Seminario Prof. Cristopher Croke
Il Prof. Cristopher Croke, University of Pennsylvania (USA) terra' un seminario dal titolo
Scattering rigidity in the presence of trapped geodesics
il giorno 26 Aprile 2011 ore 11.30 nella sala conferenze "F.Tricerri".
ABSTRACT:
We discuss how to show that the flat solid torus is scattering rigid. We will be considering 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 unitvector 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.