Does Negative Type Characterize the Round Sphere?

Simon Lyngby Kokkendorff

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    Abstract

    We discuss the measure theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio 1 must be a round sphere, was put forward in a previous paper. We resolve this conjecture in the class of Riemannian symmetric spaces by showing, that a Riemannian manifold with symmetry ratio 1 must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.
    Original languageEnglish
    JournalProceedings of the American Mathematical Society
    Volume135
    Issue number11
    Pages (from-to)3695-3702
    ISSN0002-9939
    DOIs
    Publication statusPublished - 2007

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