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Generalized Helices in an Ordinary Vn

Published online by Cambridge University Press:  24 October 2008

Yung-Chow Wong
Yung-Chow Wong
Affiliation:
King's CollegeUniversity of London
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Extract

A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 1 , January 1941 , pp. 14 - 28
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

REFERENCES

(1)Bocher, . Introduction to higher algebra (1907).Google Scholar
(2)Borüvka, O. “Sur les hypercirconférences et certaines surfaces paraboliques dans l'espace euclidien à quatre dimensions.” C.R. Acad. Sci., Paris, 193 (1931), 633–4.Google Scholar
(3)Cutler, E. H. “On the curvatures of a curve in Riemann space.” Trans. Amer. Math. Soc. 33 (1931), 833–8.Google Scholar
(4)Duschek, A. and Mayer, W.Lehrbuch der Differentialgeometrie, 2 (1930).Google Scholar
(5)Goursat, E.A course in mathematical analysis, translated by Hedrick, , 1 (Ginn, 1904).Google Scholar
(6)Hayden, H. A. “Deformations of a curve in a Riemannian n-space which displace certain vectors parallelly at each point.” Proc. London Math. Soc. (2), 32 (1931), 321–36.Google Scholar
(7)Hayden, H. A. “On a generalized helix in a Riemannian n-space.” Proc. London Math. Soc. (2), 32 (1931), 337–45.CrossRefGoogle Scholar
(8)Havelka, B. “Sur les courbes dans les espaces euclidiens à n dimensions dont les courbures sont liées par des relations linéaires à coefficients constants.” C.R. Acad. Sci., Paris, 200 (1935), 432–4.Google Scholar
(9)Schouten, J. A.Ricci Kalkül (1924).Google Scholar
(10)Schouten, J. A. and Struik, J. D.Einführung in die neueren Methoden der Differential-geometrie, 1 (1935), 2 (1938).Google Scholar
(11)Struik, J. D.Grundzüge der mehrdimensionalen Differential-geometrie in direkter Darstellung (1922).CrossRefGoogle Scholar
(12)Syptak, M. “Sur les hypercirconférences et hyperhélices dans les espaces euclidiens à p dimensions.” C.R. Acad. Sci., Paris, 195 (1932), 298–9.Google Scholar
(13)Syptak, M. “Sur les hypercirconférences et hyperhélices généralisées dans les espaces euclidiens à p dimensions.” C.R. Acad. Sci., Paris 198 (1934), 1665–7.Google Scholar
(14)Yung-Chow, Wong. “On the generalized helices of Hayden and Syptak in an n-space.” (In the Press.)Google Scholar