Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

Arturo Echeverría-Enríquez; Alberto Ibort; Miguel C. Muñoz-Lecanda; Narciso Román-Roy

doi:10.3934/jgm.2012.4.397

Article Contents

2012, Volume 4, Issue 4: 397-419. Doi: 10.3934/jgm.2012.4.397

This issue Previous Article Semi-simple generalized Nijenhuis operators Next Article Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids

Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

1.
Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Campus Norte, Ed. C-3. C/ Jordi Girona 1, E-08034 Barcelona
2.
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA, and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid
3.
Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona
4.
Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC, C/ Jordi Girona 1. 08034 Barcelona

Received: April 30, 2012

Revised: September 30, 2012

Published: November 2012

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Abstract

It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree $> 1$, which is locally homogeneous of degree $k$ with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.

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Mathematics Subject Classification: Primary: 53C15, 53D35; Secondary: 57R50, 57S25, 58A10.

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