Vector Bundle Morphism

A vector bundle morphism between two vector bundles $E$ and $F$ over the same base manifold $M$ is a smooth map $\mathrm{\Phi }:E\to F$ satisfying the following conditions:

1. Base map: The map $\mathrm{\Phi }$ projects to the identity on the base $M$, meaning ${\pi }_F\circ \mathrm{\Phi }={\pi }_E$, where ${\pi }_E:E\to M$ and ${\pi }_F:F\to M$ are the projection maps.
2. Linear structure: For each point $p\in M$, the restriction of $\mathrm{\Phi }$ to the fibers is a linear map: ${\mathrm{\Phi }}_p:E_p\to F_p$, where $E_p={\pi }_E^{-1}(\{p\})$ and $F_p={\pi }_F^{-1}(\{p\})$.

Inducing a Bundle Morphism from a Section Map

A $C^{\mathrm{\infty }}(M)$-linear map $\mathrm{\Psi }:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)$, where $\mathrm{\Gamma }(E)$ and $\mathrm{\Gamma }(F)$ denote the spaces of smooth sections of $E$ and $F$ respectively, induces a vector bundle morphism $\mathrm{\Phi }:E\to F$. This is because such a map determines a smooth fiberwise-linear map between $E$ and $F$ that respects the base manifold structure.

Key idea: we can define $\mathrm{\Phi }(u):=\mathrm{\Psi }(s)(p)$, where $s$ is a section of $E$ such that $s(p)=u$. To show the independence with respect to the choice of $s$, we write everything in a local frame of sections $\{e_i\}$ around $p$, and then use the $C^{\mathrm{\infty }}(M)$-linearity of $\mathrm{\Psi }$.