Maximally symmetric homogeneous metrics on manifolds

Abstract

A metricdon a set hasmaximal symmetryprovided its isometry group is not properly contained in the isometry group of any metric equivalent tod. This concept was introduced by Janos [7] and subsequently Williamson and Janos [17] proved that the standard euclidean metric on ℝnhas 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.