# 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.

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

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

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

where *n* is the dimension of *M*.

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

written in geometrized units with *G* = *c* = 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):

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, , with the round metric is Einstein with . - Hyperbolic space with the canonical metric is Einstein with negative .

- Complex projective space, , with the Fubini–Study metric.
- Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant . 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.

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 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

## References

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