# Einstein metrics on spheres

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.

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 $$n \geq 4$$.