A Born Structure on the Tangent Bundle of a Hessian Manifold

TL;DR

This paper explores the relationship between Hessian structures and Born structures on the tangent bundle of a manifold, establishing conditions under which the almost Born structure becomes integrable.

Contribution

It demonstrates the equivalence between Hessian structures and the integrability of the induced almost Born structure on the tangent bundle.

Findings

Hessian structures induce almost Born structures on tangent bundles.

The paper establishes conditions for the integrability of the almost Born structure.

It links geometric structures with integrability conditions in differential geometry.

Abstract

The Hessian structure, introduced by Shima(1976), is a geometric structure consisting of a pair $(\nabla,g)$ of an affine connection $\nabla$ and a Riemannian metric $g$ satisfying certain conditions. On the other hand, the Born structure, introduced by Freidel et al.(2014), is a strictly stronger geometric structure than an almost (para-)Hermitian structure. Marotta and Szabo(2019) proved that for a given manifold endowed with a pair $(\nabla, g)$, one can introduce an almost Born structure on the tangent bundle. In this article, we study the equivalence between the conditions that the pair $(\nabla, g)$ defines a Hessian structure, and that the induced almost Born structure is integrable.