Poincare extension of Mobius transformations

摘要

Given sphere preserving (Mobius) transformations in n-dimensional Euclidean space one can use the Poincare extension to obtain sphere preserving transformations in a half-space of n + 1 dimensions. The Poincare extension is usually provided either by an explicit formula or by some geometric construction. We investigate its algebraic background and describe all available options. The solution is given in terms of one-parameter subgroups of Mobius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the Mobius transformations of the real line only.