Hypersurfaces In Complex Hyperbolic Space Research Articles

A twistor construction of Hopf real hypersurfaces in complex hyperbolic space

It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex 2-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with the sets of circles in totally geodesic complex hyperbolic lines in the complex hyperbolic space. As an application, we describe all classical Hopf examples. We also solve the remarkable and long-standing problem of the existence of Hopf real hypersurfaces in the complex hyperbolic space, different from the horosphere, such that the associated principal curvature is 2. We exhibit a method to obtain plenty of them.

Transversal Jacobi Operators in Almost Contact Manifolds

Along a transversal geodesic γ whose tangent belongs to the contact distribution D, we define the transversal Jacobi operator Rγ=R(·,γ˙)γ˙ on an almost contact Riemannian manifold M. Then, using the transversal Jacobi operator Rγ, we give a new characterization of the Sasakian sphere. In the second part, we characterize the complete ruled real hypersurfaces in complex hyperbolic space.

New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications

A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space $$\mathbb {CH}^n$$ . As an application, we study minimal ruled real hypersurfaces in $$\mathbb {CH}^n$$ and also those having constant scalar curvature.

Hopf hypersurfaces in complex hyperbolic space and submanifolds in indefinite complex 2-plane Grassmannian I

We define Gauss map from a real hypersurface in complex hyperbolic space to indefinite complex 2-plane Grassmannian. We show that if a real hypersurface is Hopf, then the image of the Gauss map is a half-dimensional regular submanifold and has a nice behavior under para-quaternionic Kähler structures of the Grassmannian. In particular if absolute value of the Hopf curvature of the Hopf hypersurface is greater (resp. smaller) than 2, then the Gauss image is totally complex (resp. totally para-complex) submanifold with respect to the para-quaternionic Kähler structure of indefinite complex 2-plane Grassmannian, provided that the induced metric is nondegenerate.

A d’Alembert Formula for Hopf Hypersurfaces

A Hopf hypersurface in complex hyperbolic space \({\mathbb{C}{\rm H}^n}\) is one for which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to ambient curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit parametrization.

Hopf hypersurfaces of small Hopf principal curvature in $${\mathbb{C}{\rm H}^2}$$

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space \({\mathbb{C}{\rm H}^2}\) with any specified value of the Hopf principal curvature α less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in \({\mathbb{C}{\rm H}^2}\).

Real Hypersurfaces in Complex Hyperbolic Space with Commuting Ricci Tensor

In this paper we consider a real hypersurface M in complex hyperbolic space HnC satisfying SA = AS , where , A and S denote the structure tensor, the shape operator and the Ricci tensor of M respectively. Moreover, we give a characterization of real hypersurfaces of type A in HnC by such a commuting Ricci tensor.

REAL HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES IN COMPLEX HYPERBOLIC SPACES

We present the classification of all real hypersurfaces in complex hyperbolic space $\mathbb{C}H^{n}$, $n \geq 3$, with three distinct constant principal curvatures

Real hypersurfaces in nonflat complex space forms are irreducible

The study of real hypersurfaces in complex projective space and complex hyperbolic space has been an active field over the past three decades. Although these ambient spaces might be regarded as the simplest after the spaces of constant curvature, they impose significant restrictions on the geometry of their hypersurfaces. For instance, they do not admit totally umbilical hypersurfaces and Einstein hypersurfaces. On the other hand, several important classes of real hypersurfaces in complex projective space have been constructed and investigated by many geometers. For instance, H.B. Lawson investigated real hypersurfaces of constructed by Clifford minimal hypersurfaces of +1 via Hopf fibration. R. Takagi [9] gave the list of homogeneous real hypersurfaces of . Many geometers then study the geometry from the list of Takagi and obtained various interesting geometric characterizations of homogeneous real hypersurfaces in . Another important class of real hypersurfaces in which contains the list of R. Takagi is the class of Hopf hypersurfaces. Such hypersurfaces are real hypersurfaces whose structure vector ξ is a principal curvature vector, where is the complex structure and ξ is the unit normal vector field. Examples and geometric characterizations of Hopf hypersurfaces have also been obtained by various geometers. It is known that in , is a homogeneous real hypersurface if and only if is a Hopf hypersurface with constant principal curvatures [6, 9]. The study of real hypersurfaces in complex hyperbolic space has followed developments in , often with similar results, but sometimes with differences (see [1, 7, 8] for more details). It is well-known that real projective space and real hyperbolic space admit ample hypersurfaces which are the Riemannian products of some Riemannian manifolds. It is also well-known that 3 admits a complex hypersurface which is the Riemannian

Complete minimal hypersurfaces in complex hyperbolic space

We construct new examples of embedded, complete, minimal hypersurfaces in complex hyperbolic space, including deformations of bisectors and some minimal foliations.