Projectively equivalent metrics on the torus
type: Item
instance of: scholarly article
title: Projectively equivalent metrics on the torus
zbMATH Open document ID: 1051.37030
published in: Differential Geometry and its Applications
publication date: 2004-08-16T00:00:00Z
review text: One says that two metrics \(g_1\) and \(g_2\) on a smooth manifold are {projectively equivalent} if they have the same unparametrized geodesics. One says that \(g_1\) and \(g_2\) are strictly nonproportional at a point \(P\) of \(M\) if the eigenvalues of \(g_1\) with respect to \(g_2\) are all distinct at \(P\). The authors show: Theorem 1: Suppose \(M\) is a connected closed \(n\)-dimensional manifold. Suppose there exist Riemannian metrics \(g_1\) and \(g_2\) on \(M\) which are projectively equivalent and strictly nonproportional at at least one point. Then the following hold: (1) The first Betti number \(b_1(M)\) is at most \(n\). (2) The fundamental group of \(M\) has a commutative subgroup of finite index. (3) If there exists a point where the metrics are not strictly nonproportional, then \(b_1(M)<n\). Let \(T^n\) be the \(n\) torus. Since \(b_1(T^n)=n\), this leads to the following: Corollary 1: Suppose that \(g_1\) and \(g_2\) are metrics on \(T^n\) which are projectively equivalent and strictly nonproportional at at least one point. Then they are strictly nonproportional at every point. The authors use Corollary 1 to describe and classify (in a certain sense) all projectively equivalent Riemannian metrics on the torus which are strictly nonproportional at at least one point. This enables the authors to separate variables in the equation for the eigenvalues of the Laplacian for such a metric.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53B10
zbMATH DE Number: 2091325
zbMATH Keywords: projectively equivalent metrics
zbMATH Keywords: geodesically equivalent metrics
zbMATH Keywords: integrable systems
zbMATH Keywords: Levi-Civita coordinates
zbMATH Keywords: quantum integrability
zbMATH Keywords: separation of variables
author: Vladimir S. Matveev
MaRDI profile type: Publication
full work available at URL: j.difgeo.2003.10.009
OpenAlex ID: W2090902261
Wikidata QID: Q115358100
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DOI: 10.1016/J.DIFGEO.2003.10.009
description: scientific article; zbMATH DE number 2091325