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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. ==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 a vacuum (a region of spacetime with no matter) ''T''''ab'' = 0, and one can rewrite Einstein's equation in the form (assuming ''n'' > 2): : Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with ''k'' proportional to the cosmological constant. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「einstein manifold」の詳細全文を読む スポンサード リンク
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