Polar manifolds and actions
  A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental domain, generalizing the classical construction for cohomogeneity manifolds. We give many examples showing the richness of this class of group actions. We also relate the topology of the section to the topology of the manifold.   This is a replacement of an earlier version. Small changes, and a correction in Lemma 2.4. 
Karsten Grove and Wolfgang Ziller
math.DG