Asystatic and infinitesimally polar actions

Andreas Kollross - Stuttgart.

Resumo:

Joint work with Claudio Gorodski. A homogeneous manifold is called asystatic if sufficiently close points have different isotropy groups. More generally, we say that a smooth action is asystatic if its principal orbits are asystatic homogeneous manifolds. Isometric asystatic actions on Riemannian manifolds are polar (i.e. there exists a submanifold which meets all orbits orthogonally), but the converse does not hold in general. An action is called infinitesimally polar if all of its slice representations, i. e. the local linearisations of the action around singular orbits, are polar. We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one and prove that every polar action on one of these spaces has the same orbits as an asystatic action.

© Andreas Kollross.