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On the Rank Numbers of an Arc

Published online by Cambridge University Press:  20 November 2018

J. Turgeon
J. Turgeon*
Affiliation:
University of Toronto, Toronto, Ontario Université de Montréal, Montréal, Québec
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The kth rank number, rankkB, of a differentiable arc B in real projective n-space is the least upper bound of the number of osculating k-spaces of B which meet an (n – k – l)-flat, k = 0, 1, …, n – 1. The number rank0B is called the order of B; cf. 1.1-1.3. It has been conjectured by Peter Scherk that

(0.1)

equality holding if and only if B has the order n; cf. [2, p. 396]. In this paper we prove the following results.

THEOREM 1. If B is a differentiable elementary arc, then (0.1) holds for k = 0, 1, …, n – 1.

THEOREM 2. If B is a differentiable elementary arc and order B > n, then rankkB > (k + 1) (n – k) for k = 1, …, n – 2.

By a theorem of Park [3, p. 38], every differentiable arc contains a subarc of order n. This eliminates the assumption that B is elementary from Theorem 1. We do not know whether it can be dropped from Theorem 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 789 - 802
Copyright
Copyright © Canadian Mathematical Society 1970

References

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