Intersection Rigidity

Abstract

We consider three inverse problems related to geodesic intersections. First, we consider theproblem of recovering the geometry of a Riemannian manifold with boundary from the knowledge of all pairs of inward pointing directions at the boundary that correspond to intersecting geodesics. We call this the intersection data and show that it determines the geometry when the manifold is simple and at least three-dimensional. Next, we consider the problem of recovering the geometry of a Riemannian manifold with boundary from the knowledge of how far along each geodesic you must travel to reach the intersection points of any other geodesic. We call this information a stitching data and show that it determines the geometry of the manifold, without any restrictions on the geometry. Finally, we consider the problem of recovering the geometry of a Riemannian manifold with boundary from knowledge of how to time particles shot along geodesics from the boundary so that they collide on the interior. We call this information the delayed collision data and show that it determines the geometry of the manifold with natural geometric restrictions. In particular, the stitching data and delayed collision data apply to non-compact and unbounded manifolds.