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

Loading...
Thumbnail Image
Degree type
Doctor of Philosophy (PhD)
Graduate group
Mathematics
Discipline
Subject
C-infinity jet
boundary rigidity problem
lens rigidity problem
Geometry and Topology
Funder
Grant number
License
Copyright date
Distributor
Related resources
Contributor
Abstract

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.

Advisor
Christopher Croke
Date of degree
2011-12-21
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation