Ruled surfaces in elliptic space

Abstract

Bonnet's theorem on ruled surfaces [2, p. 449] deals solely with intrinsic properties if an intrinsic definition of the line of striction is adopted. Contrasting this, our aim is to define the parameter of distribution and the line of striction in relation to the enveloping space and to show that they have the usual properties. Attention will be called occasionally to the changes required to treat the hyperbolic case. The two theorems hold in hyperbolic space with minor variations and follow by formally analogous proofs. In fact, in the first one K S -1 must be assumed and in the second the hyperbolic tangent of the distance has to be used. Notation and terminology will largely be taken from [1]. Let D be the standard connection on the Euclidean 4-space such that DvW= (VWi)ei where ei, i= 1, 2, 3, 4 constitutes the natural frame field. Our elliptic 3-space is represented by the unit hypersphere (x, x) = 1, so the position vector x serves as unit normal and we have IDvx = V. If now 'D stands for the induced connection on the elliptic space we find for vectors tangent to it