# American Institute of Mathematical Sciences

June  2019, 13(3): 597-633. doi: 10.3934/ipi.2019028

## Causal holography in application to the inverse scattering problems

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

Received  May 2018 Revised  November 2018 Published  March 2019

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.

Citation: Gabriel Katz. Causal holography in application to the inverse scattering problems. Inverse Problems & Imaging, 2019, 13 (3) : 597-633. doi: 10.3934/ipi.2019028
##### References:
 [1] A. L. Besse, Manifolds all of whose Geodesics are Closed, Springer-Verlag, Berlin Heidelberg New York, 1978. [2] G. Besson, G. Courtois and G. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergod. Th. & Dynam. Syst., 16 (1996), 623-649. doi: 10.1017/S0143385700009019. [3] C. Croke, Scattering Rigidity with Trapped Geogesics, Ergod. Th. & Dynam. Syst., 34 (2014), 826-836. [4] C. Croke, Rigidity theorems in riemannian geometry, Chapter in Geometric Methods in Inverse Problems and PDE Control, C. Croke, I. Lasiecka, G. Uhlmann, and M. Vogelius eds., IMA vol. Math Appl., 137, Springer, 2004, 47–72. doi: 10.1007/978-1-4684-9375-7_4. [5] C. Croke, Rigidity and the distance between boundary points, J. Diff. Geometry, 33 (1991), 445-464. doi: 10.4310/jdg/1214446326. [6] C. Croke, P. Eberlein and B. Kleiner, Conjugacy and rigidity for nonpositively curved manifolds of higher rank, Topology, 35 (1996), 273-286. doi: 10.1016/0040-9383(95)00031-3. [7] C. Croke and B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field, J. Differential Geometry, 39 (1994), 659-680. doi: 10.4310/jdg/1214455076. [8] G. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc., 2 (1989), 371-415. doi: 10.1090/S0894-0347-1989-0965210-7. [9] M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.E.S., 56 (1982), 5-99. [10] G. Katz, Convexity of Morse stratifications and spines of 3-manifolds, JP Journal of Geometry and Topology, 9 (2009), 1-119. [11] G. Katz, Stratified convexity and concavity of gradient flows on manifolds with boundary, Applied Mathematics, 5 (2014), 2823–2848, (SciRes. http://www.scirp.org/journal/am) (also arXiv: 1406.6907v1, [mathGT] (26 June, 2014)). [12] G. Katz, Traversally generic and versal flows: Semi-algebraic models of tangency to the boundary, Asian J. of Math., 21 (2017), 127–168 (also arXiv: 1407.1345v1, [mathGT] (4 July, 2014)). doi: 10.4310/AJM.2017.v21.n1.a3. [13] G. Katz, The stratified spaces of real polynomials and trajectory spaces of traversing flows, JP Journal of Geometry and Topology, 19 (2016), 95-160. [14] G. Katz, Causal Holography of Traversing Flows, arXiv: 1409.0588v3, [mathGT] 7 Aug 2017. [15] G. Katz, Flows in flatland: A romance of few dimensions, Arnold Math. J., 3 (2017), 281-317. doi: 10.1007/s40598-016-0059-1. [16] V. S. Matveev, Geodesically equivalent metrics in general relativity, J. Geom. Phys., 62 (2012), 675–691, arXiv: 1101.2069v2 [math.DG] 7 Apr 2011. doi: 10.1016/j.geomphys.2011.04.019. [17] M. Morse, Singular points of vector fields under general boundary conditions, Amer. J. Math., 51 (1929), 165-178. doi: 10.2307/2370703. [18] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of mathematics studies, 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. [19] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform for tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2. [20] P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7. [21] P. Stefanov and G. Uhlmann, Boundary and lens rigidity, tensor holography, and analytic microlocal analysis, Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics, 275–293, Springer, Tokyo, 2008. doi: 10.1007/978-4-431-73240-2_23. [22] P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett., 5 (1998), 83-96. doi: 10.4310/MRL.1998.v5.n1.a7. [23] P. Stefanov and G. Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409. doi: 10.4310/jdg/1246888489. [24] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299–332, arXiv: 1306.2995. doi: 10.1090/jams/846. [25] Stefanov, P., G. Uhlmann, G., Vasy, A., Local recovery of the compressional and shear speeds from the hyperbolic DN map, preprint. doi: 10.1088/1361-6420/aa9833. [26] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic $X$-ray transform on tensors, Journal d'Analyse Mathematique, 136 (2018), 151–208, arXiv: 1410.5145. doi: 10.1007/s11854-018-0058-3. [27] P. Stefanov, G. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge, arXiv: 1702.03638, 2017. [28] G. Teschi, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Math., 140, AMS publication, 2012. doi: 10.1090/gsm/140. [29] H. Wen, Simple Riemannian Surfaces are Scattering Rigid, arXiv:1405. 1712v1, [math.DG], 7 May, 2014. [30] X. Zhou, Recovery of the $C^\infty$-jet from the boundary distance function, Geometriae Dedicata, 160, (2012), 229-241. arXiv:1103. 5509v1, [mathDG] 28 March 2011.

