Geodesic X-ray injectivity beyond the absence of conjugate points

Let $n\ge 2$ and let $(M^n,g)$ be a compact, connected, nonempty smooth Riemannian manifold with smooth boundary. Let $\mathrm{\nabla }$ be the unique Levi-Civita connection of $g$. Write ${\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ for the outward unit normal, and assume $g({\mathrm{\nabla }}_w{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}},w)>0$ for every nonzero boundary-tangent vector $w$. Assume every maximal unit-speed geodesic starting in $M^{\circ }$ reaches $\mathrm{\partial }M$ in both time directions in finite time; no absence-of-conjugate-points condition is imposed. Set $SM=\{(x,v):x\in M,v\in T_{x}M,g_x(v,v)=1\}$, let ${\nu }_{\mathrm{i}\mathrm{n}}=-{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and set ${\mathrm{\partial }}_{+}SM=\{(x,v)\in SM:x\in \mathrm{\partial }M,g_x(v,{\nu }_{\mathrm{i}\mathrm{n}})>0\}$. For $f\in C^{\mathrm{\infty }}(M)$ define $If(x,v)={\int }_0^{{\tau }_{+}(x,v)}f({\gamma }_{x,v}(t))dt$, where ${\gamma }_{x,v}$ solves ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=0$ with unit speed and initial state $(x,v)$, and ${\tau }_{+}(x,v)$ is its first positive exit time. Is $I$ injective on this class; equivalently, does $If=0$ on ${\mathrm{\partial }}_{+}SM$ imply $f=0$ for every such $(M,g)$?

Definition of a Riemannian metric and manifold

Definition of the Levi-Civita connection

Existence and uniqueness of the Levi-Civita connection

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.2, Question 2.1

Paternain, Salo, and Uhlmann, Geometric inverse problems, with emphasis on two dimensions, §15.1, Open problem 5, printed p. 339; §15.2, pp. 340–342, for higher-dimensional context

Holman and Krishnan, Ray transform of symmetric tensor fields on Riemannian manifolds with conjugate points

Mikko Salo, Minicourse on Geometric Inverse Problems, §1.2, Question 1.1

Status reviewed on 2026-10-21. Salo's RICAM notes and minicourse notes both list injectivity for compact strictly convex nontrapping manifolds as an open question. The transform is injective on the simple class used by the theorem on this page; the notes also give noninjective examples when trapping is allowed, which do not settle this nontrapping question. Higher-dimensional injectivity is known under additional convex-foliation hypotheses, including cases where conjugate points may occur. The 2025 paper inspected here studies microlocal normal-operator structure and cancellation of singularities for tensor transforms with conjugate points; it does not state a general injectivity resolution for the class above. No general resolution was found in the sources checked for this review.