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Maximally symmetric homogeneous metrics on manifolds

Published online by Cambridge University Press:  24 October 2008

Philip L. Bowers
Philip L. Bowers
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida, U.S.A.
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A metric d on a set has maximal symmetry provided its isometry group is not properly contained in the isometry group of any metric equivalent to d. This concept was introduced by Janos [7] and subsequently Williamson and Janos [17] proved that the standard euclidean metric on ℝn has maximal symmetry. In Bowers [2], an elementary proof that every convex, complete, two-point homogeneous metric for which small spheres are connected has maximal symmetry is presented. This result in turn implies that the standard metrics on the classical spaces of geometry – hyperbolic, euclidean, spherical and elliptic – are maximally symmetric. In this paper we study homogeneous metrics that possess maximal symmetry and, in particular, address the problem of the existence of such metrics and, to a lesser extent, their uniqueness.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 1 , January 1990 , pp. 115 - 126
Copyright
Copyright © Cambridge Philosophical Society 1990

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