Generalized complex structures on Courant algebroids.

摘要

In this thesis we study generalized complex structures defined on Lie bialgebroids, and arbitrary Courant algebroids. This thesis consists of two parts: the first deals with the generalized complex structures on Courant algebroids, while the second discusses generalized complex submanifolds.;In the second part we introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman [38]. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of an involution preserving a twisted generalized complex structure is a twisted generalized complex submanifold. Lastly, we also discuss generalized Kahler submanifolds.;The basic examples of generalized complex structures are given, and certain classes of Poisson-Nijenhuis manifolds are described using generalized complex structures. The Poisson structure arising from a generalized complex structure is also defined explicitly. Generalized complex structures on arbitrary Courant algebroids are also described using generating operators and spinors. A generating operator for the Courant algebroid of a Lie bialgebroid is also given.