Transformations of a Surface Bearing a Family of Asymptotic Curves

G. D. Gore
1938 Transactions of the American Mathematical Society  
Introduction. It is the purpose of this paper to establish certain transformations for any non-developable surface that bears a family of asymptotic curves and is immersed in a space of n dimensions Sn (n>3). All surfaces mentioned hereafter will belong to 5". The ambient of the osculating planes at a point of a surface to all of the curves on the surface that go through the point is a space of not more than five dimensions, f The class of all surfaces in Sn for which the ambient at all points
more » ... ient at all points is a space of four dimensions is divided into two subclasses. One of these subclasses is composed of all surfaces in Sn that bear each a conjugate net of curves, while the other is composed of all surfaces in Sn that sustain each a family of asymptotic curves. In the classical transformations for a surface bearing a conjugate net, the two congruences of lines tangent to the curves of the net have played basic rôles. We shall assign a similar rôle to the 2 lines tangent to the asymptotic curves of a family. Although the congruence of these lines contains only a one parameter family of developable surfaces, it will be defined as a parabolic congruence. To facilitate discussion, a terminology for certain geometric relations is introduced.
doi:10.2307/1990045 fatcat:2gomauqko5brhevkz7jphcminm