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What is a Kähler–Einstein 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.

What is Einstein manifold in physics?

  • 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.

What is the Einstein condition for a Ricci-flat manifold?

  • 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 . In local coordinates the condition that (M, g) be an Einstein manifold is simply

What are the vacuum solutions of Einstein's equation?

  • Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant. Simple examples of Einstein manifolds include: Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.

What is Einstein manifold in physics?What is Einstein manifold in physics?

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.

image-What is a Kähler–Einstein metric?
image-What is a Kähler–Einstein metric?
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What is a Kähler–Einstein metric?What is a Kähler–Einstein metric?

In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds,...

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What are the vacuum solutions of Einstein's equation?What are the vacuum solutions of Einstein's equation?

Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant. Simple examples of Einstein manifolds include: Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.

Related

What is the Einstein condition for a Ricci-flat manifold?What is the Einstein condition for a Ricci-flat manifold?

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 . In local coordinates the condition that (M, g) be an Einstein manifold is simply

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