Lengths and volumes in Riemannian manifolds
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.
scientific article; zbMATH DE number 2135349