Ricciflat

Ricciflat, commonly written Ricci-flat, designates a Riemannian or pseudo-Riemannian manifold whose Ricci curvature tensor vanishes everywhere. Concretely, if g is the metric, the Ricci tensor Ric(g) equals zero. The term is widely used in differential geometry and mathematical physics and is often associated with structures of special holonomy and Calabi–Yau manifolds.

In mathematics, Ricci-flat metrics are solutions to the vacuum Einstein field equations and are critical points of the Einstein–Hilbert action with zero cosmological constant. For compact manifolds, the existence of Ricci-flat metrics is linked to topological and complex-geometric data, most famously the first Chern class in the Kähler setting. A vanishing first Chern class enables the existence of Ricci-flat Kähler metrics, as established by Yau’s solution to the Calabi conjecture. This underpins the study of Calabi–Yau manifolds, which are compact Kähler manifolds with Ricci-flat metrics.

Examples include the flat Euclidean space R^n with its standard metric, complex tori with flat metrics, and K3 surfaces, which admit Ricci-flat metrics in their Kähler classes. Noncompact examples important in geometric analysis and physics include gravitational instantons such as the Eguchi–Hanson and Taub–NUT spaces, all of which are complete Ricci-flat 4-manifolds. In higher dimensions, Calabi–Yau manifolds and certain manifolds with special holonomy (such as G2 and Spin(7) manifolds) are also Ricci-flat.

In physics, Ricci-flat spacetimes (with Lorentzian signature) model vacuum solutions to Einstein’s equations and are used to describe regions of spacetime devoid of matter or energy. They appear in studies of gravitational radiation, string theory compactifications, and global geometric analysis.