# Einstein space

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## Einstein space

[¦īn¦stīn ′spās]
(mathematics)
A Riemannian space in which the contracted curvature tensor is proportional to the metric tensor.
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An [eta]-Einstein manifold with [beta] vanishing and [alpha] a constant is obviously an Einstein manifold.
M] is a steady Einstein manifold with Killing V and [bar.
n],g), n = dimM [greater than or equal to] 2, is said to be an Einstein manifold if the Ricci tensor S is given by
Then either [alpha] = [+ or -] [beta] or M is an Einstein manifold in which case the curvature is given by r = n(n - 1)[epsilon]([epsilon][[alpha].
Next we define a semi-Einstein manifold which is the generalization of Einstein manifold.
2) N is an Einstein manifold and the positive function [phi] = [f.
2] - [sigma]) or, S = - 2ng which implies that the LP-Sasakian manifold is an Einstein manifold.
In particular, M' is an Einstein manifold with zero scalar curvature and by Sekigawa theorem ([30]) J' is parallel.
An n-dimensional Riemannian manifold (M,g), n > 2, is said to be an Einstein manifold if its Ricci tensor S satisfies the condition S = r/n g, where r denotes the scalar curvature of M.
2m+1] (c); c > -3 with positive sectional curvature is Einstein manifold and satisfies:
3]) = 0, then the manifold reduces to an Einstein manifold.
This shows that Riemannian manifold is an Einstein manifold.

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