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
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)