Einstein manifold

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity. Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons.

If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that

${\displaystyle \mathrm {Ric} =kg}$

for some constant k, where Ric denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds.

The Einstein condition and Einstein's equation[]

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.

Examples[]

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.

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

Applications[]

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.

See also[]

• Einstein–Hermitian vector bundle

Notes and references[]

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