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# Ricci-flat manifold

In mathematics, Ricci-flat manifolds[1][2] 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.

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.

Further reading

Matthew Randall, Almost Projectively Ricci-flat Manifolds, Dept. of Mathematics, University of Auckland, 2010.

See also

General: Flat manifold, Weyl tensor

Other: Quaternion-Kähler manifold

References

Dictionary of Distances By Michel-Marie Deza, Elena Deza. Elsevier, Nov 16, 2006. Pg 87

Arthur E. Fischer and Joseph A. Wolf, The structure of compact Ricci-flat Riemannian manifolds. J. Differential Geom. Volume 10, Number 2 (1975), 277-288.

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