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29 March 2024
 
  » arxiv » math.DG/9805040

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Invariant Forms and Automorphisms of a Class of Multisymplectic Manifolds
A. Ibort ; A. Echeverría-Enríquez ; M.C. Mu~noz-Lecanda ; N. Román-Roy ;
Date 8 May 1998
Subject Differential Geometry | math.DG
AbstractIt is shown that the geometry of a class of multisymplectic manifolds, that is, smooth manifolds equipped with a closed nondegenerate form of degree greater than 1, is characterized by their automorphisms. Such a class is distinguished by a {sl local homogeneity} property. Thus, {sl locally homogeneous multisymplectic manifolds} extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of this 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 multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a multisymplectic manifold characterizes their multisymplectic diffeomorphisms.
Source arXiv, math.DG/9805040
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