The authors explore rigidity properties of polar actions on positively curved manifolds. By a polar action they mean an isometric action for which there is an immersed submanifold that meets all orbits orthogonally. Their main result is that a polar action on a simply connected, compact, positively curved manifold of cohomogeneity 2 is equivariantly diffeomorphic to a polar action on a compact rank-1 symmetric space.
The authors also make the conjecture that an irreducible polar action on a simply connected, non-negatively curved compact manifold is equivariantly diffeomorphic to a quotient of a polar action on a symmetric space.