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. [1], [2].)

Note that [1] is an updated version of [2], while [2] 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).