Question

Someone enlighten me.

Answer 1

Let \(M\) be a topological space. My book defines a real vector bundle of rank \(k\) over \(M\) as a topological space \(E\) together with a surjective, continuous map \(\pi:E\to M\) such that for each \(p\in M\) , the set \(E_p=\pi^{-1}\left(p\right)\subseteq E\) is endowed with the structure of a \(k\) -dimensional real vector space, and such that there exists a neighborhood \(U\) of \(p\) and a homeomorphism \(\Phi:\pi^{-1}\left(U\right)\to U\times\mathbb R^k\) such that \(\pi_1\circ\Phi=\pi\) , where \(\pi_1:U\times\mathbb R^k\to U\) is canonical, and such that for each \(q\in U\) , it is the case that \(\left.\Phi\right|_q:E_q\to\left\{q\right\}\times\mathbb R^k\cong\mathbb R^k\) is a linear isomorphism.

Answer 2

My question is: What the hell is the motivation for this definition?

Answer 3

I know that given a manifold \(M\), it is the case that \(TM\) (its tangent bundle) satisfies the definition of a real vector bundle of rank \(n\), where \(n\) is the dimension of \(M\). But what motivated this construction? I cannot find documentation on the matter.

Answer 4

Hmm, if you are asking about intuition, then the wiki article on vector bundles seems to help a lot. If you are asking about the historical motivation, then i wouldn't know.