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3. Group Actions and Submersions

    1. Problem 3.05.

      Let M^n be a manifold with \sec > 0 or \sec \geq 0 or almost non-negative curvature. Does M^n have a positive symmetry degree, i.e. is there an S^1 \subset \operatorname{Diff}(M^n)?
        • Problem 3.1.

          Is there a principal T^2-bundle whose total space admits \sec > 0?
            • Problem 3.15.

              Given a fat G-principal bundle, must G be S^1, S^3 or SO(3)?
                • Problem 3.2.

                  Can one reduce the structure group of a fat principal G-bundle?
                    • Problem 3.25.

                      Given a homogeneous space G/H, with G compact, classify all homogeneous metrics with \sec \geq 0.
                        • Problem 3.3.

                          Is there a positively curved 5-manifold with a free isometric S^3 or SO(3) action?
                            • Problem 3.35.

                              Given a Riemannian submersion with positively curved total space, is the dimension of the fiber less than the dimension of the base? It is perhaps simpler to decide if there is a bound on the dimension of the fiber in terms of the dimension of the base. Is the image of the A-tensor large at some point?
                                • Problem 3.4.

                                  Classify Riemannian submersions from a Lie group with a bi-invariant metric.
                                      As was observed at the AIM workshop, they are not necessarily biquotient submersions.
                                    • Problem 3.45.

                                      Suppose (M^n,g) is simply-connected and \sec_M \geq 0. Is \operatorname{rank}( {Iso}(M^n,g) ) \leq \frac{2}{3}n?
                                        • Problem 3.5.

                                          Suppose we have an isometric group action on M. If one changes the metric on the orbit space, does it lift to an invariant metric on M?
                                            • Problem 3.55.

                                              If a group acts isometrically on a manifold of \sec \geq 1 and the fixed point set is a circle, is it’s length \leq 2 \pi?

                                                  Cite this as: AimPL: Manifolds with non-negative sectional curvature, available at http://aimpl.org/nnsectcurvature.