A smooth closed manifold $M$ is called almost Ricci-flat if $$\inf_g||\textrm{ Ric}_g||_\infty\cdot \textrm{ diam}_g(M)^2=0$$ where $\textrm{ Ric}_g$ and $\textrm{ diam}_g$ denote the Ricci tensor and the diameter of $g$ respectively and $g$ runs over all Riemannian metrics on $M$. By using Kummer-type method we construct a smooth closed almost Ricci-flat nonspin 5-manifold $M$ which is simply connected. It's minimal volume vanishes, namely it collapses with sectional curvature bounded.
Categories: math.DG