In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian 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 the dimension, as well as the signature, of the metric can be arbitrary, unlike 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.
Read more about Einstein Manifold: The Einstein Condition and Einstein's Equation, Examples, Applications
Famous quotes containing the words einstein and/or manifold:
“If my theory of relativity is proven correct, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.”
—Albert Einstein (1879–1955)
“Odysseus saw the sirens; they were charming,
Blonde, with snub breasts and little neat posteriors,”
—John Streeter Manifold (b. 1915)