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A morphism of vector bundles over one base is a smooth map between their total spaces that commutes with projection to the base and is linear on each fiber. In local vector bundle trivializations it is multiplication by a smoothly varying matrix.

A local trivialization of a rank-$r$ vector bundle $\pi :E\to M$ is a bundle isomorphism ${\pi }^{-1}(U)\cong U\times {\mathbb{R}}^r$ over an open set $U$, linear on each fiber. Overlapping trivializations differ by $(p,v)\mapsto (p,g(p)v)$ for a smooth $G{L}_r(\mathbb{R})$-valued function $g$. A tangent bundle chart uses the derivative of its base manifold chart for this fiber identification.

A smooth section of ${\bigwedge }^k(T^{\ast }X\otimes \mathbb{C})\otimes E$ is a differential form with values in the complex vector bundle $E$. A connection on a vector bundle extends to these forms by the graded Leibniz rule.

The Total Chern class of a direct sum of complex vector bundles is the product of their total Chern classes. In each degree, $c_k(E\oplus F)={\sum }_{i+j=k}{c}_i(E)⌣c_j(F)$. The degree-zero class is one, classes beyond the bundle rank vanish, and a trivial bundle has total class one.

The fiberwise direct sum of two vector bundles over the same base. It is formed from the fiber product of their total spaces, with the direct-sum linear structure in each fiber. Transition matrices are block diagonal.

// Target: geometry-and-topology.bigb