Born geometry via Künneth structures and recursion operators

TL;DR

This work reframes Born geometry within Künneth geometry by defining a Born triple $(g,h,\omega)$ together with three recursion operators $(A,B,J)$ satisfying $A^2=B^2=-J^2=\operatorname{Id}$, and proves equivalence to existing para-quaternionic and generalized-geometry formulations. It develops integrability criteria, showing that $\omega$ closed plus integrability of two of the operators implies full integrability, and introduces two natural connections: the canonical connection $\nabla^{c}$ (an average of the Levi-Civita connection under conjugation by $A$) and the Born connection $\nabla^{B}$ (an average of the Künneth connection under conjugation by $B$); in the integrable case, $\nabla^{B}$ agrees with the Born connection of Freidel–Rudolph–Svoboda and has vanishing generalized torsion. The paper establishes that every almost Künneth structure can be enhanced to a Born structure, links Born geometry to hypersymplectic geometry via $S^1$-families, and provides explicit left-invariant examples on nilmanifolds (notably $Nil^3\oplus\mathbb{R}$ in 4D and $\mathfrak{h}_4,\mathfrak{h}_9$ in 6D) demonstrating integrability and compatibility of the required structures. These results offer a clear, algebraically grounded pathway from Künneth data to integrable Born geometries and their canonical/ Born connections with concrete geometric and topological ramifications.

Abstract

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.

Theorems & Definitions (64)

• Definition 1
• Proposition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Lemma 7
• proof
• Lemma 8
• proof