Lengths and volumes in Riemannian manifolds
type: Item
instance of: scholarly article
title: Lengths and volumes in Riemannian manifolds
zbMATH Open document ID: 1073.53053
DOI: 10.1215/S0012-7094-04-12511-4
author: Christopher B. Croke
author: N. S. Dairbekov
published in: Duke Mathematical Journal
publication date: 2005-02-21T00:00:00Z
review text: The general question of how much geometry of a Riemannian manifold is determined by the lengths of geodesics is natural and has several different versions. For a compact Riemannian manifold with nonempty boundary, it asks if an inequality between the chordal distance functions of boundary points with respect to two metrics implies an inequality on the metrics and the associated volumes. For a closed manifold, it asks if an inequality between the marked length spectra implies a corresponding inequality for the metrics and the volumes. This paper answers such questions for closed surfaces of negative curvatures and for metrics which are conformal to each other, and also for certain manifolds with boundary.
reviewed by: Lizhen Ji
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 37A20
zbMATH DE Number: 2135349
zbMATH Keywords: length spectrum
zbMATH Keywords: volume
MaRDI profile type: Publication
Wikidata QID: Q115240302
cites work: Geodesic rays, Busemann functions and monotone twist maps
cites work: Entropy and rigidity of locally symmetric spaces of strictly negative curvature
cites work: Minimal entropy and Mostow's rigidity theorems
cites work: The geometry of Teichmüller space via geodesic currents
cites work: Rigidity for surfaces of non-positive curvature
cites work: Rigidity and the distance between boundary points
cites work: Q4660784
cites work: Local boundary rigidity of a compact Riemannian manifold with curvature bounded above
cites work: The marked length-spectrum of a surface of nonpositive curvature
cites work: Filling Riemannian manifolds
cites work: On two-dimensional minimal fillings
cites work: Q4840149
cites work: Sub-actions for Anosov flows
cites work: Le spectre marqué des longueurs des surfaces à courbure négative. (The spectrum marked by lengths of surfaces with negative curvature)
cites work: On the length of the geodesics of a metric of negative curvature on the disc.
cites work: Q5816389
cites work: Q4110292
cites work: On the Space of Invariant Measures for Hyperbolic Flows
full work available at URL: s0012-7094-04-12511-4
OpenAlex ID: W1507144512
description: scientific article; zbMATH DE number 2135349