Characterizing the round sphere by mean distance

Simon Lyngby Kokkendorff

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result Of this paper is Theorem 4 stating that the round sphere S-1(n) of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric >= n - 1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to pi/2.
    Original languageEnglish
    JournalDifferential Geometry and Its Applications
    Volume26
    Issue number6
    Pages (from-to)638-644
    ISSN0926-2245
    DOIs
    Publication statusPublished - 2008

    Keywords

    • Extent
    • Potential
    • Rendezvous number
    • Mean distance
    • Metric invariants
    • Curvature

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