Ray Transform and Some Rigidity Problems for Riemannian Metrics

Abstract

This is a survey of the ray transform of symmetric tensor fields on Riemannian manifolds. In the case of second rank tensor fields, the ray transform arises in the linearization of the boundary rigidity problem which is discussed in Section 1. In Section 2 we introduce a class of Riemannian manifolds, convex non-trapping manifolds (CNTM), for which the ray transform can be defined in a very natural way. In the case of positive rank tensor fields, the ray transform has a non-trivial kernel containing the space of potential fields. The principal question is: for which CNTM’s does the kernel of the ray transform coincide with the space of potential fields? For such a manifold, we can go further and ask: is there a stability estimate in the problem of recovering the solenoidal part of a tensor field from its ray transform? Some results on these questions are listed. Integral geometry is closely related to inverse problems for kinetic and linear transport equations that are discussed in Section 3. In Section 4 we present some results on the nonlinear boundary rigidity problem whose derivation is based on stability estimates for the ray transform. Section 5 is devoted to the periodic version of the ray transform, i. e., to the question: to what extent is a tensor field on a closed Riemannian manifold determined by its integrals over all closed geodesies? Anosov manifolds, i. e., closed Riemannian manifolds with geodesic flow of Anosov type constitute the most natural class for investigating the latter question.