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A PROOF OF LENS RIGIDITY IN THE CATEGORY OF ANALYTIC METRICS

By James Vargo

Abstract

Abstract. Consider a compact Riemannian manifold with boundary. If all maximally extended geodesics intersect the boundary at both ends, then to each geodesic γ(t) we can form the triple ( ˙γ(0), ˙γ(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. 1. An introduction including the result proved Let (M, g) be a compact, Riemannian manifold with boundary ∂M, and let it be non-trapping. That means all geodesics, when maximally extended, terminate at the boundary at both their ends. Let SM denote its sphere bundle. Then for any vector v ∈ ∂SM, the geodesic γv originating at v eventually leaves the manifold after some distance T. Let ℓ(v) denote the length of the geodesic, and let Σ(v) = ˙γv(T) denote its terminal vector

Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.205.8921
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