User:Silly rabbit/Sandbox/Hyperbolic space

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[edit] Hyperbolic space as a homogeneous space

An alternative model of hyperbolic space is as a homogeneous space: a quotient of a Lie group by a closed Lie subgroup. The Lie group in question is the orthochronous Lorentz group G=SO+(n,1) which is the isometry group of (for instance) the hyperboloid model, through its action on the ambient Minkowski space Rn,1. This action is transitive and effective by Witt's theorem. The closed Lie subgroup is a group K isomorphic to SO(n), which corresponds to an ordinary spacelike rotation. This is the isotropy group of the vector e0 = (1,0,...,0). Hyperbolic space is thus diffeomorphic to the quotient G/K.

The Lie algebra g of G admits an AdK-invariant direct sum decomposition

{\mathfrak g} = {\mathfrak k} + {\mathfrak m}

where k is the Lie algebra of K and m is a complementary subspace which is isomorphic to Rn equipped with the standard representation of K=SO(n). The Killing form of g restricts to an inner product μ on m, and moreover

\mu(X,Y) = 2(n-1)\langle X, Y\rangle

where <X,Y> is the standard inner product on Rn. (In particular, this inner product is AdK-invariant.)