Hodge—dirac operators for hyperbolic spaces

Abstract

In this article we extend the concept of hyperbolically harmonic functions for the upper half plane, first introduced by Leutwiler, to an arbitrary conformally flat manifold. We shall consider the special cases of the two standard models for hyperbolic spaces, namely the Poincaré model for the upper half space and the spherical model. For these two manifolds, an analogue of the Cauchy—Riemann equations, obtained via the Euclidean Dirac operator, is studied, along with particular decompositions of its solutions.