IX.—Harmonic Riemannian Spaces

Abstract

In this paper we consider a new class of Riemannian spaces which arise in the theory of the solution of the tensor generalisation of Laplace's equation ∇2V = o. To obtain this generalisation Beltrami's second differential parameter is defined in terms of the metricof the associated n-dimensional Riemannian space by the usual formulæwhere denotes the Christoffel symbol . The generalised Laplace's equation is then Δ2V = o. For simplicity the quadratic differential form (1.1) is taken to be positive definite, which involves no essential loss of generality.