AI 中文总结
本文分类了正常 Hessian 截面曲率的 Hessian 流形标准模型，通过引入 Kähler 流形在交换李群 Hamilton 作用下的商结构实现该分类，填补了正曲率情形的空白。
AI 中文摘要
流形上的 Hessian 结构是一种几何结构，由仿射联络 $\ abla$ 与黎曼度量 $g$ 组成的对 $(\ abla,g)$ 构成，并满足某些相容性条件。在信息几何中，它也被称为对偶平坦结构。Hessian 几何被视为 Kähler 几何的实类比；特别地，Hessian 截面曲率对应于 Kähler 几何中的全纯截面曲率。具有常 Hessian 截面曲率的 Hessian 流形的标准模型在非正曲率情形下已由 Furuhata 和 Kurose（2013）分类，而正曲率情形仍是开放的。本文中，我们分类了正曲率情形下的标准模型。为此，我们引入在配备有交换李群适当 Hamilton 作用的 Kähler 流形的商流形上的 Hessian 结构，并将标准模型实现为这样的商流形。
英文摘要
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.
