Duke Mathematical Journal

Lengths and volumes in Riemannian manifolds

Christopher B. Croke and Nurlan S. Dairbekov

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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.

Article information

Source
Duke Math. J., Volume 125, Number 1 (2004), 1-14.

Dates
First available in Project Euclid: 25 September 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1096128232

Digital Object Identifier
doi:10.1215/S0012-7094-04-12511-4

Mathematical Reviews number (MathSciNet)
MR2097355

Zentralblatt MATH identifier
1073.53053

Subjects
Primary: 53C22: Geodesics [See also 58E10] 53C24: Rigidity results 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
Secondary: 58C35: Integration on manifolds; measures on manifolds [See also 28Cxx] 37A20: Orbit equivalence, cocycles, ergodic equivalence relations

Citation

Croke, Christopher B.; Dairbekov, Nurlan S. Lengths and volumes in Riemannian manifolds. Duke Math. J. 125 (2004), no. 1, 1--14. doi:10.1215/S0012-7094-04-12511-4. https://projecteuclid.org/euclid.dmj/1096128232


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