Green's function for the Hodge Laplacian on some classes of Riemannian and Lorentzian symmetric spaces

TL;DR

This paper derives the Green's function for the Hodge Laplacian on certain symmetric spaces combining Riemannian and Lorentzian geometries, using advanced analytical methods involving spherical means and Riesz distributions.

Contribution

It extends the method of spherical means and Riesz distributions to compute Green's functions for differential forms on symmetric spaces of constant curvature.

Findings

Explicit Green's functions for the Hodge Laplacian on these spaces

Reduction to a fourth-order Heun-type differential equation

Application of generalized spherical means and Riesz distributions

Abstract

We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\times\Sigma, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and \Sigma is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.