Vector bundle - Knowlpedia

A smooth $\mathbb K$ -vector bundle of rank $k$ , for $\mathbb K\in\{\mathbb R,\mathbb C\}$ , over a smooth manifold $M$ is a smooth fiber bundle $\pi:E\to M$ together with the structure of a $k$ -dimensional $\mathbb K$ -vector space on each fiber $E_x=\pi^{-1}(x)$ , such that:

the typical fiber is $\mathbb K^k$ , and
there exists an open cover $\{U_i\}$ of $M$ with local trivializations $\Phi_i:\pi^{-1}(U_i)\to U_i\times\mathbb K^k$ whose restrictions $\Phi_i|_{E_x}:E_x\to \mathbb K^k$ are linear isomorphisms for each $x\in U_i$ .

Equivalent characterizations

Equivalently, the transition functions of such a bundle take values in $\mathrm{GL}(k,\mathbb K)\subset \mathrm{Diff}(\mathbb K^k)$ . The tangent bundle and cotangent bundle are the fundamental real examples; many constructions in differential geometry (e.g. a connection on a vector bundle) are formulated for vector bundles.

Examples

Trivial rank- $k$ bundle: $M\times \mathbb{R}^k\to M$ is a vector bundle with the obvious fiberwise linear structure.
Tangent and cotangent bundles: for an $n$ -manifold $M$ , $TM\to M$ and $T^*M\to M$ are rank- $n$ vector bundles.
Möbius line bundle: a nontrivial rank-1 real vector bundle over $S^1$ with transition function $-1$ on the overlap of two arcs.