SUPPORT THEOREMS FOR CERTAIN RAY TRANSFORMS
Abstract: The main object of study in this thesis are integral transforms over simple, real analytic, Riemannian manifolds. Let (M, g) be a simple, real analytic, Riemannian manifold with boundary and of dimension n >=3. The first result that we present here establishes a support theorem for the transverse ray transform of tensor fields of rank 2 defined over such manifolds. More specifically, giv... read moreen a symmetric tensor field f of rank 2, we show that if the transverse ray transform of f vanishes over an appropriate open set of maximal geodesics of M, then the support of f vanishes on the points of M that lie on the union of the aforementioned open set of geodesics. The second problem concerns integral moments transform of a symmetric m-tensor field on a simple, real analytic, Riemannian manifold as above. Integral moments of m-tensor field were first introduced by Sharafutdinov. At first we generalize a Helgason type support theorem proven by Krishnan and Stefanov in "A support theorem for the geodesic ray transform of symmetric tensor fields", Inverse Problems and Imaging, 3(3):453-464,2009. We use this extended result along with the first m+1-integral moments of the m-tensor field to prove an injectivity result and support theorem for such transforms.
Thesis (Ph.D.)--Tufts University, 2018.
Submitted to the Dept. of Mathematics.
Advisor: Eric Quinto.
Committee: Venkateswaran Krishnan, Fulton Gonzalez, and David Finch.
Keyword: Mathematics.read less