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\begin{document}
\title{Manifolds with non-negative sectional curvature}
\author{Edited by Martin Kerin}

\urlstub{nnsectcurvature}

\maketitle




\section{Diameter Pinching}





\begin{problemblock}

\begin{problem}[1.1] Does the Grove-Petersen-Wu Finiteness Theorem hold in dimension $4$, i.e. are there only finitely many diffeomorphism types in the class of $4$-dimensional Riemannian manifolds satisfying
            $$\sec_M \geq - \Lambda^2, \ \ \operatorname{diam}(M) \leq D, \ \ \operatorname{Vol}(M) \geq V ?$$
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.2] If $\sec_M \geq 1$ and $\operatorname{diam}(M) > \frac{\pi}{2}$, must $M$ be diffeomorphic to $S^n$?

 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.3] Let $M^n$ be a manifold for which $\sec_M \geq 1$ and $\operatorname{diam} (M) \geq \frac{\pi}{2} - \varepsilon(n)$.  Is $M$ homeomorphic to a manifold $M'$, where $\sec_{M'} \geq 1$, $\operatorname{diam}(M') \geq \frac{\pi}{2}$?  This problem may be easier to address if we also assume that $\operatorname{Vol}(M) \geq V$ and $\varepsilon = \varepsilon(n, V)$.
 \end{problem}




\end{problemblock}



\section{Collapse and Alexandrov Geometry}





\begin{problemblock}

\begin{problem}[.05] Perelman's Stability Theorem yields that manifolds in a given sequence of non-collapsing manifolds are eventually pairwise homeomorphic.  Are they also PL-homeomorphic or diffeomorphic? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.1] Understand DC-structures on manifolds.  In particular, does Perelman's Stability Theorem hold in the DC-category. Is PL = DC always? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.15] Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if $X$ is a positively curved Alexandrov space and $Y \subset X$ a totally geodesic subspace of codimension $k$, is it true that $X-Y$ has homology only up to dimension $2k - 2$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.2] Suppose $X$ is the non-collapsed Gromov-Hausdorff limit of $(M_i^n, g_i)$, where $|\sec_{M_i}| \leq 1$, and that along every geodesic on $M_i$ one hits a conjugate point before $t = \pi + \frac{1}{i}$.  Is $X$ rigid in any sense?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.25] Is there a sequence of simply-connected, pointwise strictly $\frac{1}{4}$-pinched manifolds $M_i^n$, $n > 2$, that collapse?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.3] Find an appropriate definition of Morse functions on Alexandrov spaces and construct examples.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.35] Study the collapse of Alexandrov spaces.
 \end{problem}




\end{problemblock}



\begin{problemblock}Consider finite towers
\begin{equation}
\xymatrix{ M_0 \ar[r]^{F_1} & M_1 \ar[r]^{F_2} & \dots \ar[r]^{F_k} & M_k }
\end{equation}
of fiber bundles, where the fibers $\{F_1, \dots, F_k\}$ and $M_k$ are fixed topological manifolds.

\begin{problem}[2.4] Loosen this notion to get a ``brotherhood'' on the manifolds $M_0$ and a property of such $M_0$ {\it not} known to be possessed by all manifolds of $\sec \geq K$, $\operatorname{diam} \leq 1$.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.45] Give an Alexandrov analogue of rational ellipticity.  In particular, are manifolds with almost non-negative sectional curvature rationally elliptic? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.5] Given a non-collapsing Gromov-Hausdorff convergence $M_i \longrightarrow X$, can one find a ``tangent bundle'' structure on $X$ that is sensitive to the diffeomorphism class of the $M_i$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.55] Is there a Gauss formula for Alexandrov spaces, i.e. must a convex hypersurface $Y$ of an Alexandrov space $X$ have $\sec_Y \geq \sec_X$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.6] Is every finite dimensional Alexandrov space a limit of Riemannian manifolds with $\sec \geq K$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.65] Study Alexandrov (almost) submetries. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.7] Is there an alternate approach to homotopy groups that is adapted to Alexandrov spaces? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.75] Study collapse to a ray. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.8] Can an $n$-dimensional torus collapse to an interval?   \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.85] Study the collapse of Riemannian manifolds with boundary which have $\sec \geq K$ on the interior and controlled boundary concavity. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.9] Find an application where infinite-dimensional Alexandrov spaces appear as limits of manifolds of increasing dimension. \end{problem}




