Jacobi structures and Spencer operators

2013/09/24 by Marius Crainic, Crainic, Marius, Maria Amelia Salazar +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG

paper · pdf · doi:10.48550/arxiv.1309.6156

This is a revised version (where a new chapter on "Jacobi structures and their associated Spencer operator" appeared) to appear in Journal de Mathématiques Pures et Appliquées

arxiv created 2014/04/27 · arxiv updated 2014/04/29

Abstract

This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses recent results on the integrability of Spencer operators and multliplicative forms on Lie groupoids with non-trivial coefficients. In Theorem 1 we show that the Spencer operator associated to a contact groupoid reveals that the base manifold carries a Jacobi structure. Theorem 2 deals with the problem of integrating Jacobi structures to contact groupoids.

Related

• Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid
• Nonlinear Spencer operators on differentiable groupoids
• Integrating Lie algebroids via stacks and applications to Jacobi manifolds
• Multiplicative Forms and Spencer Operators
• Reduction of Jacobi manifolds via Dirac structures theory