On a Lie Algebraic Characterization of Vector Bundles

Details On a Lie Algebraic Characterization of Vector Bundles On a Lie Algebraic Characterization of Vector Bundles

2011-09-22

arxiv.org/abs/1109.4772v2

SIGMA 8 (2012), 004, 10 pages

Mathematics

Differential Geometry

Scientific paper

10.3842/SIGMA.2012.004

We prove that a vector bundle $\pi : E \to M$ 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 fibration 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.

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