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arXiv · 1109.4772

On a Lie Algebraic Characterization of Vector Bundles

Pierre B. A. Lecomte·Thomas Leuther·Elie Zihindula Mushengezi
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Abstract

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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Pierre B. A. Lecomte, Thomas Leuther, Elie Zihindula Mushengezi. 2011-09-22. On a Lie Algebraic Characterization of Vector Bundles. https://doi.org/10.3842/sigma.2012.004

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