CIRCULAR CONE AND ITS GAUSS MAP

The family of cones is one of typical models of non-cylindrical ruled surface. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map G of a circular cone, one has Delta G = integral(G + C), where Delta is the Laplacian operator, integral is a non-zero function and C is a constant vector. We prove that circular cones are characterized by being the only non-cylindrical ruled surfaces with Delta G = f(G + C) for a nonzero constant vector C.