3. Group Actions and Submersions

Problem 3.05.
Let $M^n$ be a manifold with $\sec > 0$ or $\sec \geq 0$ or almost nonnegative curvature. Does $M^n$ have a positive symmetry degree, i.e. is there an $S^1 \subset \operatorname{Diff}(M^n)$? 
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 biinvariant metric. 
Problem 3.45.
Suppose $(M^n,g)$ is simplyconnected 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 itâ€™s length $\leq 2 \pi$?
Cite this as: AimPL: Manifolds with nonnegative sectional curvature, available at http://aimpl.org/nnsectcurvature.