Howard Cohl (Information Technology Laboratory, NIST, Gaithersburg, USA)

Radial fundamental solutions of Laplace's equation on spaces of constant curvature

Abstract:
   Riemannian manifolds with constant sectional curvature include d-dimensional Euclidean space, Hyperbolic space, and hyperspherical space. Hyperbolic and Euclidean space are represented by non-compact Riemannian manifolds with negative-constant and zero sectional curvature respectively, whereas hyperspherical space is represented by a compact Riemannian manifold with positive-constant sectional curvature. Due to the isotropy of these spaces, one expects there to exist a spherically symmetric fundamental solution for their corresponding Laplace-Beltrami operators. We obtain definite integral, finite summation, Gauss hypergeometric and associated Legendre function expressions for spherically symmetric fundamental solutions of Laplace's equation in hyperbolic and hyperspherical spaces in terms of their geodesic radius. In hyperbolic space this fundamental solution is given in terms of the associated Legendre function of the second kind with degree and order given by d/2-1, and real argument greater than unity. Furthermore in hyperbolic space, we are able to demonstrate uniqueness of our fundamental solution in terms of a decay at infinity. In hyperspherical space, this fundamental solution is given in terms of the associated Legendre function of the second kind on the cut (Ferrers function of the second kind) with degree given by d/2-1, order given by 1-d/2, and real argument between plus and minus one.