# A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$

It is well known that

In dimension three a metric has positive sectional curvature if and only if $${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$$. where $$r$$ denotes the scalar curvature.

Is there a higher dimensional analogues of this theorem?

• Do you mean $Ric <\frac r2 g$ in dimension $3$? – Yu Ding May 15 at 17:12
• Yes, its a typo. thanks for your precision. – C.F.G May 15 at 18:35
• I don't think there is a high dimension version. If there is, it would have said that 4d Einstein manifolds (with positive scalar curvature) is of positive sectional curvature, and this is probably not true (I don't immediately have an example, maybe the Kahler case ?). – Yu Ding May 17 at 21:19