of vector bundles is a direct sum of corre- sponding sheaves. EXAMPLE: Let B be a vector bundle equipped with a metric (that is, a positive definite symmetric form), and B 1 ⊂ B a subbundle. Consider a subset Tot B ⊥ 1 ⊂ Tot B , consisting of all v ∈ B| x orthogonal to B 1 | x ⊂ B| x . Then Tot B ⊥ 1 is a total space of a subbundle, denoted as B ⊥ 1 ⊂ B , and we have an isomorphism B = B 1 ⊕ B ⊥ 1 . REMARK: A total space of a direct sum of vector bundles B ⊕ B ′ is home- omorphic to Tot B × M Tot B ′ . EXERCISE: Let B be a real vector bundle. Prove that B admits a metric. PROPOSITION: Let A ⊂ B be a sub-bundle. Then B ∼ = A ⊕ C . Proof: Find a positive definite metric on B , and set C := B ⊥ . 8
Geometry of manifolds, lecture 9 M. Verbitsky Tangent bundle PROPOSITION: Let M ⊂ R n be a smooth submanifold of R n , and TM ⊂ R n × R n the set of all pairs ( v, x ) ∈ M × R n , where x ∈ M × R n is a point of M , and v ∈ R n a vector tangent to M in m , that is, satisfying d ( M, m + tv ) lim − → 0 . t t − → 0 Then the natural additive operation on TM ⊂ M × R n (addition of the second argument) and a multiplication by real numbers defines on TM a structure of a relative vector space over M , that is, makes TM a total space of a vector bundle. Moreover, this vector bundle is isomorphic to a tangent bundle, that is, to the sheaf Der R ( C ∞ M ). Proof. Step 1: For each z ∈ M , we can choose coordinates in a neighbour- hood of z in R n in such a way that M = R k ⊂ R n . Therefore, it would suffice to prove proposition when M = R k ⊂ R n . Proof. Step 2: In this case, TM = R k × R k is a total space of a vector bundle, of the same dimension as the tangent bundle. It remains to construct a sheaf morphism from the sheaf of sections of TM to Der R ( C ∞ M ), inducing an isomorphism. 9
Geometry of manifolds, lecture 9 M. Verbitsky Tangent bundle (cont.) Proof. Step 3: Let π x : R n − → T x M be an orthogonal projection map. By the inverse function theorem, π x | M : M − → T x M is a diffeomorphism in a neighbourhood of x ∈ M . Let U x ⊂ T x M be such an open neighbourhood π x and π − 1 x ( U x ) − → U x a diffeomphism. Proof. Step 4: For each vector v ∈ T x M , and f ∈ C ∞ M , let D v ( f ) be the f ∈ C ∞ U x along v , where ˜ f ( z ) = f ( π − 1 derivative of ˜ x ( z )). Then a section γ ∈ TM ( U ) defines a derivation D γ ( f )( z ) := D γ | z ( f ). We obtained a sheaf Ψ → Der R ( C ∞ M ) . homomorphism TM − Step 5: The vector bundles TM and Der R ( C ∞ M ) have the same Proof. dimension, and for each non-zero vector v ∈ T x M , the corresponding deriva- tion is non-zero, hence ker Ψ = 0 . DEFINITION: The tangent bundle of M , as well as its total space, is denoted by TM . When one wants to distinguish the total space and the tangent bundle, one writes Tot( TM ). 10
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