Hessian manifolds of positive constant Hessian sectional curvature

Abstract

Hessian structure on a manifold is a geometric structure consisting of a pair $(\nabla,g)$ of an affine connection $\nabla$ and a Riemannian metric $g$ satisfying certain compatibility conditions. It is also known as a dually flat structure in information geometry. Hessian geometry is considered to be a real analogue of Kähler geometry; in particular, Hessian sectional curvature corresponds to holomorphic sectional curvature in Kähler geometry. The standard models of Hessian manifolds with constant Hessian sectional curvature were classified by Furuhata and Kurose (2013) in the nonpositive curvature case, whereas the positive curvature case is open. In this paper, we classify the standard models in the positive curvature case. For this end, we introduce Hessian structure on quotient manifolds of Kähler manifolds equipped with suitable Hamiltonian actions of abelian Lie groups and realizing the standard models as such quotient manifolds.