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 are shown to be given by Frenet-like equations. Thus, finding an integral characterization for a spherical curve is identical to finding it for an Frenet curve.

For an Frenet curve we obtain: Let be a unit speed curve in so that . Then is a Frenet curve with curvature and torsion if and only if there are constant vectors and so that

**[1]**E. Kreyzig,*Differential geometry*, Univ. of Toronto Press, Toronto, 1959.**[2]**Wilhelm Klingenberg,*A course in differential geometry*, Springer-Verlag, New York-Heidelberg, 1978. Translated from the German by David Hoffman; Graduate Texts in Mathematics, Vol. 51. MR**0474045****[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****[4]**Shlomo Breuer and David Gottlieb,*Explicit characterization of spherical curves*, Proc. Amer. Math. Soc.**27**(1971), 126–127. MR**0270275**, 10.1090/S0002-9939-1971-0270275-2**[5]**Yung Chow Wong,*On an explicit characterization of spherical curves*, Proc. Amer. Math. Soc.**34**(1972), 239–242. MR**0295224**, 10.1090/S0002-9939-1972-0295224-3

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1981-0601738-1

Article copyright:
© Copyright 1981
American Mathematical Society