Manifolds with non-negative sectional curvature

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6. Quasi-positive Curvature and Positive Curvature on an Open Dense Set

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Problem 6.1.
Which theorems from $\sec > 0$ carry over to positive curvature on an open dense set?
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Problem 6.2.
Suppose $G$ is a compact Lie group with a left-invariant metric. Are there any new examples $H \backslash G$ with quasi-positive curvature?
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Problem 6.3.
Find new examples of fundamental groups in quasi-positive curvature or positive curvature on an open dense set.
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Problem 6.4.
Fix $k \in \N$ . Is there a $n_0 = n_0 (k)$ such that for any quasi-positively curved manifold $(M^n, g)$ with $n \geq n_0$ and $\operatorname{cohom}(M^n,g) \leq k$ , there exists a chain $M_0 = M^n \subset M_1^{n+k} \subset M_2^{n+2k} \subset \cdots$ such that all inclusions are totally geodesic, the manifolds $M_i$ are quasi-positively curved, $\cup M_i$ is the classifying space of a Lie group, and $M_i / {Iso}(M_i,g)$ is isometric to $M / {Iso}(M,g)$ ?

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