Institut for Matematiske Fag > Kalender
Institut for Matematiske Fags kalender
Scattering and lens rigidity on some Riemannian manifolds with trapped geodesics
Taler(e): Christopher Croke
(University of Pennsylvania )
Mandag den 25. juni 2012 kl. 15:15-16:15. Sted: Aud 8. Sprog: engelsk.
We discuss how to show that the flat solid torus is scattering rigid. We will consider compact Riemannian manifolds M with boundary N. We let IN be the unit vectors to M whose base point is on N and point inwards towards M. Similarly we define OUT. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map from IN to OUT which assigns to each unit vector V of IN a the unit vector W in OUT. W will be the tangent vector to the geodesic determined by V when that geodesic first hits the boundary N again. This may not be defined for all V since the geodesic might be trapped (i.e. never hits the boundary again). A manifold is said to be scattering rigid if any other Riemannian manifold Q with boundary isometric to N and with the same scattering data must be isometric to M.
In this talk we will discuss the scattering rigidity problem and related inverse problems. There are a number of manifolds that are known to be scattering rigid and there are examples that are not scattering rigid. All of the known examples of non-rigidity have trapped geodesics in them. In particular, we will see that the flat solid torus is scattering rigid. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in any such Q have measure 0 in the unit tangent bundle of Q. We will also consider scattering rigidity of a number of two dimensional manifolds (joint work with Pilar Herreros) which have trapped geodesics.
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