Analogue of vector for differential operators

Question

A differential operators of order one is a vector field which is defined pointwise . Differential operators of order greater than one are not. The closest analogue to a vector is given by a germ of a differential operator at a point which acts on germs of functions at a point. Can we start from germs of differential operators and create the bundle of differential operators as we do for vector fields? The differential of a smooth functions transform vectors in vectors. Is there an analogue for germs of differential operators?

Answer 1

Yes. If $E→M$ and $F→M$ are vector bundles over a smooth manifold $M$, then differential operators $E→F$ of order less than $k≥0$ can be identified with sections of a finite-dimensional vector bundle, namely, $\def\Hom{\mathop{\sf Hom}}\Hom(J^{<k}E,F)$.

Here $J^{<k}E$ denotes the jet bundle of order less than $k$ for $E$.

Given a morphism of vector bundles $S\colon J^{<k}E→F$, the corresponding differential operator is $f↦S(j^{<k}f)$, where $j^{<k}\colon Γ(E)→Γ(J^{<k}E)$ takes the jet of order less than $k$.

This map is invertible and its inverse is known as the total symbol map.