Einstein manifold

In local coordinates the condition that (M, g) be an Einstein manifold is simply

${\displaystyle R_{ab}=kg_{ab}.}$

Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by

${\displaystyle R=nk,}$

where n is the dimension of M.

In general relativity, Einstein's equation with a cosmological constant Λ is

${\displaystyle R_{ab}-{\frac {1}{2}}g_{ab}R+g_{ab}\Lambda =\kappa T_{ab},}$

where κ is the Einstein gravitational constant.[1] The stress–energy tensor T_ab gives the matter and energy content of the underlying spacetime. In vacuum (a region of spacetime devoid of matter) T_ab = 0, and Einstein's equation can be rewritten in the form (assuming that n > 2):

${\displaystyle R_{ab}={\frac {2\Lambda }{n-2}}\,g_{ab}.}$

Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.

Simple examples of Einstein manifolds include:

• Any manifold with constant sectional curvature is an Einstein manifold—in particular:
  • Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.
  • The n-sphere, ${\displaystyle S^{n}}$, with the round metric is Einstein with ${\displaystyle k=n-1}$.
  • Hyperbolic space with the canonical metric is Einstein with ${\displaystyle k=-(n-1)}$.
• Complex projective space, ${\displaystyle \mathbf {CP} ^{n}}$, with the Fubini–Study metric, have ${\displaystyle k=n+1.}$
• Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant ${\displaystyle k=0}$. Such metrics are not unique, but rather come in families; there is a Calabi–Yau metric in every Kähler class, and the metric also depends on the choice of complex structure. For example, there is a 60-parameter family of such metrics on K3, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.
• Kähler–Einstein metrics exist on a variety of compact complex manifolds due to the existence results of Shing-Tung Yau, and the later study of K-stability especially in the case of Fano manifolds.
• An Einstein–Weyl geometry is a generalization of an Einstein manifold for a Weyl connection of a conformal class, rather than the Levi-Civita connection of a metric.

A necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe inequality.

Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as gravitational instantons in quantum theories of gravity. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose Weyl tensor is self-dual, and it is usually assumed that the metric is asymptotic to the standard metric of Euclidean 4-space (and are therefore complete but non-compact). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) hyperkähler manifolds in the Ricci-flat case, and quaternion Kähler manifolds otherwise.

Higher-dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as string theory, M-theory and supergravity. Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for nonlinear σ-models with supersymmetry.

Compact Einstein manifolds have been much studied in differential geometry, and many examples are known, although constructing them is often challenging. Compact Ricci-flat manifolds are particularly difficult to find: in the monograph on the subject by the pseudonymous author Arthur Besse, readers are offered a meal in a starred restaurant in exchange for a new example.[2]

• Einstein–Hermitian vector bundle

• Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.