The tangent orientation cover preserves simple-surface X-ray data

By [F2], $\pi$ is a two-sheeted covering; applying [F3] to compact $M$ shows that $\tilde{M}$ is compact. In each boundary chart the two sheets are copies of the same half-space chart, so ${\pi }^{-1}(\mathrm{\partial }M)=\mathrm{\partial }\tilde{M}$ and ${\pi }^{-1}(M^{\circ })={\tilde{M}}^{\circ }$.

Every connected component $C$ of the smooth manifold $\tilde{M}$ is open, since each point has a connected chart neighborhood contained in its component, and it is closed, since connected components are closed. Thus $C$ is compact. Smooth boundary charts have path-connected smaller half-ball neighborhoods, so $M$ is locally path-connected and [F8] makes connected $M$ path-connected. Fix $\tilde{x}\in C$ and any $y\in M$; lift a path from $\pi (\tilde{x})$ to $y$ starting at $\tilde{x}$ by [F6]. Its connected image stays in $C$, so its endpoint lies over $y$ and $\pi (C)=M$. The canonical orientation from [F2] restricts to $C$.

The pullback $\tilde{g}={\pi }^{\ast }g$ is smooth and positive definite because $d\pi$ is a smooth isomorphism on each sheet and $g$ is smooth positive definite by [F4]. In the sheet coordinates of [F2], $\pi$ is the identity and the metric coefficient matrices of $\tilde{g}$ and $g$ coincide. The unique Levi-Civita connections therefore have identical coordinate coefficients; the geodesic equations and curvature coefficients also coincide by [F4, F5]. Boundary normals and their second fundamental forms correspond under $d\pi$, so strict convexity of $\mathrm{\partial }M$ gives strict convexity of $\mathrm{\partial }C$.

If $\tilde{J}$ is a Jacobi field along a geodesic segment in $C$, then [F5] and the identical local connection and curvature coefficients in step 2.1 show that $d\pi (\tilde{J})$ is a Jacobi field along the projected geodesic. If $\tilde{J}$ vanishes at both endpoints, its projection vanishes there; [F1] makes the projected field zero, and invertibility of $d\pi$ gives $\tilde{J}=0$. Thus $C$ has no conjugate points.

A unit-speed geodesic in $C^{\circ }$ projects by step 2.1 to a unit-speed geodesic in $M^{\circ }$. By [F1] that projected geodesic reaches $\mathrm{\partial }M$ in finite forward and backward time. Lift each finite projected segment from the specified initial point by [F6]. In every sheet the lift solves the same geodesic ODE as the original curve, so uniqueness from [F7] makes it an extension of that curve to the exit time; the endpoint lies in $\mathrm{\partial }C$ by step 1.1. Hence $C$ is nontrapping in both directions. Together with steps 1.2–2.1, $C$ is simple.

Since $f$ is supported in the interior, $\tilde{f}=f\circ \pi$ is smooth and vanishes near $\mathrm{\partial }\tilde{M}$. Its support is a closed subset of compact $\tilde{M}$, hence compact, and it lies in $C^{\circ }$ after restriction to $C$. For an incoming $\tilde{z}\in {\mathrm{\partial }}_{+}SC$, local isometry maps its inward normal component to that of $z=\pi (\tilde{z})\in {\mathrm{\partial }}_{+}SM$. The projected unit geodesic and the lifted unit geodesic have the same parameter and first boundary-exit time by steps 1.1 and 4.1; hence $I_{0,C}\tilde{f}(\tilde{z})={\int }_0^{{\tau }_{+}(z)}f({\gamma }_z(t))dt=I_{0,M}f(z)$. This proves the transform identity and the zero-data implication.