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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?

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  • $\begingroup$ The title is misleading: the question seems more to be "why are we interested in Einstein metrics" rather than "why do we call these metrics Einstein", right? $\endgroup$
    – Didier
    Jul 21, 2023 at 12:06
  • $\begingroup$ Thank you for the suggestion, I’ve updated the question title. $\endgroup$ Jul 21, 2023 at 13:23

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There are two main reasons why Einstein metrics form a hot topic:

  • The first is that they arise as critical points of the Einstein-Hilbert functional $g \mapsto \int \mathrm{scal}^g \mathrm{d}v^g$. In physical terms, the Einstein equation $\mathrm{Ric}^g = \lambda g$ is the Euler-Lagrange equation of the Einstein-Hilbert action, which is closely related to the total energy of the universe in General Relativity. I don't think that I need to give more details on why General Relativity is a relevant topic to study.
  • The second is that the study of differentiable manifolds is hard. Endowing a smooth manifold with a metric allows one to study it through the scope of geometry. To that end, it is convenient to try to find a best metric, that will reflect the differential and topological properties of the underlying manifold. There is no concrete definition of what a best metric is, but intuition tells us that on the sphere, this should be the round metric, and on the plane, it should be the Euclidean metric. Indeed, these metrics have maximal symmetries, and they characterise those manifolds. It thus appears natural to look for metrics with as many symmetries as possible. It is not hard to show that a metric with a huge amount of symmetries must have constant sectional curvature. Up to universal covering, these metrics have been classified. The next step is to ask for a less rigid condition than having constant sectional curvature: for instance, one can look for metrics with constant Ricci curvature. They are exactly the Einstein metrics. Once these metrics will be classified (I doubt this will ever happen), the next relaxed condition is to classify metrics with constant scalar curvature (which is even harder).

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.

  • Perelman solved Thurston's Geometrization conjecture that completely classified compact 3-manifolds (and as a byproduct, this solved the famous Poincaré conjecture). The proof relies on the so called Ricci flow, which (very, very) coarsely allows one to obtain Einstein metrics as limits of solutions to the analogue of the heat equation in the space of Riemannian metrics. This idea is due to Hamilton.
  • Einstein metrics enjoy elliptic properties (for instance, they have maximal regularity in harmonic coordinates). This considerably simplify the analysis of some geometric PDE's. See this for instance.
  • In some specific cases (some instances of Kähler-Einstein metrics or of conformally compact metrics), they enjoy uniqueness properties, meaning that they are very particular, and hence a good candidate to reflect properties of the underlying manifold. We often talk about rigidity properties.
  • I cannot find the reference at the moment, but I vaguely recall that a result of Olivier Biquard states that on a (some?) complex manifold(s) (admitting Kähler metrics), the sets of (all?) metrics (which is an infinite dimensional analytic manifold) locally splits into a direct decomposition "$\{$ Riemannian metrics $\}$ = $\{$ Kähler metrics $\}$ $\oplus \{$ Einstein metrics $\}$", the intersection being the Kähler-Einstein metric. Einstein metrics appear as a distinguished subspace which helps understanding the whole set of metrics.
  • There are many interactions between geometry and topology. One of these interactions occurs when a lower bound on the Ricci curvature is assumed (see Bishop-Gromov inequality or Myer's Theorem for instance). These results naturally apply to Einstein metrics, since the Ricci curvature tensor is constant and hence bounded below.
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    $\begingroup$ Nice answer. But I had never seen the statement that equality holds for extremal domains in Einstein manifolds. In the comparison argument there’s a positive term that gets tossed away, and I have the impression (I haven’t tried to write anything down) that it’s zero only if the metric has constant sectional curvature. $\endgroup$
    – Deane
    Jul 21, 2023 at 18:00
  • $\begingroup$ @Deane I was referring to equality in the assumptions, not in the result! You are right that the equality case in Bishop-Gromov is exactly for constant curvature metrics (counter example for Einstein: the Fubini-Study metric on the complex projective space). I edited with a clearer formualtion $\endgroup$
    – Didier
    Jul 21, 2023 at 18:02

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