I've got a couple of quick questions that came up after reading a peculiar statement in some article. The sentence says something like "… is the $N$-dimensional sphere with constant Ricci curvature equal to $K$…", and the questions are something like:
For $(\mathbb{S}^n,g)$ the sphere with its standard differential structure and $some$ Riemannian metric on it,
1.a. Does $g$ being an Einstein metric implies that it is actually the round metric (up to some normalization constant)?
1.b. Does the answer change if we change to an alternative differential structure (when possible)?
I guess this shouldn't be true, so in this case
2. Is there an intuitive way to understand how one could construct a metric which is Einstein but not of constant curvature?
Anyways, I thank you all in advance for sharing your knowledge.
Best Answer
No, there are Einstein metrics on spheres which are not rescalings of the round metric. See the introduction of Einstein metrics on spheres by Boyer, Galicki, & Kollár for some constructions. However, as far as I am aware, there are no known examples of Einstein metrics with non-positive Einstein constant. In particular, it is an open question as to whether $S^n$ admits a Ricci-flat metric for $n \geq 4$.
If we consider exotic spheres, they do not admit a 'round metric' or any metric of constant curvature, so I'm not sure what is meant by this. However, there are examples of Einstein metrics on exotic spheres, see Einstein Metrics on Exotic Spheres in Dimensions 7, 11, and 15 by Boyer, Galicki, Kollár, & Thomas for example. Note however that there are some exotic spheres which, if they admit Einstein metrics, must have negative Einstein constant.
Finding Einstein metrics which are not constant curvature is, in general, a hard thing to do and an area of active research.