Cohomogeneity-One Ruled Hypersurfaces in ${\mathbb{CP}}^2$ and $\mathbb{C}{H}^2$

Abstract

In this paper, we show how to construct a special class of ruled hypersurfaces in the nonflat complex space forms ${\mathbb{CP}}^n$ and $\mathbb{C}{H}^n$. This is done by taking an arbitrary smooth curve in a totally geodesic (complex) one-dimensional submanifold and erecting an orthogonal ruling over each of its points. Concentrating on the $n=2$ case, we also examine the special situation in which the base curve has constant geodesic curvature. We show that, in this case, the construction yields precisely the real-analytic hypersurfaces of cohomogeneity one that satisfy a certain transversality condition.

arXiv comment28 pages; revised version, with corrections to main theorems; published in Differential Geometry and its Applications