vector bundle

Informally, a vector bundle is a collection of vector spaces that lie above each point in a manifold. They really would have been better named vector space bundle… but oh well. We attach a vector space to each point for some smooth manifold, topological manifold, or even an abstract topological space.

Formal definition

Note the definition here is for a smooth manifold, but this same construction can be used for other categories. A vector bundle is the data , for smooth manifolds with a smooth map such that for every point , there exists an open neighborhood and a smooth map such that

commutes. So we want points in the vector bundle above the open set and in to get mapped to each other. Basically, we understand the vector bundle using since we know what that looks like.

2. , is a vector space isomorphism. So we can think of the vector space attached to a point as being just .

Morphisms

Given two smooth manifolds and that both have vector bundles and respectively along with a smooth map , a vector bundle morphism is a map such that the following diagram commutes:

Isomorphisms

Obviously, an isomorphims of vector bundles is a morphism with an inverse morphism, but in this case a bijective morphims is sufficient to be an isomorphism.

Subbundles

Given a vector bundle , a subbundle is a vector bundle such that

2. The subset is a linear subspace for each
3. The vector space structure of is inherited from as a subspace.

Since we require to be a vector bundle, all the fibers must be non-empty and all have the same dimension.

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References

@lee2013 - Chapter 10