Recovery of the Emphasis Type="Italic"C/Emphasi

Authors: Xiaochen Zhou

Publish Date: 2011/11/23

Volume: 160, Issue: 1, Pages: 229-241

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Abstract

For a compact Riemannian manifold with boundary we want to find the metric structure from knowledge of distances between boundary points This is called the “boundary rigidity problem” If the boundary is not concave which means locally not all shortest paths lie entirely in the boundary then we are able to find the Taylor series of the metric tensor C ∞ jet at the boundary see Lassas et al Math Ann 325767–793 2003 Uhlmann et al Adv Appl Math 31379–387 2003 In this paper we give a new reconstruction procedure for the C ∞ jet at nonconcave points on the boundary using the localized boundary distance function A closely related problem is the “lens rigidity problem” which asks whether the lens data determine metric structure uniquely Lens data include information of boundary distance function the lengths of all geodesics and the locations and directions where geodesics hit the boundary We give the first examples that show that lens data cannot uniquely determine the C ∞ jet The example include two manifolds with the same boundary and the same lens data but different C ∞ jets With some additional careful work we can find examples with different C 1 jets which means the boundaries in the two lensequivalent manifolds have different second fundamental forms

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