Fiber-preserving map

Let $E,M,E',M'$ be smooth manifolds and let $\pi:E\to M$ and $\pi':E'\to M'$ be smooth maps . A smooth map $F:E\to E'$ is fiber-preserving if there exists a smooth map $f:M\to M'$ such that

In this situation one says that $F$ covers $f$, or that $F$ is a map over $f$. If $\pi$ is surjective, then the base map $f$ is uniquely determined by $F$.

When $\pi$ and $\pi'$ are surjective submersions (i.e. when they define fibered manifolds ), a fiber-preserving smooth map is the same structure as a bundle map . A basic example is the differential $d\varphi:TM\to TN$ associated to a smooth map $\varphi:M\to N$, which is fiber-preserving between the tangent bundles and covers $\varphi$.

Examples

1. Maps between products: given $f:M\to M'$ and a smooth map $g:M\times F\to F'$, the map $F:M\times F\to M'\times F'$, $F(x,u)=(f(x),g(x,u))$, is fiber-preserving over $f$.
2. Differential of a map: for any $\varphi:M\to N$, the differential $d\varphi:TM\to TN$ satisfies $\tau_N\circ d\varphi=\varphi\circ\tau_M$.
3. Restriction to an open set: for an open inclusion $i:U\hookrightarrow M$, the inclusion $\pi^{-1}(U)\hookrightarrow E$ is fiber-preserving over $i$.