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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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A tale of two arc lengths: Metric notions for curves in surfaces in equiaffine space
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by Jeanne N. Clelland, Edward Estrada, Molly May, Jonah Miller, Sean Peneyra and Michael Schmidt PDF
Proc. Amer. Math. Soc. 142 (2014), 2543-2558 Request permission

Abstract:

In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine geometry, there is no invariant notion of inner product on tangent vectors that is preserved by the full equiaffine symmetry group. Nevertheless, it is possible to define an invariant notion of arc length for nondegenerate curves and an invariant first fundamental form for nondegenerate surfaces in equiaffine space. This leads to two possible notions of arc length for a curve contained in a surface, and these two arc length functions do not necessarily agree. In this paper we will derive necessary and sufficient conditions under which the two arc length functions do agree, and illustrate with examples.
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Additional Information
  • Jeanne N. Clelland
  • Affiliation: Department of Mathematics, 395 UCB, University of Colorado, Boulder, Colorado 80309-0395
  • Email: Jeanne.Clelland@colorado.edu
  • Edward Estrada
  • Affiliation: Department of Physics, 390 UCB, University of Colorado, Boulder, Colorado 80309-0390
  • Email: Edward.Estrada@colorado.edu
  • Molly May
  • Affiliation: Department of Physics, 390 UCB, University of Colorado, Boulder, Colorado 80309-0390
  • Email: Molly.May@colorado.edu
  • Jonah Miller
  • Affiliation: Department of Physics, 390 UCB, University of Colorado, Boulder, Colorado 80309-0390
  • Address at time of publication: Department of Physics, University of Guelph, Guelph, ON N1G 2W1, Canada
  • Email: jmille16@uoguelph.ca
  • Sean Peneyra
  • Affiliation: Department of Physics, 390 UCB, University of Colorado, Boulder, Colorado 80309-0390
  • Address at time of publication: USN, 404 Brookfield Lane, Goose Creek, South Carolina 29445
  • Email: peneyra.s@gmail.com
  • Michael Schmidt
  • Affiliation: Department of Physics, 390 UCB, University of Colorado, Boulder, Colorado 80309-0390
  • Address at time of publication: The Fulton School, 123 Schoolhouse Road, St. Albans, Missouri 63073
  • Email: schmidmt@gmail.com
  • Received by editor(s): May 1, 2012
  • Received by editor(s) in revised form: August 8, 2012
  • Published electronically: April 3, 2014
  • Additional Notes: This research was supported in part by NSF grant DMS-0908456.
  • Communicated by: Lei Ni
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 142 (2014), 2543-2558
  • MSC (2010): Primary 53A15, 53A55; Secondary 53A04, 53A05
  • DOI: https://doi.org/10.1090/S0002-9939-2014-11983-6
  • MathSciNet review: 3195775