The Dirac Operator on Ultrahyperbolic Manifolds

Abstract

In this paper we consider a projective model for the time- and spacelike ultrahyperbolic unit balls in the orthogonal space $\mathbf{R}^{m,m}$. By means of an associated principal fibre bundle, a Dirac operator on these mani\-folds is defined and its fundamental solution is constructed (in case $m \in 2\mathbf{N} + 1$) with the aid of generalized Riesz distributions. Using the method of descent, we then construct fundamental solutions for the Dirac operator on time- or spacelike ultrahyperbolic unit balls in spaces of signature $(m,q)$ and $(p,m)$ respectively (with $p,q < m$).