Ruled Real Hypersurfaces in the Indefinite Complex Projective Space

TL;DR

This paper introduces and classifies ruled real hypersurfaces in indefinite complex projective space, revealing their structure, shape operator, and providing explicit examples and construction methods.

Contribution

It is the first to systematically study ruled real hypersurfaces in indefinite complex projective space, including their classification and explicit construction.

Findings

Two main families of ruled hypersurfaces identified

Explicit shape operator formulas derived

Classification of minimal ruled hypersurfaces achieved

Abstract

The main two families of real hypersurfaces in complex space forms are Hopf and ruled. However, very little is known about real hypersurfaces in the indefinite complex projective space $\cpn$. In a previous work, Kimura and the second author introduced Hopf real hypersurfaces in $\cpn$. In this paper, ruled real hypersurfaces in the indefinite complex projective space are introduced, as those whose maximal holomorphic distribution is integrable, and such that the leaves are totally geodesic holomorphic hyperplanes. A detailed description of the shape operator is computed, obtaining two main different families. A method of construction is exhibited, by gluing in a suitable way totally geodesic holomorphic hyperplanes along a non-null curve. Next, the classification of all minimal ruled real hypersurfaces is obtained, in terms of three main families of curves, namely geodesics, totally real circles and a third case which is not a Frenet curve, but can be explicitly computed. Four examples of minimal ruled real hypersurfaces are described.