A Born Structure on the Tangent Bundle of a Hessian Manifold

The Hessian structure, introduced by Shima(1976), is a geometric structure consisting of a pair $(\mathrm{\nabla },g)$ of an affine connection $\mathrm{\nabla }$ 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 $(\mathrm{\nabla },g)$, one can introduce an almost Born structure on the tangent bundle. In this article, we study the equivalence between the conditions that the pair $(\mathrm{\nabla },g)$ defines a Hessian structure, and that the induced almost Born structure is integrable.