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

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

2008-03-20

arxiv.org/abs/0803.3062v1

Mathematics

Differential Geometry

Scientific paper

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 $f$ 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 $\PD M$ 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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