4. Manifolds of Cohomogeneityone and Polar Actions

Problem 4.1.
Study the existence and nonexistence of metrics with nonnegative curvature on manifolds of cohomogeneityone. 
Problem 4.2.
Find cohomogeneityone manifolds with “interesting” topology, in particular not homeomorphic to a symmetric space. Study curvature properties of these manifolds. 
Problem 4.3.
Classify cohomogeneityone manifolds with $\sec \geq 0$ and at least one totally geodesic principal orbit. 
Problem 4.4.
Compute topological invariants of cohomogeneityone manifolds. Classify topologically the new candidates for positive curvature. 
Problem 4.6.
Suppose $M$ is a polar manifold with $\sec > 0$. Must $M$ be diffeomorphic to a compact rankone symmetric space? 
Problem 4.7.
Suppose $\Sigma \subset M^n$ is the section of a polar action. If $\Sigma$ is rationally elliptic, must $M^n$ be rationally elliptic?
Cite this as: AimPL: Manifolds with nonnegative sectional curvature, available at http://aimpl.org/nnsectcurvature.