# Metric Deformations from Non-Negative to Positive Curvature

Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) > 0$?

I understand that this question is ludicrous, at best (for instance, an affirmative would prove that $S^n\times S^m$ always carries a metric of positive sectional curvature). I'm assuming that there is a counterexample, but I can't seem to figure it out. Most examples of manifolds with positive Ricci curvature that I know can't admit metric of non-negative sectional curvature for topological reasons, for instance the Sha--Yang construction of metrics with positive Ricci curvature on connected sums of $S^n \times S^m$.

I was wondering about this in view of the following two results: The first, due to Aubin and Ehrlich, asserts that a metric $g$ with $\mathrm{Ricci}(M,g) \geq 0$ everywhere and $\mathrm{Ricci}(M,g) > 0$ somewhere can be deformed into a metric $g_1$ with $\mathrm{Ricci}(M,g_1) > 0$ everywhere. The second, due to Gao--Yau, asserts that if $\mathrm{Ricci}(M,g) > 0$ everywhere, then $g$ can be deformed into a metric $g_1$ with $\mathrm{Ricci}(M,g) > 0$ everywhere and $\mathrm{sec}(M,g) > 0$ somewhere.

The Gao--Yau result is a local solution to the intial question.

As a related question: Suppose the Ricci curvature of $(M,g)$ is pinched. Does this change the answer to the question if the pinching constant is particulary small or would it seem that pinching plays no role in this? Is there any theory of pinched Ricci curvature?

T. Aubin, Metriques riemannienes et courbure. Diff. Geom., 4:383--424, 1970

P. Ehrlich, Metric deformations of curvature. Geom. Ded., 5:1--23, 1976

L. Gao, S.-T. Yau, The existence of negatively Ricci curved metrics on three manifolds. Invent. Math., 85:637--652, 1986

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In 3 dimensions, you can do this by a theorem of Hamilton. ams.org/mathscinet-getitem?mr=664497 – Ian Agol May 27 '12 at 16:07

As Benoît Kloeckner points out this is false for non simply connected manifolds with $RP^2\times RP^2$ being a counterexample (by Synge's theorem). For simply connected manifolds this is a well known open problem.

BTW, Sha-Yang examples are only known not to admit nonnegative sectional curvature for connected sums of things like $S^n\times S^m$ for very large number of summands (by Gromov's betti number estimate). For, say, connected sum of 3 copies of $S^n\times S^m$ with $n,m>1$ nothing is known.

Also, there are plenty of easier examples of manifolds of positive Ricci curvature other than those of Sha and Yang. For example homogeneous spaces and more generally biquotients of compact Lie groups with finite fundamental groups. All of them have positive Ricci curvature and nonnegative sectional curvature but almost none are known to admit positive sectional curvature.

Moreover, there are lots of such examples with quasi-positive curvature ( where sectional curvature is positive on a dense open set of points in $M$) and the question is still open for such manifolds. See for example this paper by Wilking "Manifolds with positive sectional curvature almost everywhere."

Among many other examples Wilking constructs such metrics on $S^2\times S^3$. This shows that even in the simply connected case either the Hopf conjecture (that a product of positively curved manifolds can not be positively curved) is false or the deformation conjecture for quasi-positively curvaed manifolds is false.

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I think the answer is no, because if I remeber well $\mathbb{R}\mathrm{P}^2 \times \mathbb{R}\mathrm{P}^2$ does not admit a positively curved metric. My reference for this is Gallot-Hulin-Lafontaine, but I do not have the book at hand right now.

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Yes, this is a standard corollary of Meyers' theorem. – Misha May 27 '12 at 15:58