The marked length-spectrum of a surface of nonpositive curvature
type: Item
instance of: scholarly article
title: The marked length-spectrum of a surface of nonpositive curvature
zbMATH Open document ID: 0779.53025
DOI: 10.1016/0040-9383(92)90013-8
author: Christopher B. Croke
author: Albert Fathi
author: Jacob Feldman
published in: Topology
publication date: 1993-05-16T00:00:00Z
review text: If \(M\) is a manifold and \(g_ 1\), \(g_ 2\) are Riemannian metrics, then \(g_ 1\) and \(g_ 2\) are said to have the same marked length-spectrum if in each homotopy class of closed curves in \(M\) the infimum of \(g_ 1\)- lengths of curves and the infimum of \(g_ 2\)-lengths are the same. The marked length spectrum problem in general is to show that two metrics with the same marked length spectrum are isometric. One of the results in this paper states that if \(M\) is a closed surface and if \(g_ 1\) and \(g_ 2\) are Riemannian metrics on \(M\) with \(g_ 1\) of nonpositive curvature and \(g_ 2\) without conjugate points then \(g_ 1\) and \(g_ 2\) are isometric by an isometry homotopic to the identity when they have the same marked length-spectrum. The authors also show that it is possible to obtain a similar result when the Morse correspondence preserves angles.
reviewed by: A. P. Stone
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 167788
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: conjugate points
zbMATH Keywords: isometry
zbMATH Keywords: Morse correspondence
MaRDI profile type: Publication
full work available at URL: 0040-9383(92)90013-8
OpenAlex ID: W2056145457
description: scientific article; zbMATH DE number 167788