Let $M$ be a smooth manifold and $E \to M$ a vector bundle.


I'm reading a text which says:



> 
> Recall that there is a one-to-one correspondence between:
> 
> 
> 1. linear vector fields on $E$, and
> 2. linear operators $D : \Gamma(E) \to \Gamma(E)$ such that there exists a vector field $X$ on $M$ such that
> $$D(fs) = f D(s) + X(f) s$$
> for all $f \in C^\infty(M)$ and $s \in \Gamma(E)$.
> 
> 
> 


Unfortunately, I can't recall ever seeing this. I'm not even sure what is a linear vector field (Google hasn't return anything useful).


What is this correspondence? If this is a standard fact, where is it explained (textbook/lecture notes/paper)?



Linear vector fields $\leftrightarrow$ certain differential operators
Using google (linear vector field on vecot bundle), I found the following reasonable definition for a linear vector field $\hat X$ on a vector bundle $E\to M:$ it is a vector field $\hat X\in\mathcal X(E)$ which is a vector bundle morphism $\hat X\colon E\to TE$ along a map $X\colon M\to TM$ given by a vector field on $M.$


The one-to-one correspondence is given as follows:


$1\Rightarrow2:$ if you have a linear vector field $\hat X$ on $E$, and a section $s\colon \Gamma(M,E)$ consider the difference $$\hat D\_ps:=T\_p s(X)-\hat X\_{s(p)}\in T\_{s(p)}E,$$ where $T$ denotes the differential of the smooth map $s\colon M\to E$. It can be checked using the definition that $\hat D\_ps$ is in the vertical tangent bundle $\mathcal V$, i.e., in the kernel of the differential of the projection $\pi\colon E\to M.$ On the other handd, there is a canonical isomorphism $$\pi^\*E=\mathcal V\to E.$$ Then, for any section $s\colon M\to E$ we obtain $$s^\*\pi^\*\mathcal V=E.$$ Using this observation, we can identify the map $$p\in M\mapsto \hat D\_ps\in \mathcal V\_{s(p)}$$ with a section $Ds\in\Gamma(M,E).$$


$2\Rightarrow1:$ given a first order differential operator $D$ the prescribed properties, one can define the vector field $\hat X$ by reversing the above process.


Unfortunately, I do not know a detailed reference for the above, but it is closely related to Ehresmann-type treatmeant of linear connections (by replacing directional derivatives with the differential respectively $D$ with a linear connection $\nabla$ on $E$).


