4. Riemannian Geometry

Problem 4.1.
[W. Tuschmann] What are closed manifolds with almost nonnegative curvature operator? $$ \mbox{Rm} \cdot \mbox{diam}^2 \geq \epsilon.$$ 
Problem 4.2.
[W. Ziller] Let $M^3$ be a closed manifold. Outside flat points, it splits locally as $\Sigma \times \mathbb{R}$. Does there exist a collection of totally geodesic flat surfaces $S_1, \cdots, S_k, \cdots $ such that $$M\setminus (S_1 \cup \cdots \cup S_k \cup \cdots ) \mbox{ is } (\Sigma \times \mathbb{R} )/\mbox{group}?$$ where $\Sigma$ is a surface (with possibly finitely or infinitely many components) with boundary a union of closed geodesics. Moreover, if $M$ has nonnegative curvature, then is it diffeomorphic to a lens space? 
Problem 4.3.
[J. Lott] Let $M_\epsilon =M\times_{S^1} (\epsilon S^1)$. As $\epsilon \to 0$, is Riem(Curvature Operator) bounded below? 
Problem 4.5.
[J. Lott] What are ANCO(almost nonnegative curvature operator, i. e., there exists metric with $\mbox{Riem} \cdot \mbox{diam}^2 \geq \epsilon$) manifolds? Constructions: \\ 1. compact symmetric spaces\\ 2.almost flat manifolds\\ 3. products\\ 4. finite covers, finite quotients\\ 5. total space of principal $S^1$bundle over an ANCO manifold 
Problem 4.6.
[J. Lott] Can we smooth metrics with $\mbox{Riem} \geq \mbox{Id}$ on volume scale, i.e., want nearby metrics with $\nabla^k Rm \leq C_k C_p^{k2}$?
Cite this as: AimPL: Geometric flows and Riemannian geometry, available at http://aimpl.org/flowriemannian.