The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n . These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n . In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.

Real Hypersurfaces in Complex Space Forms

This paper aims at being a starting point for the investigation of the global sub-Lorentzian (or more generally sub-semi-Riemannian) geometry, which is a subject completely not known. We present an adaptation to the fiber bundles setting (and state its elementary proof in the real analytic case) of the decomposition result for indefinite metrics. Using it and the result by Sussmann on global spanning of distributions, we state and prove some global facts on sub-Lorentzian manifolds. We also construct a global control system describing nonspacelike curves and study its controllability properties in case of compact manifolds.

/pdf/remarks-on-global-sub-lorentzian-geometry-1pboa9nqq5.pdf

Remarks on global sub-Lorentzian geometry

Foliations, submanifolds, and mixed curvature

Publisher Summary This chapter discusses the research on four relevant topics on Lorentzian geometry. Several Lorentzian results are compared to Riemannian or indefinite (non-Lorentzian) ones, emphasizing on mathematical behaviors, which are specific of Lorentzian geometry. Degenerate tangent planes play an important role in the study of the geometry of Lorentzian manifolds. Harris introduced the notion of null sectional curvature for degenerate tangent planes, which has shown to be fruitful to get some comparison theorems and to characterize Robertson–Walker spacetimes. The chapter discusses Bochner technique on Lorentzian manifolds focusing aims and difficulties.

Chapter 8 Certain actual topics on modern Lorentzian geometry

We characterize real hypersurfaces with constant holomorphic sectional curvature of a non flat complex space form as the ones which have constant totally real sectional curvature.

/pdf/real-hypersurfaces-with-constant-totally-real-sectional-11x8v3ydj3.pdf

Real hypersurfaces with constant totally real sectional curvature in a complex space form

The purpose of this paper is to give another new characterization of ruled real hypersurfaces in a complex space form (c), c0 in terms of the covariant derivative of its Weingarten map in the direction of the structure vector .

A new characterization of ruled real hypersurfaces in complex space forms

The purpose of this paper is to give a necessary and sufficient condition for a smooth manifold to admit a Lorentzian metric As an application of this result, on Lorentzian manifolds we have shown that the existence of a 1-dimensional distribution is equivalent to the existence of a non-vanishing vector field

Young-Suk Choi

Papers

A new characterization of ruled real hypersurfaces in complex space forms

Remarks on the topology of lorentzian manifolds