On a Lie Algebraic Characterization of Vector Bundles

We prove that a vector bundle E -> M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole f ibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229{239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1.