Some uniqueness results in tensor tomography

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 \(f\).