Doubly ruled submanifolds in space forms

Abstract

In this paper we extend a classical result, namely, the one that states that the only doubly ruled surfaces in $\mathbb R^3$ are the hyperbolic paraboloid and the hyperboloid of one sheet, in three directions: for all space forms, for any dimensions of the rulings and manifold, and to the conformal realm. We show that all this can be reduced, with the help of quite natural constructions, to just one simple example, the rank one real matrices. We also give the affine classification in Euclidean~space. To deal with the conformal case, we make use of recent developments on Ribaucour transformations.