geometric interpretation of derivation between two algebras

Question

Given a smooth manifold $M,$ it is well known that any derivation of the algebra of smooth functions $C^{\infty}(M)$ can be seen (or it is associated to) a smooth vector field on $M.$ I am looking for a similar geometric meaning for a derivation $D: C^{\infty}(M)\rightarrow C^{\infty}(T^*M)$ along $\rho^*,$ where $\rho$ is the contangent bundle projection, it means: $D(fg) = \rho^*(f)D(g)+\rho^*(g)D(f),$ for every $f, g\in C^{\infty}(M).$ Thanks in advance.

Answer 1

In general, given a map of smooth manifolds $\phi:N\to M$, derivations $C^\infty M \to C^\infty N$ are in canonical one to one correspondence with sections of the pullback bundel $\phi^* TM$ on $N$. Such a section is a smooth map that associates to each point $p\in N$ a tangent vector in $T_{\phi(p)}M$. These are sometimes called vector fields relative to the map $\phi$ and can also be pictured as infinitesimal deformations of the map $\phi$. In your case $N$ is the cotangent bundle of $M$ with its canonical projection.