Duke Mathematical Journal

Lengths and volumes in Riemannian manifolds

Christopher B. Croke and Nurlan S. Dairbekov

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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.

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

First available in Project Euclid: 25 September 2004

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Zentralblatt MATH identifier

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


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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