and write $X\sim_f Y$. We have $X\sim_f Y$ iff the maps ${\mathcal{L}}_X\circ f^*$ and $f^*\circ{\mathcal{L}}_Y$ from ${\mathcal{F}}(M)$ to ${\mathcal{F}}(N)$ coincide. If $X_1,X_2\in{\mathfrak{X}}(N)$ are $f$-related with $Y_1,Y_2$ respectively, then $X_1+X_2$ and $[X_1,X_2]$ are also $f$-related, respectively with $Y_1+Y_2$ and $[Y_1,Y_2]$. A given $X\in{\mathfrak{X}}(N)$ will not be $f$-projectable in general. However, if $f$ is a diffeomorphism, then every vector field $X\in{\mathfrak{X}}(N)$ is projectable onto a unique vector field on $M$, to wit, $Y=Tf\circ X\circ f^{-1}$, and we say $X$ is the pull-back of $Y$. A vector field $X\in{\mathfrak{X}}(M)$ is *invariant* under a diffeomorphism $f$ of $M$ iff $X\sim_f X$. On ${\mathbb{R}}_t$ (or on an open interval $I\subset{\mathbb{R}}_t$) there is a canonical vector field $d/dt$. A curve $\gamma:{\mathbb{R}}_t\to M$ is said to be an integral curve for a vector field $X\in{\mathfrak{X}}(M)$ if $d/dt$ and $X$ are $\gamma$-related: $\dot\gamma:=T\gamma\circ d/dt=X\circ \gamma$. Well-known theorems assert that the integral curves of a vector field define a local ${\mathbb{R}}_t$-action or flow [@Carniceria].

By a *vector field along* $f$ we understand a map $Y:N\to
TM$ such that $\tau_M\circ Y=f$: $$\xymatrix{&TM\ar[d]^{\tau_M} \\ N\ar[r]_f\ar[ur]^Y &M}$$ It is clear that the concept is just a particular case of a more general one: section along the map $f$ over a general bundle 