Chapter 10 - Vector Bundles Flashcards
Define: vector bundle, smooth vector bundle, line bundle, total space, base, protection, trivial bundle
Let M be a topological space. A REAL VECTOR BUNDLE OF RANK K OVER M is a topological space E together with a surjective continuous map pi : E –> M satisfying the following conditions:
- For each p in M, the fiber E_p = pi^-1(p) over p is endowed with the structure of a k-dimensional real vector space
- For each p in M, there exists a neighborhood U of p in M and a homeomorphism phi : pi^-1(U) –> U x R^k called a LOCAL TRIVIALIZATION OF E OVER U satisfying the following conditions:
a. pi_U o phi = pi (where pi_U : U x R^k –> U is the projection)
b. For each q in U, the restriction of phi to E_q is a vector space isomorphism from E_q to q x R^k = R^k
If M, E smooth manifolds, pi smooth, and local trivializations can be chosen to be diffeomorphisms, then E is called a SMOOTH VECTOR BUNDLE. Local trivializations are called smooth local trivializations.
A rank-1 vector bundle is often called a real LINE BUNDLE.
E = TOTAL SPACE OF BUNDLE M = BASE pi = PROJECTION
If there exists a local trivialization of E over all of M (called a GLOBAL TRIVIALIZATION OF E) then E is said to be a TRIVIAL BUNDLE. In this case E is homeomorphic/diffeomrophic to M x R^k
Discuss the Mobius Bundle
pg 251
Prove the tangent bundle is a vector bundle
pg 252
What does the composition of two smooth local trivializations look like? Proof? Transition function?
What is the transition function between two local trivializations of TM?
pg 252 - 253
What is vector bundle chart lemma? Proof?
pg 253 - 254
Define: section of vector bundle, local vs. global, smooth, rough, zero section, support
Let pi : E –> M be a vector bundle. A SECTION OF E is a section of the map pi (ie a continuous map sigma : M –> E satisfying pi o sigma = Id_M). This means sigma(p) is an element of the fiber E_p for each p in M.
pg 255-256
Examples of sections of vector bundles?
- Sections of TM are vector fields on M
- Vector field along S
- Sections of product bundle M x R^k = Continuous functions from M to R^k. or smooth sections - smooth functions. Example: C^inf(M) = smooth sections of trivial line bundle.
pg 256
Let E –> M be a smooth vector bundle. Discuss the algebraic structure of the set of smooth sections of E, Lambda(E)
Lambda(E) is a C^inf(M) - module
pg 257
What is the extension lemma for vector bundles? Relation to other extension lemmas?
The following are roughly the same and proved similarly:
- Extension lemma for smooth functions
- Extension lemma for smooth vector fields
- Extension lemma for smooth vector bundles
In each of the above, we consider a map defined on a closed subset and smooth in the sense that around each point we can extend to a smooth function on an open neighborhood. Partition of unity allows us to glue together to get a nice global extension
DIFFERENT: We also have thought about extending a function from a submanifold S < M. Here we were concerned with extending functions in C^inf(S) - i.e. functions which are smooth in all coordinate charts of S - unrelated to ambient space. This is a more constrained problem - in general only can extend to a neighborhood of S if S embedded and can only extend to all of M if properly embedded.
pg 257
Define: linearly independent sections of a vector bundle, span E, local frame for E over U, global frame, smooth frame
Discuss completion of local frames for vector bundles. Relate to proposition for vector fields
pg 257 -258
- We can extend independent sections to local frame
- We can extend independent vectors in one fiber Ep to a local frame
- Can extend smooth linearly independent sections that span a closed subset to a frame over some open subset
This is identical to proposition early for extending vector fields
Discuss the relationship between local frames and local trivializations. Proof? Global?
Relationship of smooth triviality with bundle homomorphisms? Proof?
For each smooth local trivialization phi we can define the LOCAL FRAME ASSOCIATED WITH PHI.
For each smooth local frame, we can define the smooth local trivialization associated with the local frame.
Thus we have a bijection: local frames <–> local trivializations
Cor. A smooth vector bundle is smoothly trivial <=> it admits a smooth global frame.
Applied to the tangent bundle this says TM is trivial <=> M is parallelizable.
pg 258-260
A smooth rank-k vector bundle over M is smoothly trivial <=> it is smoothly isomorphic over M to the product bundle M x R^k pg. 262
What are the component functions of a section w.r.t. a local frame? Local frame criterion for smoothness?
Use fact that local frame yields local trivialization
pg 260
Define: bundle homomorphism, isomorphism
If pi : E –> M and pi’ : E’ –> M’ are vector bundles, a continuous map F : E –> E’ is called a BUNDLE HOMOMORPHISM if there exists a map f : M –> M’ satisfying pi’ o F = f o pi with the proper that for each p in M, the restricted map F|Ep : E_p –> E’_f(p) is linear. Say that F COVERS f.
SPECIAL CASE: E and E’ vector bundles over the same base space M. A BUNDLE HOMOMORPHISM OVER M is a bundle homomorphism covering the identity map of M - a continuous map E –> E’ such that pi’ o F = pi whose restriction to each fiber is linear.
What is the relationship between smooth bundle homomorphisms F: E –> E’ over M and C^inf(M)-module homomorphisms Lambda(E) –> Lambda(E’)? Proof?
A map F : Lambda(E) –> Lambda(E’) is linear over C^inf(M) <=> there is a smooth bundle homomorphism f : E –> E’ over M such that F(sigma) = f o sigma for all sigma in Lambda(E)
So we have a bijection smooth bundle homomorphisms <–> C^inf(M)-module homomorphism.
Bundle => module easy
<= harder
pg 262 - 263
Examples of bundle homomorphisms?
- global differential dF : TM –> TN
- Inclusion of subbundle
pg 262
Examples induced by homomorphisms of spaces of sections
- Multiplication of vector field X by f in C^inf(M)
- Cross product
- dot product
