Bundle morphism

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

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

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

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

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.

Examples

1. 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 $f$.
2. Differentials: for any $f:M\to N$, the differential $df:TM\to TN$ is a bundle morphism covering $f$ (and is fiberwise linear).
3. 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 $f$.