Abstract:
For a given smooth compact manifold \$M\$, we introduce a massive class \$\mathcal G(M)\$ of Riemannian metrics. We call them 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. Moreover, for every \$g \in \mathcal G(M)\$, the geodesic scattering along the boundary \$\d M\$ can be expressed in terms of a discontinuous map \$C_{v(g)}: \d_1^+(SM) \to \d_1^-(SM)\$. It acts from a domain \$\d_1^+(SM)\$ in the boundary \$\d(SM)\$ to the complementary domain \$\d_1^-(SM)\$, both domains being diffeomorphic.
We prove that, for a boundary generic metric \$g\$ of the gradient type, the scattering map \$C_{v(g)}\$ allows for a reconstruction of \$SM\$ and of the geodesic flow on it, up to a homeomorphism (often a diffeomorphism). Also, for such \$g\$, the knowledge of the scattering map \$C_{v(g)}\$ allows to reconstruct the homology of \$M\$, the Gromov simplicial semi-norm on it, and the fundamental group of \$M\$.
We aim to understand the constraints on \$(M, g)\$ under which the scattering map allows for a reconstruction of \$M\$ and of the metric \$g\$ on it. In particular, we consider a closed Riemannian \$n\$-manifold \$(N, g)\$ which is locally symmetric and of a negative sectional curvature. Let \$M\$ is obtained from \$N\$ by removing several \$n\$-balls so that the metric \$g|_M\$ is boundary generic and of the gradient type. Then we prove that the scattering map \$C_{v(g|_M)}\$ makes it possible to recover \$N\$ and the metric \$g\$ on it.