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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.