Properties
The isometry group of de Sitter space is the Lorentz group O(1,n). The metric therefore then has n(n+1)/2 independent Killing vectors and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by
De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:
This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
The scalar curvature of de Sitter space is given by
For the case n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.
Read more about this topic: De Sitter Space
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)