Hypersurfaces of Order Two

Abstract

A hypersurface ${S^{n - 1}}$ of order two in the real projective n-space is met by every straight line in maximally two points; cf. [1, p. 391]. We develop a synthetic theory of these hypersurfaces inductively, basing it upon a concept of differentiability. We define the index and the degree of degeneracy of an ${S^{n - 1}}$ and classify the ${S^{n - 1}}$ in terms of these two quantities. Our main results are (i) the reduction of the theory of the ${S^{n - 1}}$ to the nondegenerate case; (ii) the Theorem (A.5.11) that a nondegenerate ${S^{n - 1}}$ of positive index must be a quadric and (iii) a comparison of our theory with Marchaud’s discussion of “linearly connected” sets; cf. [3].