Let $(M,g)$ be a compact $n$-dimensional Riemannian space with boundary $\partial M$. Consider different types of "scattering" functions:
1) \begin{equation} y:TM \to \partial M: (x,v_x) \mapsto y(x,v_x) \end{equation} where $y(x,v_x)$ is the point of $\partial M$ where a geodesic emanating from $x$ with velocity $v_x$ "intersects" the boundary $\partial M$ after some strictly positive time has passed.
2) \begin{equation} d:TM \to \mathbb{R}^+: (x,v_x) \mapsto d(x,v_x) \end{equation} is the length of the same geodesic between $x$ and $y(x,v_x)$. Let $K=\left\{(x,v_x) \in TM \left.\right|x \in \partial M\right\}$ be the restriction of the tangent bundle to the boundary and $y'=y\left.\right|_{\partial M}$, $d'=d\left.\right|_{\partial M}$
Q1: What conditions do arbitrary functions $\tilde{y'}$ and $\tilde{d'}$ have to satisfy in order for there to exist a metric $g$ which turns them into the above scattering functions $y'$, $d'$ for that metric?
Q2: Are there conditions on $M$, $y'$ and/or $d'$ which are necessary or sufficient for the uniqueness of such a metric $g$?
Q3: Are there conditions on $M$, $y'$ and/or $d'$ which force $(M,g)$ to be conformally flat (or which are necessary in that respect)?
(Maybe, by negligence, I have overlooked some other interesting "scattering functions" or "scattering questions". The answers to the questions which I ask here definitely have some nice applications in classical mechanics.)
(Q1 and Q2 can also be modified, removing $d'$ from their formulation, to ask after the existence and uniqueness of an affine connection only for which $y'$ is the above described scattering function.)