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Mean-value theorems for Riemannian manifolds

Published online by Cambridge University Press:  14 November 2011

A. Gray  and
T. J. Willmore
A. Gray
Affiliation:
Department of Mathematics, University of Maryland, College Park, Md 20742, U.S.A.
T. J. Willmore
Affiliation:
Department of Mathematics, University of Durham, Durham
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Synopsis

Let Mm (r, f) denote the mean-value of a real-valued integrable function f over a geodesic sphere with centre m and radius r in an n-dimensional Riemannian manifold M. We obtain an expansion of Mm (r, f) in powers of r, thereby generalizing Pizzetti's formula valid in euclidean space. From this expansion we prove that the property

for every harmonic function near m, characterizes Einstein spaces. We define super-Einstein spaces and prove that they are characterized by the property

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 3-4 , 1982 , pp. 343 - 364
Copyright
Copyright © Royal Society of Edinburgh 1982

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