Para-Holomorphic Algebroids and Para-Complex Connections

The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold has an almost para-complex structure, and use this to define a notion of para-holomorphic algebroid. We investigate connections on para-holomorphic algebroids and determine an appropriate sense in which they can be para-complex. Finally, we show through a series of examples how the theory of exact para-holomorphic algebroids with a para-complex connection is a generalization of both para-K\"{a}hler geometry and the theory of Poisson-Lie groups.