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 \(f\).