# Causal Holography in Application to the Inverse Scattering Problems

## TL;DR

This paper introduces a class of Riemannian metrics called gradient type metrics on smooth compact manifolds, and demonstrates how their geodesic scattering maps enable reconstruction of manifold topology, geodesic foliation, and fundamental group.

## Contribution

It establishes that for boundary generic metrics of the gradient type, the scattering map uniquely determines the manifold's topology, geodesic foliation, and algebraic invariants, advancing inverse scattering theory.

## Key findings

- Scattering maps allow reconstruction of the manifold and geodesic foliation.
- Knowledge of the scattering map recovers the fundamental group and homology.
- The results apply to a class of non-trapping, gradient type metrics.

## 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 $M$.