The C-infinity Jet of Non-Concave Manifolds and Lens Rigidity of Surfaces

In this thesis we work on the boundary rigidity problem, an inverse problem on a manifold with boundary, which studies the unique determination of, and algorithms towards total recovery of, the metric tensor, based on the information of distances between boundary points. There are three main results in this thesis. The rst result is an algorithm to recover the Taylor series of the metric tensor (C-infinity jet) at the boundary. The data we use are the distances between pairs of points on the boundary which are close enough to each other, i:e: the \localized" distance function. The restriction we impose on the shape of the manifold near the boundary is the minimal possible, i.e., the localized distance function does not completely coincide with the localized in-boundary distance function at any point. Here "in-boundary" distance means the length of the shortest path lying entirely on the boundary. Such a boundary we call "non-concave". A different algorithm has already been published in [26], but our result in this thesis is much more elementary. Our second result is a counter-example to the statement "Lens data uniquely determine the C-infinity jets at boundary points". It is the rst known pair of manifoldswith identical lens data but different C-infinity jets. Our first example is easy to construct, but the jets of the metrics only dier in the second order. With a careful modification to preserve smoothness, we can construct a pair with different C1 jets, meaning dierent second fundamental forms of the boundaries. The results above are already published in the author's paper [28]. Our third result is, if two surfaces with the same boundary are conformal, have the same lens data, and have no trapped geodesic or conjugate points, then they are isometric. The proof applies techniques in integral geometry, and uses results in [3] and [14]. If we combine this with not yet published results in [10] of Croke, Pestov, and Uhlmann, then we can drop the conformal assumption.