Injective geodesic ray transform on a transversal manifold

Let $(M_0,g_0)$ be a compact connected smooth Riemannian manifold with smooth boundary. A geodesic segment in $M_0$ satisfies the geodesic equation for the Levi-Civita connection of $g_0$ on $(0,L)$ and extends smoothly to its endpoints, where derivatives are interpreted one-sided. A nontangential maximal geodesic segment is a unit-speed geodesic $\gamma :[0,L]\to M_0$, with $L>0$, whose initial velocity points strictly inward at $\mathrm{\partial }{M}_0$, whose terminal velocity points strictly outward at $\mathrm{\partial }{M}_0$, and whose points for $0<t<L$ lie in $M_0^{\circ }$. For $f\in C_c^{\mathrm{\infty }}(M_0^{\circ },\mathbb{R})$ define its geodesic ray transform by $I_{g_0}f(\gamma )={\int }_0^{L}f(\gamma (t))dt$. The unweighted scalar ray transform is called injective on compactly supported smooth functions if $I_{g_0}f(\gamma )=0$ for every such segment implies $f=0$. For complex-valued $f$, injectivity means injectivity of its real and imaginary parts separately.

Riemannian metric and unit-speed convention

Boundary and interior of a smooth manifold with boundary

Geodesics of a connection

Levi-Civita connection of the metric