Inverse kinematic problem and boundary rigidity of Riemannian surfaces

TL;DR

This paper addresses the inverse problem of reconstructing Riemannian metrics on surfaces from boundary geodesic data, establishing conditions for uniqueness and stability, and exploring related integral geometry problems.

Contribution

It proves that the full set of geodesic lengths determines the metric up to boundary-fixing automorphisms, even without knowing the conformal class.

Findings

Unique determination of Riemannian metric from geodesic lengths.

Integral inequality implies stability and uniqueness.

Analysis of related integral geometry problems.

Abstract

Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is stated which implies uniqueness and stability for this problem. If the conformal class is not known a unique reconstruction is not possible since of shortage of information. It is proved that the list of all geodesic lengths is sufficient for unique determination of a Riemannian metric in a compact surface with boundary up to an automorphism which fix the boundary. Some related problems of integral geometry are studied. Key words: Geodesic curve, Travel-time, Conjugate point, Geodesic flow, Hodograph, Geodesic integral transform.