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##### References:
 [1] A. L. Besse, Manifolds all of whose Geodesics are Closed, Springer-Verlag, Berlin Heidelberg New York, 1978. [2] G. Besson, G. Courtois and G. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergod. Th. & Dynam. Syst., 16 (1996), 623-649. doi: 10.1017/S0143385700009019. [3] C. Croke, Scattering Rigidity with Trapped Geogesics, Ergod. Th. & Dynam. Syst., 34 (2014), 826-836. [4] C. Croke, Rigidity theorems in riemannian geometry, Chapter in Geometric Methods in Inverse Problems and PDE Control, C. Croke, I. Lasiecka, G. Uhlmann, and M. Vogelius eds., IMA vol. Math Appl., 137, Springer, 2004, 47–72. doi: 10.1007/978-1-4684-9375-7_4. [5] C. Croke, Rigidity and the distance between boundary points, J. Diff. Geometry, 33 (1991), 445-464. doi: 10.4310/jdg/1214446326. [6] C. Croke, P. Eberlein and B. Kleiner, Conjugacy and rigidity for nonpositively curved manifolds of higher rank, Topology, 35 (1996), 273-286. doi: 10.1016/0040-9383(95)00031-3. [7] C. Croke and B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field, J. Differential Geometry, 39 (1994), 659-680. doi: 10.4310/jdg/1214455076. [8] G. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc., 2 (1989), 371-415. doi: 10.1090/S0894-0347-1989-0965210-7. [9] M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.E.S., 56 (1982), 5-99. [10] G. Katz, Convexity of Morse stratifications and spines of 3-manifolds, JP Journal of Geometry and Topology, 9 (2009), 1-119. [11] G. Katz, Stratified convexity and concavity of gradient flows on manifolds with boundary, Applied Mathematics, 5 (2014), 2823–2848, (SciRes. http://www.scirp.org/journal/am) (also arXiv: 1406.6907v1, [mathGT] (26 June, 2014)). [12] G. Katz, Traversally generic and versal flows: Semi-algebraic models of tangency to the boundary, Asian J. of Math., 21 (2017), 127–168 (also arXiv: 1407.1345v1, [mathGT] (4 July, 2014)). doi: 10.4310/AJM.2017.v21.n1.a3. [13] G. Katz, The stratified spaces of real polynomials and trajectory spaces of traversing flows, JP Journal of Geometry and Topology, 19 (2016), 95-160. [14] G. Katz, Causal Holography of Traversing Flows, arXiv: 1409.0588v3, [mathGT] 7 Aug 2017. [15] G. Katz, Flows in flatland: A romance of few dimensions, Arnold Math. J., 3 (2017), 281-317. doi: 10.1007/s40598-016-0059-1. [16] V. S. Matveev, Geodesically equivalent metrics in general relativity, J. Geom. Phys., 62 (2012), 675–691, arXiv: 1101.2069v2 [math.DG] 7 Apr 2011. doi: 10.1016/j.geomphys.2011.04.019. [17] M. Morse, Singular points of vector fields under general boundary conditions, Amer. J. Math., 51 (1929), 165-178. doi: 10.2307/2370703. [18] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of mathematics studies, 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. [19] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform for tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2. [20] P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7. [21] P. Stefanov and G. Uhlmann, Boundary and lens rigidity, tensor holography, and analytic microlocal analysis, Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics, 275–293, Springer, Tokyo, 2008. doi: 10.1007/978-4-431-73240-2_23. [22] P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett., 5 (1998), 83-96. doi: 10.4310/MRL.1998.v5.n1.a7. [23] P. Stefanov and G. Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409. doi: 10.4310/jdg/1246888489. [24] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299–332, arXiv: 1306.2995. doi: 10.1090/jams/846. [25] Stefanov, P., G. Uhlmann, G., Vasy, A., Local recovery of the compressional and shear speeds from the hyperbolic DN map, preprint. doi: 10.1088/1361-6420/aa9833. [26] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic $X$-ray transform on tensors, Journal d'Analyse Mathematique, 136 (2018), 151–208, arXiv: 1410.5145. doi: 10.1007/s11854-018-0058-3. [27] P. Stefanov, G. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge, arXiv: 1702.03638, 2017. [28] G. Teschi, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Math., 140, AMS publication, 2012. doi: 10.1090/gsm/140. [29] H. Wen, Simple Riemannian Surfaces are Scattering Rigid, arXiv:1405. 1712v1, [math.DG], 7 May, 2014. [30] X. Zhou, Recovery of the $C^\infty$-jet from the boundary distance function, Geometriae Dedicata, 160, (2012), 229-241. arXiv:1103. 5509v1, [mathDG] 28 March 2011.
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).
The sets $St(\theta) \cap \Delta$ (left diagram) and $St(\theta) \cap \Delta' \subset \beta \Delta$ (right diagram)
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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