An Introduction to Calabi-Yau Manifolds

TL;DR

This paper develops a theory of Courant algebroids with para-Hermitian structures, linking them to Lie bialgebroids, and generalizing para-Kähler geometry and Poisson-Lie groups.

Contribution

It introduces para-holomorphic algebroids and explores their connections, extending existing geometric frameworks with new structures and examples.

Findings

Defined para-holomorphic algebroids on manifolds with almost para-complex structures

Established connections on para-holomorphic algebroids and their para-complex properties

Showed that the theory generalizes para-Kähler geometry and Poisson-Lie groups

Abstract

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.