On Pursell-Shanks type results

We prove a Lie-algebraic characterization of vector bundle for the Lie algebra \(\mathcal{D}(E,M),\) seen as \({\rm C}^\infty(M)-\)module, of all linear operators acting on sections of a vector bundle \(E\to M\). We obtain similar result for its Lie subalgebra \(\mathcal{D}^1(E,M)\) of all linear first-order differential operators. Thanks to a well-chosen filtration, \(\mathcal{D}(E,M)\) becomes \(\mathcal{P}(E,M)\) and we prove that \(\mathcal{P}^1(E,M)\) characterizes the vector bundle without the hypothesis of being seen as \({\rm C}^\infty(M)-\)module. We prove that the Lie algebra \(\mathcal{S}(\mathcal{P}(E,M))\) of symbols of linear operators acting on smooth sections of a vector bundle \(E\to M,\) characterizes it. To obtain this, we assume that \(\mathcal{S}(\mathcal{P}(E,M))\) is seen as \({\rm C}^\infty(M)-\)module. We obtain a similar result with the Lie algebra \(\mathcal{S}^1(\mathcal{P}(E,M))\) of symbols of first-order linear operators without the hypothesis of being seen as a \({\rm C}^\infty(M)-\)module.