Tópicos de Geometria Diferencial

Tópicos de Geometria Diferencial

Topic: Dirac structures and generalized geometry:
In the first part of this course we will look at how and why the notions of Dirac structure, Courant algebroid and generalized complex structure were introduced. We will focus on explaining the basic concepts and their interplay. In the second part we will move to more specialized topics.

Part I – Basics:

  • Presymplectic and Poisson structures in Mechanics.
  • Dirac structures.
  • Courant algebroids.
  • Generalized complex geometry.

Part II: Topics

  • Generalized metrics and generalized Kaehler geometry.
  • Generalized reduction.
  • Normal forms.
  • Supergeometric viewpoint.

T. Courant, Dirac manifolds, Trans. AMS, 1990.
N. Hitchin, Generalized Calabi-Yau manifolds, Quaterly Math. J., 2003.
M. Gualtieri, Generalized complex geometry. PhD thesis, Oxford. Published version at Annals of Math., 2011.
M. Gualtieri, Generalized Kahler geometry. Comm. Math. Phys., 2014.
H. Bursztyn, G. Cavalcanti, M. Gualtieri. Reduction of Courant algebroids and generalized complex structures. Advances in Math, 2007.
M. Bailey. Local classification of generalized complex structures. J. Differential Geom., 2013. .