Recovery of the C ∞ jet from the boundary distance function

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 325:767–793, 2003), Uhlmann et al. (Adv Appl Math 31:379–387, 2003)). In this paper we give a new reconstruction procedure for the C ∞ jet at non-concave 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 lens-equivalent manifolds have different second fundamental forms.