on $M$ is endowed with the standard commutative and associative algebra structure. Let $\tau_M:TM\to M$ denote the tangent bundle to $M$. Vector fields on $M$ can be defined either as sections for $\tau_M$, that is, maps $X:M\to TM$ such that $\tau_M\circ X={{\mathrm{id}}}_M$, or as derivations of ${\mathcal{F}}(M)$. When we wish to distinguish between those roles, we denote by ${\mathcal{L}}_X$ the differential operator corresponding to the vector field $X$. Because the commutator of two derivations is again a derivation, the space ${\mathfrak{X}}(M)$ of vector fields has a Lie algebra structure; we choose to define the bracket there as the opposite of the usual one: in local coordinates, $$[X, Y]^i = Y^j \,{\partial}_j X^i - X^j \,{\partial}_j Y^i;$$ so ${\mathfrak{X}}(M)\equiv{\mathfrak{diff}}(M)$, the Lie algebra of the infinite-dimensional Lie group $\operatorname{Diff}(M)$ [@Omori]. Also ${\mathfrak{X}}(M)$ becomes a faithful ${\mathcal{F}}(M)$-module when one defines $hX(x)=h(x)X(x)$ for $h\in{\mathcal{F}}(M)$. Together, ${\mathcal{F}}(M)$ and ${\mathfrak{X}}(M)$ constitute a Lie–Rinehart algebra in the sense of [@Bigotes1].

Given a smooth map $f:N\to M$, the pull-back $f^*:{\mathcal{F}}(M)\to{\mathcal{F}}(N)$ is defined as $f^*h=h\circ f$. A vector field $X\in{\mathfrak{X}}(N)$ is said to be $f$-related with the vector field $Y\in{\mathfrak{X}}(M)$ if $Tf\circ X=Y\circ f$; we then say that the vector field $X$ is $f$-projectable onto the vector field $Y$, 