Born geometry via K\"unneth structures and recursion operators

TL;DR

This paper introduces a new, simplified definition of Born geometry within K"unneth geometry, demonstrating its equivalence to existing frameworks and exploring integrability and examples on nilmanifolds.

Contribution

It provides a novel, straightforward definition of Born geometry in K"unneth terms and analyzes integrability and connections, including explicit examples.

Findings

Born connection is obtained by averaging the K"unneth connection under conjugation.

The new definition is equivalent to para-quaternionic and generalized geometries.

Examples of integrable Born geometries are constructed on nilmanifolds.

Abstract

We propose a simple definition of a Born geometry in the framework of K\"unneth 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 K\"unneth connection. We also give examples of integrable Born geometries on nilmanifolds.