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

Venkateswaran P. Krishnan; Plamen Stefanov

doi:10.3934/ipi.2009.3.453

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2009, Volume 3, Issue 3: 453-464. Doi: 10.3934/ipi.2009.3.453

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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
2.
Department of Mathematics, Purdue University, West Lafayette, IN 47907

Received: February 29, 2008

Revised: February 28, 2009

Published: July 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 53C65; Secondary: 35R30.

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