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

Integral characterizations and the theory of curves


Author: Victor Dannon
Journal: Proc. Amer. Math. Soc. 81 (1981), 600-602
MSC: Primary 53A07
MathSciNet review: 601738
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Abstract: Spherical curves in $ {E^4}$ are shown to be given by Frenet-like equations. Thus, finding an integral characterization for a spherical $ {E^4}$ curve is identical to finding it for an $ {E^3}$ Frenet curve.

For an $ {E^3}$ Frenet curve we obtain: Let $ \alpha (s)$ be a unit speed $ {C^4}$ curve in $ {E^3}$ so that $ \alpha '(s) = T$. Then $ \alpha $ is a Frenet curve with curvature $ \kappa (s)$ and torsion $ \tau (s)$ if and only if there are constant vectors $ {\mathbf{a}}$ and $ {\mathbf{b}}$ so that

$\displaystyle {\mathbf{T'}}(s) = \kappa (s)\{ {{\mathbf{a}}\cos \xi (s) + {\mat... ...[\xi (s) - \xi (\delta )]{\mathbf{T}}(\delta )\kappa (\delta )\;d\delta } } \},$

where $ \xi (s) = \int_0^s {\tau (\delta )\;d\delta } $.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0601738-1
PII: S 0002-9939(1981)0601738-1
Article copyright: © Copyright 1981 American Mathematical Society