A generalized Helmholtz-type decomposition of symmetric tensor fields and applications to ray transforms

TL;DR

This paper extends a Helmholtz-type decomposition for symmetric tensor fields in 2D, demonstrating its use in proving injectivity of momentum and elastic ray transforms, with implications for tensor tomography.

Contribution

It generalizes a tensor decomposition to two dimensions under a mean-zero condition and applies it to establish injectivity of specific ray transforms.

Findings

Extended tensor decomposition to 2D with mean-zero assumption

Proved injectivity of momentum and elastic ray transforms in 2D

Established a connection between the two integral transforms

Abstract

We study a solenoidal-potential type decomposition of a symmetric $m$-tensor field in $\Rb^2$, and its implications to injectivity questions for the momentum and elastic ray transforms. For symmetric tensor fields, a general decomposition with a restriction on the dimension and order of the decomposition was proved in~\cite{Rohit_Suman}. We extend the result to dimension $2$ under a mean-zero assumption. We use the decomposition in $2$ dimensions to prove the injectivity of the momentum and elastic ray transforms. We also prove a connection between the two integral transforms for $2$-tensors.