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On the generalized helices of Hayden and Sypták in an N-space

Published online by Cambridge University Press:  24 October 2008

Yung-Chow Wong  and
W. V. D. Hodge
Yung-Chow Wong
Affiliation:
King's CollegeLondon
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Extract

In this paper we are concerned with curves (C) of the following types:

where k1, k2, …, kh−1, kh = 0 (h ≤ n) are the curvatures of (C) relative to the space Vn in which (C) lie. Hayden proved that a curve in a Vn is an (A)2m, h = 2m + 1, if and only if it admits an auto-parallel vector along it which lies in the osculating space of the curve and makes constant angles with the tangent and the principal normals. Independently, Sypták∥ stated without proof that a curve in an Rn is a (B)n if and only if it admits a certain number of fixed R2's having the same angle properties; he also gave to such a curve a set of canonical equations from which many interesting properties follow as immediate consequences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 3 , July 1941 , pp. 229 - 243
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

REFERENCES

(1)Borůvka, 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
(2)Butchart, J. H. ‘Helices in euclidean n-space.’ Amer. J. Math. 55 (1933), 126–30.Google Scholar
(3)Duschek, A. and Mayer, W.Lehrbuch der Differentialgeometrie, 2 (1930).Google Scholar
(4)Eisenhart, L. P.Riemannian geometry (1926).Google Scholar
(5)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
(6)Hayden, H. A. ‘On a generalized helix in a Riemannian n-space.’ Proc. London Math. Soc. (2), 32 (1931), 337–45.Google Scholar
(7)Havelka, B. ‘Sur les courbes dans les espaces euclidiens à n dimensions dont les courbures sont liées pas des relations lineaires à coefficients constants.’ C.R. Acad. Sci., Paris, 200 (1935), 432–4.Google Scholar
(8)Schouten, J. A. and Struik, J. D.Einführung in die neueren Methoden der Differentialgeometrie, 2 (1938).Google Scholar
(9)Sypták, M. ‘Sur les hypercirconférences et hyperhélices dans les espaces euclidien à p dimensions.’ C.R. Acad. Sci., Paris, 195 (1932), 298–9.Google Scholar
(10)Sypták, M. ‘Sur les hypercirconférences et hyperhélices généralisées dans les espaces euclidean à p dimensions.’ C.R. Acad. Sci., Paris, 198 (1934), 1665–7.Google Scholar
(11)Yung-Chow, Wong. ‘On a certain matrix occurring in the theory of helices.’ J. London Math. Soc. 15 (1940), 168–73.Google Scholar