Geometry of Pseudospheres I.

Abstract

The n-dimensional pseudospheres are the surfaces in Rn+1 given by the equations [see pdf for equations]. We consider the pseudospheres as surfaces in En+1,k, where Em,k = Rk × (iR)m−k, and investigate their geometry in terms of the linear algebra of these spaces. Each of the spaces Em,k has a natural (not generally positive definite) metric, which is inherited by the pseudospheres. We prove that each matrix with columns in Em,k can be put into a canonical form by premultiplying by an orthogonal matrix (a matrix which effects an isometry of Em,k). We term a matrix in this form bitriangular. This generalizes upper triangular form for real square matrices.