Abstract:
We consider the problem of recovering the solenoidal part of a symmetric tensor field $f$ on a compact Riemannian manifold $(M,g)$ with boundary from the integrals of $f$ over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary $\partial M$ is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if $\partial M$ is convex and there is no geodesic of infinite length in $\partial M$. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of $\partial M$ we assume that $\partial M$ is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for $\partial M$.
Keywords:ntegral geometry, ray transform, tensor field.
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