 Vector bundle

A rank-$r$ vector bundle over a topological space $B$ is a fiber bundle $\pi :E\to B$ whose fibers are vector spaces over a fixed field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. Each point has an open neighborhood $U$ and a vector bundle trivialization ${\pi }^{-1}(U)\cong U\times {\mathbb{F}}^r$ that is a linear isomorphism on each fiber.

On overlaps the vector bundle trivializations change by $(x,v)\mapsto (x,g_{ij}(x)v)$, where the invertible matrices $g_{ij}(x)$ vary continuously and satisfy the Čech cocycle condition. For a smooth vector bundle, the base and total space are smooth manifolds and these functions are smooth. The fiberwise vector addition and scalar multiplication are therefore globally well-defined.

The tangent bundle collects all tangent spaces of a smooth manifold. Its transition functions of a vector bundle are Jacobian matrices, so the chain rule supplies the cocycle identity. The Möbius line bundle supplies another example: every small patch is a product, but no global frame of a vector bundle exists. A global fiber-linear product isomorphism is the extra condition defining a trivial vector bundle.