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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

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