Spinor Green's functions via spherical means on products of space forms

TL;DR

This paper develops a method to explicitly compute the Green's function for the spinor Klein-Gordon equation on product manifolds of constant curvature, extending spherical means to spinor fields using Riesz distributions.

Contribution

It introduces an extension of spherical means for spinor fields and computes explicit Green's functions on product space forms with constant curvature.

Findings

Explicit formulas for spinor Green's functions on product manifolds.

Extension of spherical means method to spinor fields.

Application of Riesz distributions in the computation.

Abstract

We explicitly compute the Green's function of the spinor Klein-Gordon equation on the Riemannian and Lorentzian manifolds of the form $M_0 \times ... \times M_N$, with each factor being a space of constant sectional curvature. Our approach is based on an extension of the method of spherical means to the case of spinor fields and on the use of Riesz distributions.