A new foundation of the projective differential theory of curves in five-dimensional space

Abstract

Introduction.The purpose of the present paper is to develop a purely geometric theory of the projective differential geometry of curves in a space of five dimensions, the methods hitherto adopted by various authors(2) being more or less analytic and artificial.The projective differential theory of plane curves is one of the cornerstones upon which our theory of curves in five dimensions rests. The problem of finding a covariant figure to associate projectively with the neighborhood of order five at an ordinary point of a plane curve is an interesting one as has been pointed out by . We have succeeded in solving this problem in an elementary way. Then using neighborhoods of order six we have obtained, besides a covariant triangle, three covariant points /,• (i = 1, 2, 3) any one of which can be selected as unit point.In five-dimensional space the osculating plane p at an ordinary point P oí a curve V intersects the developable hypersurface of T in a plane curve C, of which P is either an ordinary point or a &-ic (k = 6, 7, 8) point [2]. In any case a covariant unit point / and a covariant triangle {PPiP2} can be determined in p, PPi being the tangent to T at P.When a plane p, passing through PPi and lying in the osculating threespace of r at P, rotates about PPi, the Bompiani osculant On [l], associated with the point of inflexion P of the projection produced by projecting V on the plane p from the point aP+ßPi, constitutes a generator of a covariant quadric Q3. Since P, Pi, P2 are three vertices of a quadrilateral on Q3, we may take the fourth vertex for P3. In a similar way we define two other vertices Pi and P6 of a covariant pyramid {PPiPiP3PiPf,} and two other covariant quadrics Q4 and Q$ which pass through the quadrilaterals PPiP^PJ* and