Simple Riemannian surfaces are scattering rigid

Wen, Haomin

doi:10.2140/gt.2015.19.2329

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Simple Riemannian surfaces are scattering rigid

Haomin Wen

Geometry & Topology 19 (2015) 2329–2357

DOI: 10.2140/gt.2015.19.2329

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/

Volume 19, issue 4 (2015)