The isometry groups of manifolds admitting nonconstant convex functions
type: Item
instance of: scholarly article
title: The isometry groups of manifolds admitting nonconstant convex functions
zbMATH Open document ID: 0611.53039
author: Robert E. Greene
author: Katsuhiro Shiohama
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: The main result of this paper is the following theorem: If M is a complete Riemannian manifold with noncompact isometry group, and if there is a convex function \(\phi\) : \(M\to {\mathbb{R}}\) with \(\{x\in M| \quad \phi (x)=\inf_ M\phi \}=\emptyset\) and \(\{x\in M| \quad \phi (x)=\alpha \}\) compact for all \(\alpha\in {\mathbb{R}}\), then M is isometric to the product \(N\times {\mathbb{R}}\) of a compact \(C^{\infty}\) Riemannian manifold N and the real line \({\mathbb{R}}\). The elaborate argument proving this theorem makes use of two lemmas of independent interest, concerning also the properties of a complete Riemannian manifold having special convex functions.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3988175
zbMATH Keywords: isometry group
zbMATH Keywords: product decomposition
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: convex functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03910001
OpenAlex ID: W2079566998
DOI: 10.2969/JMSJ/03910001
description: scientific article; zbMATH DE number 3988175