Is it always possible that to perturb the metric to a metric of positive Ricci curvature at some points?

Question

This is a follow-up of my previous post.

Question1: Is it always possible that to perturb the metric to a metric of positive Ricci curvature at some points (or neighborhood) and without change in other points?

Maybe the answer is obviously no. because we can do this process over and over to obtain a metric of positive Ricci curvature which seems to be impossible. Am I right?

Question2: Is there a closed manifold $M$ of positive Ricci curvature so that $M\text{#} M$ (or arbitrary number connected sum) admit no metric of positive Ricci curvature?

Answer 1

For your first question, you can't do this for all points simultaneously otherwise you would end up with a metric with positive Ricci curvature, but these do not always exist. For example, it follows from Myers' Theorem that if a compact manifold $M$ admits a complete metric of positive Ricci curvature, then $\pi_1(M)$ is finite. However, the proposition in Jason DeVito's answer to your linked question shows you can always perturb the metric in a neighbourhood of a point $p$ so that it has positive sectional curvature (and hence positive Ricci curvature) at $p$.

As for your second question, let $M$ be a manifold with positive Ricci curvature which is not simply connected. If $n \geq 3$, then $\pi_1(M\# M) \cong \pi_1(M)\ast\pi_1(M)$ which is infinite, so by Myers' Theorem, the manifold $M\# M$ does not admit a metric of positive Ricci curvature, however it does admit a metric of positive scalar curvature by a theorem of Gromov and Lawson.

Answer 2

Michael's answer is fine, but I wanted to add some results in the opposite direction.

First, the following is a result of Aubin (see here)

Suppose $M$ is a complete Riemannian manifold for which $Ric\geq 0$ and for which there is some point for which $Ric_p > 0$. Then $M$ admits a metric of positive Ricci curvaure at each point.

So, as Michael points out, in general you cannot deform a metric to one with positive Ricci at some point without also messing things up elsewhere. However, if $M$ has non-negative Ricci curvature which is positive at some point, then you can. (By the way, the analogous statement for sectional curvature is FALSE in general, but is open for simply connected closed manifolds.)

Second, there are closed manifolds with positive Ricci curvature for which $M\sharp M\sharp...$ admits a metric of positive Ricci curvature for any number of connect summands (including infinitely many). One can, for example, take $M = S^2\times S^2$. This is a result of Sha and Yang: see here.