Bundle morphism - Knowlpedia

Let $\pi:E\to M$ and $\pi':E'\to M'$ be smooth fiber bundles. A bundle morphism from $E$ to $E'$ covering a map $f:M\to M'$ is a smooth map $\Phi:E\to E'$ such that

\pi'\circ \Phi \;=\; f\circ \pi.

Thus $\Phi$ is a fiber-preserving map whose domain and codomain carry fiber-bundle structures.

A bundle morphism is an isomorphism precisely when it is a bundle morphism whose total-space map is a diffeomorphism and whose base map is a diffeomorphism.

Equivalent characterizations

Equivalently, for every point $x\in M$ the map $\Phi$ restricts to a smooth map of fibers

\Phi_x:E_x\to E'_{f(x)},

and this fiberwise map depends smoothly on $x$ (a fact that can be made explicit using local trivializations).

Examples

Maps between products: a map $(x,u)\mapsto(f(x),g(x,u))$ from $M\times F$ to $M'\times F'$ is a bundle morphism covering $f$ .
Differentials: for any $f:M\to N$ , the differential $df:TM\to TN$ is a bundle morphism covering $f$ (and is fiberwise linear).
Pullback projection: for $f:N\to M$ and a bundle $E\to M$ , the canonical map $f^*E\to E$ from the pullback bundle is a bundle morphism covering $f$ .