Lower and Upper Bound for Ricci Curvature

Question

As mentioned in first chapter of John M. Lee: Riemannian Geometry, one of our goal in differential geometry is connecting geometry and topology. For this reason it is natural to compare curvature quantities with its correspondence in model spaces; e.g. $Ric \geq k g, Ric\geq0$. But I never seen any result about $Ric\leq kg$ or $Ric \geq -k g$ for some positive constance $k$.

Does this conditions deduced from earlier one? or topology of this manifolds are so complicated to handle?

Answer 1

There can be no topological consequences of estimates of the form $Ric \le k g$ or $Ric \ge -k g$ (at least in dimensions greater that $2$), because this paper by Joachim Lohkamp shows that every smooth manifold of dimension at least $3$ admits a complete metric whose Ricci curvature is bounded between two negative constants.

Addition:

Your comments suggest that you may have some misunderstanding about what these inequalities mean. Here are some remarks that might help to clarify what's going on.

• First, an equality like $Ric\le k$ or $Ric \ge k$ is really shorthand for $Ric\le kg$ or $Ric \ge kg$. Equivalent statements are that $Ric(v,v)\le k$ (resp. $\ge k$) for every unit vector $v$; or $Ric(v,v) \le k|v|^2$ (resp. $\ge k|v|^2$) for every tangent vector $v$. For example, if an $n$-dimensional Riemannian manifold has sectional curvatures bounded above by a constant $c$, then its Ricci curvature is bounded above by $(n-1)c$.
• Second, note that the metrics whose existence is guaranteed by Lohkamp satisfy $Ric \le -k_2g$ for some positive constant $k_2$. By transitivity, such a metric therefore also satisfies $Ric \le k g$ for every positive constant $k$, so there is no topological consequence that can be deduced from any upper bound on Ricci curvature.
• Next, it certainly does not follow from Lohkamp's theorem (and it's not true) that no manifold admits a metric whose Ricci curvature is positive somewhere and negative somewhere else, as you seemed to suggest in a comment. In fact, every manifold admits such a metric -- you can just use a bump function to "paste in" a positively curved metric in one open set, and paste in a negatively curved metric in a different open set.
• Finally you asked what topological consequences can be deduced from estimates of the form $K\le k$ or $K\ge -k$ (where $K$ represents sectional curvature). The answer is none -- this paper by R. E. Greene showed that every manifold admits a complete metric satisfying both of these inequalities.

Hope this helps.