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

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

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

Venkateswaran P. Krishnan ^1, and Plamen Stefanov ^2,

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

Abstract
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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.

Keywords: integral geometry, support theorem., X ray transform, tensors.

Mathematics Subject Classification: Primary: 53C65; Secondary: 35R3.

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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