A uniqueness result for light ray transform on symmetric 2-tensor fields (Q2187040)

The main result of the paper is a characterization theorem for the kernel of a symmetric tensor field of rank 2 under the light ray transform. The space of such tensors is denoted by \(S^2\mathbb{R}^{1+n}\) and \(\Omega\) denotes a bounded domain in \(\mathbb{R}^{1+n}\). The light ray transform of \(F\in C^{\infty}(\overline{\Omega}; S^2\mathbb{R}^{1+n})\) is given by \[LF(t,x,\tilde\theta)=\int_{\mathbb{R}}\sum_{i,j=0}^{n}\tilde\theta^i\tilde\theta^jF_{ij}(t+s,x+s\theta)ds\] where \((t,x)\in\overline{\Omega}\subset\mathbb{R}^{1+n}\) and \(\tilde\theta=(1,\theta)\) with \(\theta\in S^{n-1}\). The authors prove the following result: If for a fixed \(\theta_0\in S^{n-1}\), for all \((t,x)\in\mathbb{R}^{1+n}\) and for all \(\theta\) near \(\theta_0\), \(LF(t,x,\tilde\theta)=0\), then \(F=\lambda g+dv\), where \(\lambda\in C^{\infty}(\overline\Omega)\), \(g\) is the Minkowksi metric with \((-1,1,\dots,1)\) along the diagonal, and \(v\) is a vector field with components in \(C^{\infty}(\overline\Omega)\) with \(v|_{\partial \Omega}=0\).