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Article · Duke Mathematical Journal · October 1, 2004

Lengths and volumes in Riemannian manifolds

Christopher B. Croke, N. S. Dairbekov

Nurlan 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.

Geometric Analysis and Curvature FlowsGeometric and Algebraic TopologyAdvanced Differential Geometry ResearchRiemannian metricsvolume inequalitiescompact manifolds

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