On the curvature of affine homogeneous convex domains
type: Item
instance of: scholarly article
title: On the curvature of affine homogeneous convex domains
zbMATH Open document ID: 0618.53041
author: Tadashi Tsuji
published in: Tokyo Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Omega\) be a convex domain in \({\mathbb{R}}^ n\) not containing affine lines and let G(\(\Omega)\) be the group of all affine automorphisms of \(\Omega\). \(\Omega\) is homogeneous when G(\(\Omega)\) acts transitively. It is known that any homogeneous convex domain is affinely equivalent to \(\Omega\) (U) where \(U=\sum_{1\leq i,j\leq r}\quad U_{ij}\) is a T- algebra of rank r \((r>1)\), * is an involutive automorphism, \[ T=\{(t_{ij});\quad t_{ii}>0\quad (1\leq i\leq r-1),\quad t_{rr}=1,t_{ij}=0\quad (1\leq j<i\leq r)\},\quad \Omega (U)=\{tt^*;\quad t\in T\}. \] Moreover, since the rank of the T-algebra \(U_ 1r_ kU\), is an affine invariant of \(\Omega\), the rank of \(\Omega\) is defined to be \(r_ kU-1\). Also, when [t,t] is Abelian, t the Lie algebra of T, \(\Omega\) (U) is said to be metabelian. Throughout this paper the author considers homogeneous convex domains given by T-algebras and determines those of non-positive sectional curvature (with respect to the canonical metric) among homogeneous convex domains of low rank (section 3), or of metabelian type (sections 2,3). Furthermore, in section 4, homogeneous convex domains over cones of square or dual type of nonpositive curvature are characterized completely.
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 4002983
zbMATH Keywords: homogeneous convex domain
zbMATH Keywords: T-algebra of rank
zbMATH Keywords: involutive automorphism
zbMATH Keywords: non-positive sectional curvature
reviewed by: Isabel Dotti Miatello
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270150721
OpenAlex ID: W2074617662
DOI: 10.3836/TJM/1270150721
description: scientific article; zbMATH DE number 4002983