\end{problemblock}



\section{Group Actions and Submersions}





\begin{problemblock}

\begin{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 $S^1 \subset \operatorname{Diff}(M^n)$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.1] Is there a principal $T^2$-bundle whose total space admits $\sec > 0$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.15] Given a fat $G$-principal bundle, must $G$ be $S^1$, $S^3$ or $SO(3)$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.2] Can one reduce the structure group of a fat principal $G$-bundle? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.25] Given a homogeneous space $G/H$, with $G$ compact, classify all homogeneous metrics with $\sec \geq 0$. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.3] Is there a positively curved $5$-manifold with a free isometric $S^3$ or $SO(3)$ action? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{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?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.4] Classify Riemannian submersions from a Lie group with a bi-invariant metric.   \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.45] Suppose $(M^n,g)$ is simply-connected and $\sec_M \geq 0$.  Is $\operatorname{rank}( {Iso}(M^n,g) ) \leq \frac{2}{3}n$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{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 $M$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{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 $\leq 2 \pi$? \end{problem}




\end{problemblock}



\section{Manifolds of Cohomogeneity-one and Polar Actions}





\begin{problemblock}

\begin{problem}[4.3] Classify cohomogeneity-one manifolds with $\sec \geq 0$ and at least one totally geodesic principal orbit.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[4.4] Compute topological invariants of cohomogeneity-one manifolds.  Classify topologically the new candidates for positive curvature.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[4.5] Study the existence of Einstein metrics on manifolds of cohomogeneity-one.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[4.6] Suppose $M$ is a polar manifold with $\sec > 0$.  Must $M$ be diffeomorphic to a compact rank-one symmetric space?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[4.7] Suppose $\Sigma \subset M^n$ is the section of a polar action.  If $\Sigma$ is rationally elliptic, must $M^n$ be rationally elliptic?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[4.2] Find cohomogeneity-one manifolds with ``interesting'' topology, in particular not homeomorphic to a symmetric space.  Study curvature properties of these manifolds.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[4.1] Study the existence and non-existence of metrics with non-negative curvature on manifolds of cohomogeneity-one.
 \end{problem}




\end{problemblock}



\section{Vector Bundles}





\begin{problemblock}

\begin{problem}[5.1] Is there a metric with $\sec \geq 0$ on $\R^6$-bundles over $S^3 \times S^3$ with non-trivial Euler class?  Wilking has shown that the answer is ``No'' if the soul is $S^3 \times S^3$ with the product metric.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[5.2] Which vector bundles over $S^2 \times S^2$ or $\C P^2 \# \pm \C P^2$ where the structure group does not reduce to a torus admit $\sec \geq 0$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[5.3] Classify metrics with $\sec \geq 0$ on $S^2 \times \R^4$ (or, more generally, on $S^n \times \R^k$). \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[5.4] Suppose $E \longrightarrow M$ is a vector bundle over a compact, simply-connected manifold $M$ for which $\sec_M \geq 0$.  Does $E \oplus \R^k \longrightarrow M$ have $\sec \geq 0$ for $k$ large?
 \end{problem}




\end{problemblock}



\section{Quasi-positive Curvature and Positive Curvature on an Open Dense Set}





\begin{problemblock}

\begin{problem}[6.1] Which theorems from $\sec > 0$ carry over to positive curvature on an open dense set?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{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?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[6.3] Find new examples of fundamental groups in quasi-positive curvature or positive curvature on an open dense set.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{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 $M / {Iso}(M,g)$?
 \end{problem}




\end{problemblock}



\section{Ricci Flow}





\begin{problemblock}

\begin{problem}[7.1] Let $M^n$ be a compact manifold with positive isotropic curvature.  Does any blow-up limit of the Ricci flow have non-negative curvature operator?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[7.2] Suppose $G$ acts freely and isometrically on $M$.  What kind of flows on $M/G$ are induced by the Ricci flow on $M$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[7.3] Can one improve the Hsiang-Kleiner theorem on positively curved $4$-manifolds with symmetry from homeomorphism to diffeomorphism by using the Ricci flow?
 \end{problem}




\end{problemblock}



\section{Miscellaneous Problems}





\begin{problemblock}

\begin{problem}[8.1] Is a ``generic'' manifold a $K(\pi, 1)$-space (where ``generic'' is to be determined)? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[8.2] Does positive sectional curvature imply that the manifold is formal (in the sense of Sullivan's minimal model)?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[8.3] For compact, odd dimensional, positively curved manifolds is there a cyclic subgroup of the fundamental group whose index is bounded only in terms of the dimension?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[8.4] Is there a finiteness result for $n$-dimensional, positively curved manifolds with $\pi_1 = \pi_2 = 0$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[8.5] Is there a $\delta(n) > 0$ such that any $n$-dimensional, positively curved manifold carries a $\delta(n)$-pinched metric?
 \end{problem}




\end{problemblock}





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\end{document}
