Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1981-0601738-1
MathSciNet review: 601738
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 } $.

References [Enhancements On Off] (What's this?)

  • [1] E. Kreyzig, Differential geometry, Univ. of Toronto Press, Toronto, 1959.
  • [2] Wilhelm Klingenberg, A course in differential geometry, Springer-Verlag, Berlin and New York, 1978. MR 0474045 (57:13702)
  • [3] Yung Chow Wong, A global formulation of the condition for a curve to lie in a sphere, Monatsh. Math. 67 (1963), 363-365. MR 0155237 (27:5173)
  • [4] S. Breuer and D. Gottlieb, Explicit characterizations of spherical curves, Proc. Amer. Math. Soc. 27 (1971), 126-127. MR 0270275 (42:5165)
  • [5] Yung Chow Wong, On an explicit characterization of spherical curves, Proc. Amer. Math. Soc. 34 (1972), 239-242. MR 0295224 (45:4292)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53A07

Retrieve articles in all journals with MSC: 53A07


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0601738-1
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society