Existence and uniqueness theorem for Frenet frame supercurves

Abstract

In the first part of this paper,using the Banach Grassmann algebra $B_L$ given by Rogers in her paper [10],a new scalar product and a new definition of the orthogonality are introduced on the $(m,n)$-dimensional total supereuclidean space ${B_L}^m+n$. Using the GH^∈fty functions given by Rogers in [10], the new definitions of the supercurve, of the supersmooth supercurve, of the supersmooth supercurve in general position and of the Frenet frame associated to a supersmooth supercurve in general position are given. In second part of this paper, using the classical results described in [9], the new existence and uniqueness theorem for some supercurves which admit Frenet frame is proved.