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Nonexistence of skew loops on ellipsoids

Author: Mohammad Ghomi
Journal: Proc. Amer. Math. Soc. 133 (2005), 3687-3690
MSC (2000): Primary 53A04, 53A05; Secondary 53C45, 52A15
Published electronically: June 3, 2005
MathSciNet review: 2163608
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Abstract: We prove that every $C^1$ closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve.

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Additional Information

Mohammad Ghomi
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georia 30332
Address at time of publication: Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Keywords: Tantrix, skew loop, ellipsoid, quadric surface
Received by editor(s): April 21, 2004
Received by editor(s) in revised form: August 17, 2004
Published electronically: June 3, 2005
Additional Notes: The author’s research was partially supported by NSF Grant DMS-0336455, and CAREER award DMS-0332333.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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