Clifford analysis on spheres and hyperbolae

We develop aspects of Clifford analysis over the sphere and hyperbolae. We focus primarily on the hyperbola lying in the Minkowski type space ℝ^n,1. We show that in order to give a proper extension of basic results on Clifford analysis in Euclidean space to this context one needs to consider both hyperbolae lying in ℝ^n,1. We also introduce Bergman spaces of L^p left monogenic sections in this context and consider the decomposition of square integrable sections over suitable bundles constructed over subdomains of spheres and hyperbolae. The results presented here cover the necessary background to enable one to set up and solve boundary value problems for field-type equations over hyperbolae. In particular, one can study analogues of the Dirichlet problem for analogues of the Laplacian over hyperbolae and spheres. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.