Entropy and rigidity of locally symmetric spaces of strictly negative curvature
type: Item
instance of: scholarly article
title: Entropy and rigidity of locally symmetric spaces of strictly negative curvature
zbMATH Open document ID: 0851.53032
DOI: 10.1007/BF01897050
author: G. Besson
author: Gilles Courtois
author: Sylvestre (Sylvain) Gallot
published in: Geometric and Functional Analysis. GAFA
publication date: 1996-03-18T00:00:00Z
full work available at URL: https://eudml.org/doc/58209
review text: In this article the authors prove a major new result whose corollaries settle several difficult outstanding problems or provide new proofs of other significant known results. Theorem. Let \(Y\) and \(X\) denote two compact oriented manifolds of the same dimension \(n\), and let \(f : Y \to X\) be a continuous map of nonzero degree. Suppose that \(X\) admits a locally symmetric Riemannian metric \(g_0\) with strictly negative sectional curvature, and let \(g\) be any Riemannian metric on \(Y\). Let \(h(Y,g)\) and \(h(X,g_0)\) denote the volume entropy of \((Y,g)\) and \((X,g_0)\), respectively. Then \(h^n(Y,g) \text{Vol} (Y,g) \geq |\text{deg } f|h^n(X,g_0) \text{Vol} (X,g_0)\), and equality holds if and only if there exists a positive constant \(c\) and a Riemannian covering \(f^* : (Y,cg) \to (X,g_0)\) that is homotopic to \(f\). Note that in a Riemannian manifold \((Z,g)\) the constant \(h^n(Z,g) \text{Vol}(Z,g)\) is unchanged if one multiplies the metric \(g\) by any positive constant. As easy consequences of the result above the authors obtain several important corollaries. Corollary 1. Let \(Y\) and \(X\) denote two compact oriented manifolds of the same dimension \(n\), and let \(f : Y \to X\) be a continuous map of nonzero degree. Let \(g_0\) be a metric on \(X\) with sectional curvature \(\equiv -1\), and let \(g\) be a metric with \(\text{Ricci}(g) \geq -(n -1)g\). Then \(\text{Vol} (Y,g) \geq |\text{deg }f |\text{Vol}(X,g_0)\). If equality holds and \(n \geq 3\), then \((Y,g)\) has constant negative sectional curvature and there exists a positive constant \(c\) and a Riemannian covering \(f^* : (Y,cg) \to (X,g_0)\) that is homotopic to \(f\). Corollary 2 (Mostow rigidity theorem). Let \((Y,g)\) and \((X,g_0)\) be homotopically equivalent, compact oriented Riemannian manifolds with negative sectional curvature. Then \((Y,cg)\) and \((X,g_0)\) are isometric for some positive constant \(c\). Corollary 3 (Corlette, Siu, Thurston). Let \((Y,g)\) and \((X,g_0)\)…
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 847576
zbMATH Keywords: negative curvature
zbMATH Keywords: harmonic manifolds
zbMATH Keywords: locally symmetric Riemannian metric
zbMATH Keywords: entropy
zbMATH Keywords: rigidity
zbMATH Keywords: geodesic flows
zbMATH Keywords: Busemann function
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
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OpenAlex ID: W2064013450
description: scientific article; zbMATH DE number 847576