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Reconstruction of a compact manifold from the scattering data of internal sources

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  • Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.

    Mathematics Subject Classification: Primary: 53C22, 35R30.

    Citation:

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  • Figure 1.  Here is a schematic picture about our data $R_{\partial M}(p)$, where the point $p$ is the blue dot. Here the black arrows are the exit directions of geodesics emitted from $p$ and the blue arrows are our data

    Figure 2.  Here is a visualization of the set up in the definition of the function $\varrho_k$ in Lemma 2.8. The blue dot is $p$ and the red&blue dot is $q$. The black curve is the geodesic $\gamma_{p, \eta}$. The red line is the hypersurface $\tilde S_1$ and the blue line is the hypersurface $\tilde S_2$. The small blue and red segments indicate the intervals $(s_k-\delta, s_k+\delta)$ where the function $\varrho_k(\cdot, \eta)$ is defined

    Figure 3.  Here is a schematic picture about $K(p)$, where the point $p \in M$ is the blue dot. The black curves represent the geodesics $\gamma_{z, \xi}$, and $\gamma_{w, \eta}$ respectively, where vectors $(z, \xi), (w, \eta) \in R_{\partial M}^E(p)$. Notice that only $(w, \eta)\in K(p)$

    Figure 4.  Here is a schematic picture about the map $\Theta_{q, \tilde q, v}$ evaluated at point $p\in M$, where the point $p \in M$ is the blue dot and $V_q\cap V_{\tilde q}$ is the blue ellipse. The higher red&blue dot is $q$ and the lower is $\tilde q$. The blue vector is the given direction $v \in T_{\tilde q}N$

    Figure 5.  Here is a schematic picture about the situation where the boundary normal geodesic $\gamma_{p, \nu}$ (the black curve) is self-intersecting at $p \in \partial M$ (red&blue dot). Here the blue curve is the geodesic $\gamma_{p, W(p)}$, where $W(p)\notin I_p$. For the point $w \in M$ (blue dot) the point $p$ satisfies $p = \Pi_W(w)$

    Figure 6.  Here is a schematic picture about the map $(\tilde Q_{q, v}, \Pi^E_{q, W})$ evaluated at a point $x\in M$ that is close to $\partial M$, where the point $x \in M$ is the blue dot. The right hand side red&blue dot is $\Pi^E_{q, W}(x)$ and the left hand side red&blue dot is $q$. The blue arrow is the given vector $v \in T_qN$

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