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 (∇,g) of an affine connection ∇ 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 (∇, 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 (∇, g) defines a Hessian structure, and that the induced almost Born structure is integrable.