Vector bundle morphism

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 fiberwise surjective morphism of constant-rank vector bundles has a kernel vector subbundle. For a morphism from a trivial rank-$N$ bundle, a local full-row-rank matrix $F(x)$ gives the continuous kernel projection $I-F(x{)}^T(F(x)F(x{)}^T{)}^{-1}F(x)$. Changing the local frame in the target leaves this projection unchanged. Thus its kernel defines a continuous map into the appropriate Grassmannian.

A smooth vector bundle morphism from a vector bundle $E$ to itself, covering the identity on the base, is a vector bundle endomorphism. It is fibrewise linear and is equivalently a smooth section of $\mathrm{End}E$. In a local frame its matrix is smooth and changes by conjugation under changes of frame. An almost complex structure is a real endomorphism of the tangent bundle satisfying $J^2=-I$.

A holomorphic bundle map is a holomorphic map between total spaces of holomorphic vector bundles that covers a holomorphic base map and is complex-linear on each fibre. Over a fixed base, its matrix in holomorphic local frames has holomorphic entries.

Every $C^{\infty }(M)$-linear map between the modules of smooth sections of finite-rank smooth vector bundles is induced by a unique vector bundle morphism. A smooth bump function shows that the map is local. A bumped local frame then proves that a section vanishing at $x$ has image vanishing at $x$. Evaluating the image therefore defines a well-defined fiberwise linear map. The images of the bumped frame give smooth matrix columns, proving the resulting morphism is smooth. This elementary argument is also described in Brian Conrad's bundle notes.

A vector bundle morphism is an isomorphism if it admits an inverse of the same kind. A smooth fiberwise bijective bundle morphism automatically has a smooth inverse, because inverse matrices in vector bundle trivializations depend smoothly on their entries wherever their determinants are nonzero.