Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 033, 44 pages      arXiv:1406.2138      https://doi.org/10.3842/SIGMA.2017.033
Contribution to the Special Issue “Gone Fishing”

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

María Amelia Salazar a and Daniele Sepe b
a) IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil 22460-320
b) Universidade Federal Fluminense, Instituto de Matemática, Departamento de Matemática Aplicada, Rua Mário Santos Braga S/N, Campus do Valonguinho, Niterói, Brazil 24020-140

Received October 07, 2016, in final form May 17, 2017; Published online May 25, 2017

Abstract
Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones.

Key words: Jacobi structures; contact manifolds; Poisson structures; projective structures; contact actions.

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