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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Small volume on big $n$-spheres
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by Christopher B. Croke PDF
Proc. Amer. Math. Soc. 136 (2008), 715-717 Request permission

Abstract:

We consider Riemannian metrics on the $n$-sphere for $n\geq 3$ such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the $2$-dimensional case where Berger has shown that $Area\geq 1/2$.
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Additional Information
  • Christopher B. Croke
  • Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
  • MR Author ID: 204906
  • Email: ccroke@math.upenn.edu
  • Received by editor(s): November 9, 2006
  • Published electronically: November 6, 2007
  • Additional Notes: The author was supported by NSF grants DMS 02-02536 and 07-04145
  • Communicated by: Jon G. Wolfson
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 715-717
  • MSC (2000): Primary 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-07-09079-X
  • MathSciNet review: 2358513