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# Some uniqueness results in tensor tomography

Mr. Rohit Kumar Mishra TIFR-CAM,Bangalore
 Speaker Mr. Rohit Kumar Mishra TIFR-CAM,Bangalore Jul 26, 2017 from 02:00 PM to 03:00 PM LH 006 vCal iCal

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 Helgason-type support theorem given the first $$m+1$$ integral moments of a symmetric $$m-$$tensor field $$f$$.