User:Silly rabbit/Sandbox/Hyperbolic space

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 R^n,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 Ad_K-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 Ad_K-invariant.)