Invariant distributions and the geodesic ray transform

Paternain, Gabriel P.; Zhou, Hanming

doi:10.2140/apde.2016.9.1903

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Invariant distributions and the geodesic ray transform

Gabriel P. Paternain and Hanming Zhou

Vol. 9 (2016), No. 8, 1903–1930

DOI: 10.2140/apde.2016.9.1903

Abstract

We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric $m$ -tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal $m$ -tensors. We work with compact simple manifolds, but several of our results apply to nontrapping manifolds with strictly convex boundary.

Keywords

geodesic ray transform, first integral, tensor tomography, invariant distribution

Mathematical Subject Classification 2010

Primary: 53C65

Secondary: 58J40

Milestones

Received: 18 November 2015

Revised: 28 July 2016

Accepted: 12 September 2016

Published: 11 December 2016

Authors

Gabriel P. Paternain
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB3 0WB United Kingdom

Hanming Zhou
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB3 0WB United Kingdom

Vol. 9, No. 8, 2016