A Born structure on the tangent bundle of a Hessian manifold

Abstract

Abstract The Hessian structure is a geometric structure consisting of a pair $$(\nabla ,g)$$ ( ∇ , g ) of an affine connection $$\nabla $$ ∇ and a Riemannian metric g satisfying certain compatibility conditions. In information geometry, it is also known as a dually flat structure, and the tangent bundle of a Hessian manifold is known to admit a natural Kähler structure. However, the Kähler structure does not necessarily distinguish the underlying Hessian manifolds. In this article, we introduce a richer geometric structure, called a strongly integrable Born structure, on the tangent bundle of a Hessian manifold. We prove that the strongly integrable Born structure can distinguish the underlying Hessian manifolds.