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 $x\in M$ 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 $(E,M,\pi ,k)$, for $E,M$ smooth manifolds with a smooth map $\pi :E⟶M,$ such that for every point $p\in M$, there exists an open neighborhood $U\subset M$ and a smooth map $\phi :{\pi }^{-1}(U)⟶U\times {\mathbb{R}}^k$ such that

commutes. So we want points in the vector bundle above the open set and in $U\times {\mathbb{R}}^k$ to get mapped to each other. Basically, we understand the vector bundle using $U\times {\mathbb{R}}^k$ since we know what that looks like.

2. $\forall p\in M$, $\phi {|}_{{\pi }^{-1}(\{p\})}:{\pi }^{-1}(\{p\})⟶{\sigma }^{-1}(\{p\})=\{p\}\times {\mathbb{R}}^k$ is a vector space isomorphism. So we can think of the vector space attached to a point as being just ${\mathbb{R}}^k$.

Morphisms

Given two smooth manifolds $M$ and $M^{\prime }$ that both have vector bundles $E$ and $E^{\prime }$ respectively along with a smooth map $f:M\to M^{\prime }$, a vector bundle morphism is a map $F:E\to E^{\prime }$ 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 $\pi :E\to M$, a subbundle is a vector bundle ${\pi }_D:D\to M$ such that

1. ${\pi }_D=\pi {|}_D$
2. The subset $D_p=D\cap E_p$ is a linear subspace for each $p\in M$
3. The vector space structure of $D_p$ is inherited from $E_p$ as a subspace.

Since we require $D$ to be a vector bundle, all the fibers ${\pi }_D^{-1}(p)$ must be non-empty and all have the same dimension.

---

References

@lee2013 - Chapter 10