
## 4. Manifolds of Cohomogeneity-one and Polar Actions

1. #### Problem 4.1.

Study the existence and non-existence of metrics with non-negative curvature on manifolds of cohomogeneity-one.
• #### Problem 4.2.

Find cohomogeneity-one manifolds with “interesting” topology, in particular not homeomorphic to a symmetric space. Study curvature properties of these manifolds.
• #### Problem 4.3.

Classify cohomogeneity-one manifolds with $\sec \geq 0$ and at least one totally geodesic principal orbit.
• #### Problem 4.4.

Compute topological invariants of cohomogeneity-one manifolds. Classify topologically the new candidates for positive curvature.
• #### Problem 4.5.

Study the existence of Einstein metrics on manifolds of cohomogeneity-one.
• #### Problem 4.6.

Suppose $M$ is a polar manifold with $\sec > 0$. Must $M$ be diffeomorphic to a compact rank-one 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 non-negative sectional curvature, available at http://aimpl.org/nnsectcurvature.