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Does anyone know if there exists a scalar-flat metric on the $n$-sphere, $n>4$, such that it is not Ricci flat. This should be easy, because it seems doubtful that spheres can carry a Ricci flat metric, but I'm having trouble proving existence.

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See also the MO version of this question. –  Joseph O'Rourke Jun 24 '11 at 5:42

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up vote 3 down vote accepted

This doesn't quite answer your question, but perhaps gives a few directions worth pursuing.

First, there is the Kazdan-Warner theorem which states the following:

Consider the class of all compact $n$-manifolds with $n\geq 3$ and divide them into 3 subclasses:

  1. Those that admit a metric with scalar curvature $\geq 0$ and positive at some point.

  2. Those that admit a metric with scalar curvature identically 0, but which are not in class 1.

  3. Everything else.

Then, if $M$ is of type 1, any smooth function $f:M\rightarrow \mathbb{R}$ is the scalar curvature of some metric. If $M$ is of type 2, a function $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric iff it is identically 0 or negative somewhere. If $M$ is of type 3, a function $f:M\rightarrow \mathbb{R}$ is the scalar curvature of some metric iff $f$ is negative somewhere.

In particular, since spheres are clearly in class 1, it must be the case that they all have scalar flat metrics. I am not sure how constructive the proof of the Kazdan-Warner theorem is, so I'm not sure whether or not one can verifywhether these scalar flat metrics are not Ricci flat or not.

Second, here's an actual attempt at an answer, though it only works on odd dimensional spheres and has a gap.

The odd dimensional spheres can all be expressed as homogeneous spaces $SU(n+1)/SU(n)$. The left action of $SU(n)$ on the tangent space at the identity coset is not irreducible, and in fact factors into 2 irreducible summands. The upshot is that there is one parameter family of homogeneous metrics on an odd dimensional sphere, all normalized to, say, have volume 1. Intuitively, one keeps the metric the same in directions orthogonal to the Hopf fibration and scales the metric in the direction of the Hopf circles. Call this scaling factor $\lambda$. Since the metrics are homogeneous, they have constant scalar curvature - it's only an issue of figuring out what the constant is.

Now, I know for a fact that for $\lambda < 1$, the metrics one obtains all have strictly positive sectional curvature, and in fact as $\lambda \rightarrow 0$, the spheres converge (in the Gromov-Hausdorff sense) to complex projective space with Fubini-Study metric.

I also know for a fact that for $\lambda$ large enough (4/3 if I remember correctly, or possible even anything bigger than 1), one begins to get negative sectional curvature. Here comes the gap: I'd guess that as $\lambda\rightarrow \infty$ these sectional curvatures tend to $-\infty$ and hence, for some choice of $\lambda$, we get a scalar flat metric (which one could verify is not Ricci flat, probably without too much effort).

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I should have said that spheres with the metrics I'm considering in the second part go by the name of "Berger spheres", just to give you a key word to look up. –  Jason DeVito Jun 24 '11 at 5:07

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