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4. Manifolds of Cohomogeneity-one and Polar Actions


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      Problem 4.1.

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


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          Problem 4.2.

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


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              Problem 4.3.

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


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                  Problem 4.4.

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


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                      Problem 4.5.

                      Study the existence of Einstein metrics on manifolds of cohomogeneity-one.


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                          Problem 4.6.

                          Suppose $M$ is a polar manifold with $\sec > 0$. Must $M$ be diffeomorphic to a compact rank-one symmetric space?


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                              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.