Vector bundle morphism

Let $\pi_E:E\to M$ and $\pi_F:F\to N$ be smooth real or complex vector bundles. A vector bundle morphism (also called a fiberwise linear bundle map) from $E$ to $F$ is a pair $(\Phi,f)$ consisting of a smooth map $\Phi:E\to F$ and a smooth map $f:M\to N$ such that:

1. Covers the base map: $\pi_F\circ \Phi = f\circ \pi_E$.

2. Fiberwise linearity: for every $x\in M$, the induced map on fibers

   $$\Phi_x:E_x\to F_{f(x)},\qquad \Phi_x(v):=\Phi(v),$$

   is $\mathbb F$-linear (with $\mathbb F=\mathbb R$ or $\mathbb C$ according to the bundles).

If $M=N$ and $f=\mathrm{id}_M$, one often says $\Phi:E\to F$ is a bundle map over $M$.

Composition of vector bundle morphisms is defined by composition of the total-space maps and the base maps, and yields a category of smooth vector bundles and bundle morphisms.

Examples

1. Differential of a smooth map. For any smooth map $f:M\to N$, the differential

   $$df:TM\to TN$$

   is a vector bundle morphism covering $f$ between the tangent bundles .

2. Projection from a direct sum. For bundles $E,F\to M$, the projection $\mathrm{pr}_E:E\oplus F\to E$ is a bundle morphism over $\mathrm{id}_M$, fiberwise the linear projection $E_x\oplus F_x\to E_x$.

3. Inclusion of a subbundle. If $E\subseteq F$ is a smooth vector subbundle over the same base $M$, then the inclusion map $E\hookrightarrow F$ is a vector bundle morphism over $\mathrm{id}_M$.