# American Institute of Mathematical Sciences

August  2009, 3(3): 453-464. doi: 10.3934/ipi.2009.3.453

## A support theorem for the geodesic ray transform of symmetric tensor fields

 1 110, 8th Street, Rensselaer Polytechnic Institute, Troy, NY 12180, United States 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

Received  March 2008 Revised  March 2009 Published  July 2009

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of such a field $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary ∂$M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques.
Citation: Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453
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