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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A tale of two arc lengths: Metric notions for curves in surfaces in equiaffine space


Authors: Jeanne N. Clelland, Edward Estrada, Molly May, Jonah Miller, Sean Peneyra and Michael Schmidt
Journal: Proc. Amer. Math. Soc. 142 (2014), 2543-2558
MSC (2010): Primary 53A15, 53A55; Secondary 53A04, 53A05
Published electronically: April 3, 2014
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Abstract | References | Similar Articles | Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11983-6
PII: S 0002-9939(2014)11983-6
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.