Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Isoperimetric inequalities for immersed closed spherical curves
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by Joel L. Weiner
Proc. Amer. Math. Soc. 120 (1994), 501-506
DOI: https://doi.org/10.1090/S0002-9939-1994-1163337-4

Abstract:

Let $\alpha :{S^1} \to {S^2}$ be a ${C^2}$ immersion with length $L$ and total curvature $K$. If $\alpha$ is regularly homotopic to a circle traversed once then ${L^2} + {K^2} \geqslant 4{\pi ^2}$ with equality if and only if $\alpha$ is a circle traversed once. If $\alpha$ has nonnegative geodesic curvature and multiple points then $L + K \geqslant 4\pi$ with equality if and only if $\alpha$ is a great circle traversed twice.
References
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Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 120 (1994), 501-506
  • MSC: Primary 53A04; Secondary 53C42
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1163337-4
  • MathSciNet review: 1163337