7 problems

A closed manifold $M$ is almost flat if, for every $\epsilon>0$, it admits a Riemannian metric $g_\epsilon$ satisfying … where $K_{g_\epsilon}$ denotes sectional curvature. Farrell…

An almost flat manifold is a closed manifold admitting metrics of arbitrarily small sectional curvature and uniformly bounded diameter; a flat manifold is a closed manifold with a…

An $\mathrm{RCD}(-(N-1),N)$ space $(X,d,\mathcal{H}^N)$ is considered in the setting of local bounded covering geometry, where $\mathrm{RCD}$ denotes the Riemannian curvature-dimen…

Let $(M,g)$ be a closed $m$-dimensional Riemannian manifold whose Ricci curvature is almost non-negative, and suppose that its first Betti number equals its dimension, $b_1(M)=m$.…

Let $Q$ be a closed almost flat manifold. A complete finite-volume noncompact manifold with pinched negative curvature has cusps diffeomorphic to $Q\times [b,\infty)$; $Q$ bounds g…

A closed almost flat manifold is a closed manifold whose Riemannian metric has almost zero curvature in the standard sense. To bound smoothly means to be the boundary of a compact…

A closed almost flat manifold is a closed manifold admitting a Riemannian metric whose sectional curvatures and diameter can be made simultaneously arbitrarily small. A closed almo…