Ricci-flat manifold

In mathematics, Ricci-flat manifolds[1][2] are Riemannian manifolds whose Ricci curvature tensor 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.

Applications

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.

References

Deza, Michel-Marie; Deza, Elena (2006). "7 Riemannian and Hermitian Metrics § Ricci-flat metric". Dictionary of Distances. Elsevier. p. 87. ISBN 978-0-08-046554-8.
Fischer, Arthur E.; Wolf, Joseph A. (1975). "The structure of compact Ricci-flat Riemannian manifolds" (PDF). J. Differential Geom. 10 (2): 277–288. doi:10.4310/jdg/1214432794.

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[1] Deza, Michel-Marie; Deza, Elena (2006). "7 Riemannian and Hermitian Metrics § Ricci-flat metric". Dictionary of Distances. Elsevier. p. 87. ISBN 978-0-08-046554-8.

[2] Fischer, Arthur E.; Wolf, Joseph A. (1975). "The structure of compact Ricci-flat Riemannian manifolds" (PDF). J. Differential Geom. 10 (2): 277–288. doi:10.4310/jdg/1214432794.

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Ricci-flat manifold

Applications

Further reading

See also

References