Title: Causal Holography in Application to the Inverse Scattering Problems,
Abstract: For a given smooth compact manifold $M$, we introduce an open class $\mathcal
G(M)$ of Riemannian metrics, which we call \emph{metrics of the gradient type}.
For such metrics $g$, the geodesic flow $v^g$ on the spherical tangent bundle
$SM \to M$ admits a Lyapunov function (so the $v^g$-flow is traversing). It
turns out, that metrics of the gradient type are exactly the non-trapping
metrics.
For every $g \in \mathcal G(M)$, the geodesic scattering along the boundary
$\partial M$ can be expressed in terms of the \emph{scattering map} $C_{v^g}:
\partial_1^+(SM) \to \partial_1^-(SM)$. It acts from a domain
$\partial_1^+(SM)$ in the boundary $\partial(SM)$ to the complementary domain
$\partial_1^-(SM)$, both domains being diffeomorphic. We prove that, for a
\emph{boundary generic} metric $g \in \mathcal G(M)$ the map $C_{v^g}$ allows
for a reconstruction of $SM$ and of the geodesic foliation $\mathcal F(v^g)$ on
it, up to a homeomorphism (often a diffeomorphism).
Also, for such $g$, the knowledge of the scattering map $C_{v^g}$ makes it
possible to recover the homology of $M$, the Gromov simplicial semi-norm on it,
and the fundamental group of $M$. Additionally, $C_{v^g}$ allows to reconstruct
the naturally stratified topological type of the space of geodesics on $M$.