The sectional curvature is one of the main invariants of Riemannian manifolds. But despite its importance, there is actually little known about questions like: What is the distinction (for closed manifolds) between admitting a metric of positive sectional curvature vs admitting a metric of nonnegative sectional curvature?
Another problem is also to distinguish positive sectional curvature from positive Ricci curvature. If we only count those results which hit into the above mentioned distinction between positive and nonnegative sectional curvature and which at the same time do not also apply to positive Ricci curvature, we arrive at exactly two theorems:
- (Synge 1936) Even-dimensional manifolds of positive curvature are simply connected if and only if they are orientable; especially, the only nontrivial fundamental group can be \(\small \mathbb{Z}/2\mathbb{Z}\). In the odd-dimensional case positively curved manifolds must be orientable.
- (Schoen 2023) In \(\small 4k\)-dimensional manifolds of positive curvature the only possible non-trivial element of \(\pi_1\) must act trivially on \(\pi_2\). In odd-dimensional manifolds of positive curvature no element of \(\pi_1\) can reverse the orientation of a nontrivial element of \(\pi_2\).
Synge’s result rules out positive curvature on the manifold \(\small RP^n \times RP^n\), whereas Schoen’s recent result rules out positive curvature on \(\small RP^2 \times S^{4k-2}\). Note that these products admit metrics of nonnegative sectional curvature.
What about the case of, say, \(\small S^2 \times S^2\)? Well, this is still open and known as Hopf’s conjecture. This conjecture can be actually extended to the following version: There are no positively curved metrics on the product of two closed manifold.