A Riemannian manifold $(M,g)$ is Einstein if its Ricci curvature tensor satisfies
$$ \operatorname{Ric} = \lambda g $$
for some real constant $\lambda$.
In Besse's "Einstein Manifolds" I read that this condition is the nicest extension of the concept of constant curvature in dimensions higher than two, and so Einstein Manifolds generalize constant curvature smooth surfaces without being too rigid.
I understand this motivation but it seems a bit weak to me to justify all of the efforts in finding existing results of these structures: do Einstein Manifolds have some special properties which make them so interesting?
Best Answer
There are two main reasons why Einstein metrics form a hot topic:
These are the two main motivations to look for Einstein metrics. So far, so good, we have motivations, but they do not say whether these metrics have nice properties. If you read (and understand) in details Besse's book, you will realise that indeed, they do. Here is a non-exhaustive list of such nice properties. Anyone willing to extend the list is kindly encouraged to do so.