Hyperboloid Model - Isometries

Isometries

The indefinite orthogonal group O(1,n), also called the (n+1)-dimensional Lorentz group, is the Lie group of real (n+1)×(n+1) matrices which preserve the Minkowski bilinear form. In a different language, it is the group of linear isometries of the Minkowski space. In particular, this group preserves the hyperboloid S. Recall that indefinite orthogonal groups have four connected components, corresponding to reversing or preserving the orientation on each subspace (here 1-dimensional and n-dimensional), and form a Klein four-group. The subgroup of O(1,n) which preserves the sign of the first coordinate is the orthochronous Lorentz group, denoted O+(1,n), and has two components, corresponding to preserving or reversing the spacial dimension. Its subgroup SO+(1,n) consisting of matrices with determinant one is a connected Lie group of dimension n(n+1)/2 which acts on S+ by linear automorphisms and preserves the hyperbolic distance. This action is transitive and the stabilizer of the vector (1,0,…,0) consists of the matrices of the form

Where belongs to the compact special orthogonal group SO(n) (generalizing the rotation group SO(3) for n=3). It follows that the n-dimensional hyperbolic space can be exhibited as the homogeneous space and a Riemannian symmetric space of rank 1,

In fact, the group SO+(1,n) is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space.