## Geometry & Topology

### Simple Riemannian surfaces are scattering rigid

Haomin Wen

#### 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.

#### Article information

Source
Geom. Topol., Volume 19, Number 4 (2015), 2329-2357.

Dates
Accepted: 5 October 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858806

Digital Object Identifier
doi:10.2140/gt.2015.19.2329

Mathematical Reviews number (MathSciNet)
MR3375529

Zentralblatt MATH identifier
1323.53041

Subjects
Primary: 53C24: Rigidity results
Secondary: 57M27: Invariants of knots and 3-manifolds

#### Citation

Wen, Haomin. Simple Riemannian surfaces are scattering rigid. Geom. Topol. 19 (2015), no. 4, 2329--2357. doi:10.2140/gt.2015.19.2329. https://projecteuclid.org/euclid.gt/1510858806

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