Cohomogeneity-one ruled hypersurfaces in CP2 and CH2

Abstract

In this paper, we show how to construct a special class of ruled hypersurfaces in the nonflat complex space forms ℂ ⁢ ℙ 𝑛 and 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 certain transversality and nonvanishing conditions.