Simple surface and geodesic X-ray transform

Let $(M,g)$ be a compact, connected, nonempty smooth two-dimensional Riemannian manifold with smooth boundary. For $x\in \mathrm{\partial }M$, let ${\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}(x)$ be the outward unit normal. The boundary is strictly convex if

$$g({\mathrm{\nabla }}_w{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}},w)>0$$

for every $x\in \mathrm{\partial }M$ and every nonzero $w\in T_x\mathrm{\partial }M$. A Jacobi field along a unit-speed geodesic $\gamma$ is a smooth vector field $J$ along $\gamma$ satisfying

$$D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma }=0,$$

where $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. The manifold has no conjugate points if, on every unit-speed geodesic segment $\gamma :[a,b]\to M$, a Jacobi field with $J(a)=J(b)=0$ must be identically zero. It is nontrapping if every maximal unit-speed geodesic starting in $M^{\circ }$ reaches $\mathrm{\partial }M$ in finite forward and backward time. The pair $(M,g)$ is a simple surface if its boundary is strictly convex, it has no conjugate points, and it is nontrapping.

The unit tangent bundle is $SM=\{(x,v):x\in M,v\in T_{x}M,g_x(v,v)=1\}$. Write ${\nu }_{\mathrm{i}\mathrm{n}}=-{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and

$${\mathrm{\partial }}_{+}SM=\{(x,v)\in SM:x\in \mathrm{\partial }M,g_x(v,{\nu }_{\mathrm{i}\mathrm{n}})>0\}$$

for inward-pointing unit boundary vectors. For $(x,v)\in {\mathrm{\partial }}_{+}SM$, let ${\gamma }_{x,v}$ be the unit-speed geodesic with ${\gamma }_{x,v}(0)=x$ and ${\dot{\gamma }}_{x,v}(0)=v$, and let ${\tau }_{+}(x,v)$ be its first positive exit time from $M$. For $f\in C^{\mathrm{\infty }}(M)$, its scalar geodesic X-ray transform is the function on ${\mathrm{\partial }}_{+}SM$ defined by

$$I_{0}f(x,v)={\int }_0^{{\tau }_{+}(x,v)}f({\gamma }_{x,v}(t))dt\mathrm{.}$$

The geodesic equation is interpreted up to first boundary exit; local smooth extensions of the metric are used only to solve the coordinate ODE at boundary initial data.

Riemannian metric and riemannian manifold

Levi civita connection

Fundamental theorem of riemannian geometry

Jacobi field

Local existence, uniqueness, and smooth dependence for manifold integral curves