A necessary and sufficient condition that a curve lie on a hyperquadric

Abstract

The purpose of this paper is to give a necessary and sufficient condition in terms of the ordinary curvatures of a curve for it to lie on any real central hyperquadric in a Euclidean space 9Î". This problem was solved for the hypersphere in 9în by Karel Havlicek in his paper Contact des courbes et des hypersphères dans un espace euclidean à n dimensions.-Courbes sphériques, which appears in Casopis pro Pestování Matematiky a Fysiky vol. 72 (1947). First we find a necessary and sufficient condition in terms of socalled hyperquadric curvatures for a curve to lie on a given real central hyperquadric. Then by means of invariant functions in §3 we find the solution of our problem.