The non-abelian X-ray transform on surfaces (Q6561318)

Given a matrix-valued function \(\mathbb{A}\) on a bounded domain \(M\) with boundary and a curve \(\gamma : [a, b] \rightarrow M\) connecting boundary points, one can consider the linear matrix differential equation\N\[\N\begin{cases} \dot{U}+\mathbb{A}(\gamma(t))U = 0, \\\NU(b)=\mathrm{Id}. \end{cases}\N\]\NThe matrix \(C_{\mathbb{A}}(\gamma) := U (a)\) at the boundary is called the non-abelian X-ray transform of \(\mathbb{A}\).\N\NIn this paper, the authors show that it is possible to recover \(\mathbb{A}\) from \(C_{\mathbb{A}}\) when the curves \(\gamma\) are the geodesics of a Riemannian surface \((M,g)\) with strictly convex boundary and non-trapping (i.e., for all \((x,v)\) in the unit tangent bundle of \(M\), the geodesic starting at \(x\) with velocity \(v\) exits \(M\) is finite).