3. Group Actions and Submersions
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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 3.25.
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. -
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.