Annali di Matematica Pura ed Applicata

, Volume 58, Issue 1, pp 303–315 | Cite as

A characteristic property of spheres

  • A. D. Alexandrov


We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that\({}_{\partial \xi _i }^{\partial \varphi } > 0\). Then if φ(k1, ..., kn)=const on S and S is subject to some additional general conditions (those(II0) or(II) no 1), S is a sphere.


General Condition Characteristic Property Constant Curvature Principle Curvature Additional General Condition 
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Copyright information

© Nicola Zanichelli Editore 1962

Authors and Affiliations

  • A. D. Alexandrov
    • 1
  1. 1.LeningradU. R. S. S.

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