Geodesic
On the Rank Numbers of an Arc
Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 789-802

Voir la notice de l'article provenant de la source Cambridge University Press

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 rank0 B 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. 
Turgeon, J. On the Rank Numbers of an Arc. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 789-802. doi: 10.4153/CJM-1970-089-6
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