An interesting special case of Kaehler-Einstein manifolds are the Calabi-Yau manifolds, which like all other Kaehler-Einstein manifolds (and in general, Kaehler manifolds), have \(2k\) real dimensions.
Calabi-Yau manifolds are of interest in string theory, where extra dimensions in 10-dimensional superstring theories are compatified on Calabi-Yau manifolds with 3 complex dimensions (= 6 real dimensions) and holonomy group \(SU(3)\).
(A similiar anlog exists for the 11-dimensional M-theory, which can be compactified on a G2-manifold. A G2 manifold is an Einistein manifold, because it's Ricci-flat (and all Ricci-flat manifolds are Einstein manifolds), but not a Kaehler manifold, because it's 7-dimensional (while Kaehler manifolds have an even number of real dimensions). However, one may say that it is "analogous" to Calabi-Yau manifolds, see e.g. , .)
Note that  is an updated version of , while  has some more interesting answers, so I reference it too.
- Spiro Karigiannis, What is a G2 manifold?. AMS Notices - 58 (4), 2011.
- Spiro Karigiannis, Answer to "\(G_2\) and geometry" (MathOverflow, 2010).
- Kaehler manifold (nLab wiki)
- Robert J Berman, Kahler-Einstein metrics emerging from free fermions and statistical mechanics. ArXiV (2010).