Study of a Surface by Means of Certain Associate Ruled Surfaces in Affine Space

Abstract

In this paper we shall deduce some results concerning the theory of surfaces in affine space. Some of these results may be regarded as analogues of the projective theory of surfaces immersed in ordinary space. Others seem to be properties of surfaces immersed in affine space. In what follows we shall use the same iotations as in Blaschke's Vorlesungen jiber Differentialgeometrie II. First, we define the canonical quadric Q at an ordinary point P of a surface ar in affine space by means of the Bompiani-Klouboucek's asymptotic osculating quadrics. Using this quadric Q we give a simple geometrical interpretation of the Pick invariant J and the Gaussian curvature S of the fundamental quadric form = 2Fdudv of ai. In particular if S =0, Q is a paraboloid, and conversely. Next we study the Moutard quadrics Q. (u) and Q. (v) belonging to the tangent t. and the asymptotic ruled surfaces R (u) and R(v) generated by the asymptotic uand v-tangents along the vand u-curves respectively. Using these two quadrics Q, (u) and Q. (v) we give a characteristic property of the tangents of Segre. We have shown that the loci of the diameters dn (u) anid dn (v) of Qn (u) and Q. (i) are two quadric cones r2 ( and r2(v) passing respectively through the asymptotic uand v-tangents and having the affine surface normal as a common generator. The cones r2 (u) and r2(V) intersect the quadric of Lie in the asymptotic tangents and in a pair of non-composite twisted cubics C3 (u) and C3 (v). The tangent plane Xr of cr at P cuts the tangent surface of U3(u) and C3 (V) in a pair of parabolas, mutually intersecting, besides the point P, and at three other points lying on the tangents of Segre. This property of the Segre tangents is similar to properties demonstrated by Cech ' and Su.2 Analogous to a theorem of Transon we prove that the loci of the affine