Knowlpedia

Definition

Einstein manifold

A Riemannian manifold whose Ricci tensor is a constant multiple of the metric.

Definition

An Einstein manifold is a (Mn,g)(M^n,g) for which there is a real constant λ\lambda, on each , such that

Ric⁡g=λg,\operatorname{Ric}_g=\lambda g,

where Ric⁡g\operatorname{Ric}_g is the . The metric gg is then called an Einstein metric, and λ\lambda its Einstein constant. Equivalently, all tangent directions have the same Ricci curvature after normalization by squared length. The case λ=0\lambda=0 is called Ricci-flat. This is a Riemannian definition; pseudo-Riemannian Einstein metrics use the same tensor equation but allow indefinite gg.

Equivalent characterizations and scaling

Taking the metric trace gives constant scalar curvature

Scal⁡g=nλ.\operatorname{Scal}_g=n\lambda.

Conversely, constant scalar curvature alone does not imply the Einstein equation when n≥3n\geq3, because it controls only the trace of the Ricci tensor. If gg is Einstein and c>0c>0 is constant, then cgcg is Einstein with constant λ/c\lambda/c. The Levi–Civita connection and Ricci tensor as a covariant 22-tensor remain unchanged under this rescaling.

Examples and non-examples

An nn-manifold of constant sectional curvature kk is Einstein with λ=(n−1)k\lambda=(n-1)k. Hence the round sphere is Einstein, while and flat tori are Ricci-flat. In dimension two, Ric⁡=Kg\operatorname{Ric}=Kg, so an Einstein surface is exactly a surface of constant Gaussian curvature.

A product of two Einstein manifolds with the product metric is Einstein when their Einstein constants agree. If the constants differ, the product has different Ricci eigenvalues along the two factors and is a decisive non-example.

Structure and scope

The contracted implies that a relation Ric⁡=fg\operatorname{Ric}=f g already forces ff to be constant on connected manifolds of dimension at least three. In dimension two this conclusion fails without imposing constancy, which is why the constant is included in the definition. Einstein metrics are not generally metrics of constant sectional curvature: the Ricci tensor is only a trace of the full curvature tensor.

References
  1. Arthur L. Besse, Einstein Manifolds, Springer, 1987. Publisher record. Relevant: Chapter 1, “Basic Material,” pp. 20–65, for curvature conventions and the Einstein condition.
  2. John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. Publisher record. Relevant: Chapter 7, “Curvature,” for Ricci and scalar curvature and the contracted Bianchi identity.