Vector bundle

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

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

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

Examples

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