Mostow rigidity and the Bishop-Steger dichotomy for surfaces of variable negative curvature
type: Item
instance of: scholarly article
title: Mostow rigidity and the Bishop-Steger dichotomy for surfaces of variable negative curvature
zbMATH Open document ID: 0782.53032
DOI: 10.1215/S0012-7094-92-06810-4
author: Steven P. Lalley
published in: Duke Mathematical Journal
publication date: 1992-01-01T00:00:00Z
review text: The author finds necessary and sufficient conditions for two compact surfaces of negative Gaussian curvature to have the same marked length spectrum. The first result stated below generalizes a result of Bishop and Steger for compact surfaces of constant negative curvature. If \((M,g)\) denotes a compact Riemannian manifold of sectional curvature \(K<0\), then each free homotopy class \(\beta_ \delta\) of closed curves in \(M\) admits a unique closed geodesic of length \(\lambda_ g(\beta_ \delta)\). The collection of pairs \(\{(\beta_ \delta,\lambda_ g(\beta_ \delta))\}\), as \(\delta\) ranges over the set \(\Delta\) of free homotopy classes of \(M\), is called the marked length spectrum of \((M,g)\). If two compact negatively curved manifolds \((M_ 1,g_ 1)\) and \((M_ 2,g_ 2)\) have isomorphic fundamental groups, then there is a natural bijection between the free homotopy classes of \(M_ 1\) and \(M_ 2\). If this bijection preserves the lengths of the closed geodesics in corresponding free homotopy classes, then \((M_ 1,g_ 1)\) and \((M_ 2,g_ 2)\) are said to have the same marked length spectrum. \textit{C. Croke} [Comment. Math. Helv. 65, 150-169 (1990; Zbl 0704.53035)] has shown that two compact surfaces of nonpositive Gaussian curvature with the same marked length spectrum are isometric. \textit{U. Hamenstädt} in [Ergodic Theory Dyn. Syst. 12, No. 1, 67-74 (1992; Zbl 0766.58045)] has shown that two compact manifolds with \(K<0\) and arbitrary dimension have the same marked length spectrum if and only if there is a homeomorphism between the unit tangent bundles that conjugates the geodesic flows in time preserving fashion. In stating the results of this article we let \(M\) denote a compact surface with Riemannian metrics \(g_ 1\), \(g_ 2\) of negative curvature. For \(i=1,2\), let \((\widetilde M,\widetilde g_ i)\) denote the universal cover of \((M,g_ i)\), and let \(\widetilde d_ i\) denote the Riemannian distance function in \((\widetilde M,\widetilde g_ i)\)…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 179214
zbMATH Keywords: marked length spectrum
zbMATH Keywords: fundamental groups
zbMATH Keywords: free homotopy classes
zbMATH Keywords: Liouville measure classes
zbMATH Keywords: Patterson measure classes
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 179214