Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

3. Group Actions and Submersions


    1. Add remark

      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)?


        • Add remark

          Problem 3.1.

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


            • Add remark

              Problem 3.15.

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


                • Add remark

                  Problem 3.2.

                  Can one reduce the structure group of a fat principal G-bundle?


                    • Add remark

                      Problem 3.25.

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


                        • Add remark

                          Problem 3.3.

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


                            • Add remark

                              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?


                                • Add remark

                                  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.


                                    • Add remark

                                      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?


                                        • Add remark

                                          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?


                                            • Add remark

                                              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.