Journal of the Graduate Research Center


In the projective differential geometry of ordinary space a problem of fundamental importance is that of obtaining a covariantly determined reference frame for the definition of local point coordinates and associated power series developments for the equations of curves and surfaces. Much of the celebrated memoir [1] of G. M. Green was devoted to this problem for a surface or 2-dimensional Cartan variety. However, the complete geometric characterization of the reference frame used by Green was not completed until sixteen years later by Bell [2]. In this paper, an extension of Green's "relation R" is given for a linear (n-2)-space L and a line l. This extended relation will also be known as the relation R. The power series expansion in local non-homogeneous coordinates for the equation of an (n-1)-dimensional variety Vo in n space is obtained using the method introduced by Bell [3]. This power series is simplified by choosing a suitable reference frame. The complete geometric characterization of this reference frame is given and a generalization of the edges of Green for n-dimensional space is obtained by use of the relation R. The power series obtained and also the reference frame chosen are shown to be generalizations of those obtained by Green [1] using Wilczynski' s normal coordinates.

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