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The Kahler-Einstein metric is an example of a canonical metric on a Kahler manifold. We say that a metric $\omega$ is Kahler-Einstein if $Ric(\omega)=\lambda\omega$, where $\lambda\in\{-1,0,+1\}$.

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In what sense exactly are the Einstein metrics distinguished?

OK I think I understand the question now. You want to look at Lovelock gravities. These are the most general theories (not bothering with a list of qualifiers here, see wikipedia) that produce 2nd ord …
AlexArvanitakis's user avatar