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Proceedings of the American Mathematical Society

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

 

 

Small volume on big $ n$-spheres


Author: Christopher B. Croke
Journal: Proc. Amer. Math. Soc. 136 (2008), 715-717
MSC (2000): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-07-09079-X
Published electronically: November 6, 2007
MathSciNet review: 2358513
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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
Email: ccroke@math.upenn.edu

DOI: https://doi.org/10.1090/S0002-9939-07-09079-X
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.