i'm would like to know application (or applications) for the following exercise:
Let $F:M^{m}\to N^{n}$ a submersion between smooth manifolds, let $X \in \mathfrak{X}(M)$ and $Z \in \mathfrak{X}(N)$, we say that $X$ is lift for $Z$ if $X$ and $Z$ are $F$-related.
OBS: If in addition $F$ is surjective with connected fibers. A vector field $X \in \mathfrak{X}(M)$ is a lift of smooth vector field on $N$ if and only $[V,X]$ is vertical whenever $V \in \mathfrak{X}(M)$ is vertical.
Thanks!