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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Simple Riemannian surfaces are scattering rigid

Haomin Wen

Geometry & Topology 19 (2015) 2329–2357
Abstract

Scattering rigidity of a Riemannian manifold allows one to recognize the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to recognize 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 implies that two-dimensional simple manifolds (such as the flat disk) are scattering rigid since they are lens/boundary rigid.

Keywords
scattering rigidity, lens rigidity, knot
Mathematical Subject Classification 2010
Primary: 53C24
Secondary: 57M27
References
Publication
Received: 16 June 2014
Accepted: 5 October 2014
Published: 29 July 2015
Proposed: Dmitri Burago
Seconded: Tobias H Colding, Bruce Kleiner
Authors
Haomin Wen
Max Planck Institute for Mathematics
Vivatsgasse 7
53111 Bonn
Germany
http://www.math.upenn.edu/~weh/