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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A finite version of Schur's theorem


Author: R. S. Kulkarni
Journal: Proc. Amer. Math. Soc. 53 (1975), 440-442
MSC: Primary 53C20
MathSciNet review: 0383295
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Abstract: A necessary and sufficient condition for constancy of curvature in terms of umbilicity of small metric spheres is given.


References [Enhancements On Off] (What's this?)

  • [1] E. Cartan, Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris, 1963.
  • [2] Oldřich Kowalski, Properties of hypersurfaces which are characteristic for spaces of constant curvature, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 233–245. MR 0362156 (50 #14598)
  • [3] Katsumi Nomizu, Generalized central spheres and the notion of spheres in Riemannian geometry, Tôhoku Math. J. (2) 25 (1973), 129–137. Collection of articles dedicated to Shigeo Sasaki on his sixtieth birthday. MR 0328798 (48 #7140)
  • [4] Dominic S. Leung and Katsumi Nomizu, The axiom of spheres in Riemannian geometry, J. Differential Geometry 5 (1971), 487–489. MR 0290288 (44 #7472)
  • [5] F. Schur, Über den Zusammenhang der Räume konstanten Krümmungsmasses mit den projectiven Räumen, Math. Ann. 27 (1886), 537-567.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0383295-8
PII: S 0002-9939(1975)0383295-8
Article copyright: © Copyright 1975 American Mathematical Society