Definition
De Sitter space can be defined as a submanifold of a Minkowski space of one higher dimension. Take Minkowski space R1,n with the standard metric:
De Sitter space is the submanifold described by the hyperboloid of one sheet
where is some positive constant with dimensions of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. The induced metric is nondegenerate and has Lorentzian signature. (Note that if one replaces with in the above definition, one obtains a hyperboloid of two sheets. The induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space.)
De Sitter space can also be defined as the quotient O(1,n)/O(1,n−1) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.
Topologically, de Sitter space is R × Sn−1 (so that if n ≥ 3 then de Sitter space is simply-connected).
Read more about this topic: De Sitter Space
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