Ricci-flat manifold

In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish. In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish. Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their 'shape' may vary from the shape of the standard ball in Euclidean space. For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. This is due to Weyl curvature. Ricci-flat manifolds often have restricted holonomy groups. Important cases include Calabi–Yau manifolds and hyperkähler manifolds. In physics, Ricci-flat manifolds represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant.