Métodos de promedios para estructuras geométricas y sus aplicaciones

Abstract

This thesis is devoted to the study of geometric structures with symmetries, in particular Dirac and Poisson structures. We describe an averaging procedure on Dirac and Poisson manifolds, with respect to a class of compatible actions of compact Lie groups. Necessary conditions for the existence of invariant Poisson structures around (singular) symplectic leaves are presented. We also introduce a generalization of the Hannay-Berry connection in the context of foliated manifolds carrying Lie group actions with premomentum map. The main application of our approach is related to the construction of Dirac structures with symmetry on Poisson foliations. Moreover, we apply the averaging method due to Jotz, Ratiu and Sniatycki to construct invariant Dirac and Poisson structures with respect to the proper action of a Lie group (not necessarily compact).