An Einstein manifold is a Riemannian manifold (M,g) whose Ricci tensor is proportional to the metric tensor: In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

Taking a trace shows that k is equal to s/n, where n is the dimension of M and s is the scalar curvature. Einstein manifolds with k = 0 are also called Ricci-flat manifolds. In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci tensor vanishes. ...

In general relativity, these manifolds (in the pseudo-Riemannian case) can be thought of as vacuum solutions of Einstein's equations with a cosmological constant proportional to k. Two-dimensional visualization of space-time distortion. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... For other uses, see vacuum cleaner and Vacuum (musical group). ... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einsteins theory of general relativity. ...

Examples

• The n-sphere, Sⁿ, with the round metric is Einstein with k = n − 1.
• Hyperbolic space with the canonical metric is Einstein with negative k.
• Complex projective space, CPⁿ, with the Fubini-Study metric.

A sphere is a perfectly symmetrical geometrical object. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ...

Literature

• Arthur L. Besse, "Einstein Manifolds", Springer-Verlag.