1 October 2004 Lengths and volumes in Riemannian manifolds
Christopher B. Croke, Nurlan S. Dairbekov
Duke Math. J. 125(1): 1-14 (1 October 2004). DOI: 10.1215/S0012-7094-04-12511-4

Abstract

We consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes. We prove this in a number of cases for compact manifolds with and without boundary. In particular, we show that for two Riemannian metrics of negative curvature on a compact surface without boundary, an inequality between the marked length spectra implies the same inequality between the areas, with equality implying isometry.

Citation

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Christopher B. Croke. Nurlan S. Dairbekov. "Lengths and volumes in Riemannian manifolds." Duke Math. J. 125 (1) 1 - 14, 1 October 2004. https://doi.org/10.1215/S0012-7094-04-12511-4

Information

Published: 1 October 2004
First available in Project Euclid: 25 September 2004

zbMATH: 1073.53053
MathSciNet: MR2097355
Digital Object Identifier: 10.1215/S0012-7094-04-12511-4

Subjects:
Primary: 53C22 , 53C24 , 53C65
Secondary: 37A20 , 58C35

Rights: Copyright © 2004 Duke University Press

Vol.125 • No. 1 • 1 October 2004
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