Generalized complex geometry and nilmanifolds

The generalized complex geometry is a relatively new and highly popular subject which also has applications in theoretical physics. It studies the geometric structures on TM directsum T*M. That is to say, this new geometry develops a new language that treats tangent and contangent bundles of a manifold simultaneously. In this thesis, we will first study essential properties of generalized complex geometry. On the way we will see, how this approach gives a new way to study complex and symplectic structures. This observation directs us to investigate the generalized complex and generalized Kahler structures on nilmanifolds since these spaces contain interesting examples of complex and symplectic geometries. Although it is known that some six dimensional nilmanifolds do no admit neither symplectic nor complex structures, they all admit generalized complex structure. In order to understand these structures in details, we explicitly construct a generalized complex structure on a nilmanifold. Moreover, discussing the Hodge theory and the formality property of generalized complex structures, we will show that if a nilmanifold admits a generalized Kahler structure then it has a trivial nilpotent Lie algebra.