Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Details Decomposition of symmetric tensor fields in the presence of a flat contact projective structure Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

2007-03-30

arxiv.org/abs/math/0703922v2

Mathematics

Differential Geometry

Scientific paper

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $\alpha$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities, the vertical cotangent lift of contact form $\alpha$ is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the symbol space is invariant with respect to the action of the projective contact algebra $sp(2n+2)$. The preceding invariant operators lead to a decomposition of the symbol space (expect for some critical density weights), which generalizes a splitting proposed by V. Ovsienko.

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