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.
Categories: math.DG