A proof of Lens Rigidity in the category of Analytic Metrics

TL;DR

This paper proves that in the category of analytic Riemannian manifolds with boundary, the lens data uniquely determine the metric up to isometry, even without convexity assumptions and allowing conjugate points under certain restrictions.

Contribution

It establishes a lens rigidity result for analytic Riemannian manifolds with boundary, extending previous results by removing convexity constraints and handling conjugate points.

Findings

Lens data uniquely determine the metric in the analytic category.

No convexity assumptions are needed on the boundary.

Conjugate points are permitted with restrictions.

Abstract

Consider a compact Riemannian manifold with boundary. Assume all maximally extended geodesics intersect the boundary at both ends. Then to each maximal geodesic segment one can form a triple consisting of the initial and final vectors of the segment and the length of the segment. The collection of all such triples comprises the lens data. In this paper, it is shown that in the category of analytic Riemannian manifolds, the lens data uniquely determine the metric up to isometry. There are no convexity assumptions on the boundary, and conjugate points are allowed, but with some restriction.