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4. Riemannian Geometry


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      Problem 4.1.

      [W. Tuschmann] What are closed manifolds with almost nonnegative curvature operator? \mbox{Rm} \cdot \mbox{diam}^2 \geq -\epsilon.


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          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?


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              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?


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                  Problem 4.4.

                  [J. Lott] For the above problem, what about complex sectional curvature?


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                      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


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                          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^{-k-2}?

                              Cite this as: AimPL: Geometric flows and Riemannian geometry, available at http://aimpl.org/flowriemannian.