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IX.—Harmonic Riemannian Spaces

Published online by Cambridge University Press:  15 September 2014

E. T. Copson  and
H. S. Ruse
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Extract

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 metric

of the associated n-dimensional Riemannian space by the usual formulæ

where 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.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 60 , Issue 1 , 1940 , pp. 117 - 133
Copyright
Copyright © Royal Society of Edinburgh 1940

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