Triads of Ruled Surfaces
Transactions of the American Mathematical Society
The projective differential geometry of a configuration composed of two ruled surfaces whose generators are in one-to-one correspondence has been discussed by E. P. Lanef. He uses for defining system of equations a set of four ordinary linear first order differential equations in four dependent variables which, with slightly changed notation, we will write in the form y' = any + aXîz + aX3a + aXiß, z' = a2iy + aaz + aî3a + a24ft a' = lay + I12Z + k3a + laß, ß' = /2iy + luz + h*a + hiß,
... h*a + hiß, differentiation being with respect to the variable x. A system of simultaneous solutions yit zi} a¡, ßi(i = l, ■ • • , 4), with nonvanishing determinant is interpreted geometrically as determining four points Fy, (yi, ya> y», y*); P*, Oi, ** *»> «0; Pa, (ax, a2, as, a4); Pß, (ft, ft, ft, ft), which in turn determine two non-intersecting lines lvz, laß-As x varies these points trace four curves C", Ct, Ca, Cß, while the lines lyi, laß generate two ruled surfaces Ryz, Raß, on the first of which lie the curves Cv, Cz, and on the second, the curves Ca, Cß. In this paper we propose to develop the projective differential theory of triads of ruled surfaces whose generators are in one-to-one correspondence. We will determine the defining system of differential equations, calculate certain of the invariants and covariants, and exhibit their geometric significance in a number of instances.