Scattering and Lens Rigidity

Publication Year:
2014
Usage 19
Downloads 13
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Repository URL:
https://repository.upenn.edu/edissertations/1498
Author(s):
Wen, Haomin
Tags:
boundary rigidity; inverse problem; lens rigidity; Riemannian geometry; scattering rigidity; Mathematics
thesis / dissertation description
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equivalent for two-dimensional simple manifolds with boundaries. In particular, this impliesthat two-dimensional simple manifolds (such as theat disk) are scattering rigid since they are lens/boundary rigid (Pestov--Uhlmann, 2005).