Born Geometry via Künneth Structures and Recursion Operators

We propose a simple definition of a Born geometry in the framework of Künneth geometry. While superficially different, this new definition is equivalent to the known definitions in terms of para-quaternionic or generalized geometries. We discuss integrability of Born structures and their associated connections. In particular we find that for integrable Born geometries the Born connection is obtained by a simple averaging under a conjugation from the Künneth connection. We also give examples of integrable Born geometries on nilmanifolds.