Question

This questions asks for your intuition and insight as I'm surprised by how little is known about the difference between nonnegative and positive curvature. I don't want to be completely vague, so I could ask: What are the difficulties and currently blocked paths to solving the Hopf Conjecture? (Does $S^2\times S^2$ support a metric of positive curvature?). But in general, I would like to know what others might know on why it's difficult to determine if a given closed simply-connected space of nonnegative curvature can also admit positive curvature. As far as I know, there are no obstructions, how come? The amount of examples of nonnegative curvature compared to that of examples of positive curvature seem to suggest there should be something distinguishing the two.

Answer 1

Yau asked in 1982 if there is any compact simply connected manifold with nonnegative curvature for which one can prove that it does not admit a metric of positive curvature. This question opens his list of unsolved problems in geometry (see "Seminar on Differential Geometry", p. 670.)

Let me quote from "A Panoramic View of Riemannian Geometry" by Berger (Springer 2003, p. 579):

It is not surprising that many people tried to address Yau’s remark, starting with the Hopf conjecture on $S^2 × S^2$, by trying to deform such a metric with $K ≥ 0$ into one with $K >0$. This means considering some one parameter family $g(t)$ of metrics and computing the various derivatives at $t = 0$ of the sectional curvature. Technically it is very easy to compute such a derivative for a given tangent plane, but what is difficult is to find a variation for which all the derivatives would be positive. Today this approach still does not work.

A major difficulty is that it is not clear how to find the critical set of the sectional curvature (as a function on the set of tangent planes).

The earlier short survey by Bourguignon contains a discussion of the reasons why some seemingly natural approaches fail.