A proof of Lens Rigidity in the category of Analytic Metrics

Consider a compact Riemannian manifold with boundary. If all maximally extended geodesics intersect the boundary at both ends, then to each geodesic $\gamma(t)$ we can form the triple $(\dot\gamma(0),\dot\gamma(T),T)$, consisting of the initial and final vectors of the segment as well as the length between them. 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.

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Published 1 January 2009