On Hessian structures on the Euclidean space and the hyperbolic space
type: Item
instance of: scholarly article
title: On Hessian structures on the Euclidean space and the hyperbolic space
zbMATH Open document ID: 0933.53012
author: Hideyuki Kito
published in: Osaka Journal of Mathematics
publication date: 2000-03-29T00:00:00Z
review text: For a manifold \(M\) with flat affine connection \(D\), one considers a Riemannian metric \(g\) which stands as the real version of a Kähler metric and calls \((M,g)\) a Hessian structure. Since the set of almost complex structures making \(g\) a Kähler metric is finite dimensional, the question arises whether a set of flat affine connections making \(g\) a Hessian metric is finite dimensional or not. In this paper, it is shown that if \((M,g)\) is Euclidean space \((\mathbb{R}^n,g_0)\) or hyperbolic space \((H^n,g_0)\), then the former has at least the freedom of \(n\) functions on \(\mathbb{R}\) and the latter of \(n-1\) functions. Accordlingly, the set of all flat connections \(D\) admitting these Hessian structures is infinite dimensional, for both cases.
reviewed by: Tanjiro Okubo
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53B15
zbMATH DE Number: 1270301
zbMATH Keywords: Hessian connection
zbMATH Keywords: flat connection
zbMATH Keywords: Hessian structure
zbMATH Keywords: almost complex structure
zbMATH Keywords: Euclidean-space
zbMATH Keywords: hyperbolic-space
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1270301