Ricci-flat manifold

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In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. In physics, they represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish.

Ricci-flat manifolds, in general, have restricted holonomy groups. Important cases include Calabi-Yau manifolds and hyperkähler manifolds.