Causal holography in application to the inverse scattering problems

Gabriel Katz

doi:10.3934/ipi.2019028

Article Contents

2019, Volume 13, Issue 3: 597-633. Doi: 10.3934/ipi.2019028

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Causal holography in application to the inverse scattering problems

Gabriel Katz

MIT, Department of Mathematics, 77 Massachusetts Ave., Cambridge, MA 02139, USA

Received: April 30, 2018

Revised: October 31, 2018

Published: March 2019

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Abstract

For a given smooth compact manifold $ M $, we introduce an open class $ \mathcal G(M) $ of Riemannian metrics, which we call 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 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 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 $.

We aim to understand the constraints on $ (M, g) $, under which the scattering data allow for a reconstruction of $ M $ and the metric $ g $ on it, up to a natural action of the diffeomorphism group $ \mathsf{Diff}(M, \partial M) $. In particular, we consider a closed Riemannian $ n $-manifold $ (N, g) $ which is locally symmetric and of negative sectional curvature. Let $ M $ is obtained from $ N $ by removing an $ n $-domain $ U $, such that the metric $ g|_M $ is boundary generic, of the gradient type, and the homomorphism $ \pi_1(U) \to \pi_1(N) $ of the fundamental groups is trivial. Then we prove that the scattering map $ C_{v^{g|_M}} $ makes it possible to recover $ N $ and the metric $ g $ on it.

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Mathematics Subject Classification: 37C10, 58E10, 53C22.

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Figure 1. The flat metric on torus becomes of the gradient type on the complement to a curvy disk (whose "center" is at the corners of the square fundamental domain).

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Figure 2. The sets $St(\theta) \cap \Delta $ (left diagram) and $St(\theta) \cap \Delta' \subset \beta \Delta$ (right diagram)

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Figure 3. The vertical traversing vector field $ v $ on a surface $ X $ divides its boundary $ \delta_1X $ into two loci, $ \delta_1^+X $ and $ \delta_1^-X $. The causality map $ C_v: \delta_1^+X \to \delta_1^-X $ is shown by the vertical arrows. The figure emphasizes the discontinuous nature of the map $ C_v $.

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