Some uniqueness results in tensor tomography
Speaker 
Mr. Rohit Kumar Mishra
Research Scholar, TIFRCAM, Bangalore


When 
Apr 20, 2017
from 02:00 PM to 03:00 PM 
Where  LH 006 
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Abstract: We consider a generalization of geodesic ray transforms, called integral moment transforms, of symmetric \(m\)tensor fields in Riemannian and Euclidean geometries.
The \(q^{\mathrm{th}}\) integral moment transform of a symmetric \(m\)tensor field \(f=f_{i_{1}\cdots i_{m}} dx^{i_{1}}\cdots dx^{i_{m}}\) on a Riemannian manifold \((M,g)\) is defined as follows:
$$ I^qf(x, \xi) = \int_{\mathbb{R}} t^q\langle f(\gamma_{x,\xi}(t)), \dot{\gamma}_{x,\xi}^m(t) \rangle_{g} dt = \int_{\mathbb{R}}t^q f_{i_1\dots i_m}(\gamma_{x,\xi}(t))\dot{\gamma}_{x,\xi}^{i_1}(t)\cdots \dot{\gamma}_{x,\xi}^{i_m}(t) dt. $$
where \(\gamma_{x,\xi}(t)\) is the geodesic starting from \(x\) in the direction \(\xi\). The special case when \(q=0\) in the above definition is called the longitudinal geodesic ray transform.
We are interested in the question of recovery of the symmetric \(m\)tensor field \(f\) from the knowledge of its integral moments. We first consider the Euclidean setting and show that a vector field in \(\mathbb{R}^{n}\) can be uniquely recovered with an explicit inversion formula from the knowledge of the first two integral moments restricted to lines passing through a fixed curve. Next we consider a restricted longitudinal ray transform of symmetric \(m\)tensor fields and show that this transform can be inverted microlocally recovering a component of the field \(f\) modulo a known error term and smoothing terms. Finally, we consider the integral moment transforms in a Riemannian manifold setting and prove a Helgasontype support theorem given the first \(m+1\) integral moments of a symmetric \(m\)tensor field \(f\).