Hessian manifolds of constant Hessian sectional curvature
type: Item
instance of: scholarly article
title: Hessian manifolds of constant Hessian sectional curvature
zbMATH Open document ID: 0845.53033
author: Hirohiko Shima
published in: Journal of the Mathematical Society of Japan
publication date: 1996-06-30T00:00:00Z
review text: Let \(M\) be a flat affine manifold with a flat affine connection \(D\). If \(M\) admits a Riemannian metric \(g\) such that \(g\) is locally expressed by \(g= D^2 u\), then \((M, D, g)\) is said to be a Hessian manifold. We define Hessian sectional curvatures (these correspond to holomorphic sectional curvatures for Kählerian manifolds) and construct Hessian manifolds of constant Hessian sectional curvatures. We show that the family of normal distributions and the family of multinomial distributions are Hessian manifolds of constant Hessian sectional curvatures. We define affine Chern classes for flat affine manifolds (analogous to Chern classes for complex manifolds), and using affine Chern classes we give a characterization of constant Hessian sectional curvatures being similar to a result of Chen and Ogiue for complex space forms.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 894107
zbMATH Keywords: affine Chern class
zbMATH Keywords: constant Hessian sectional curvature
zbMATH Keywords: Hessian manifold
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04740735
OpenAlex ID: W2047125792
DOI: 10.2969/JMSJ/04740735
description: scientific article; zbMATH DE number 894107