Complex Space Form Research Articles

Real hypersurface of a complex space form

The purpose of the present paper is to give characterization of real hyper- surface of a complex space form. We find conditions for these hypersurfaces to be phi- symmetric and to have eta- parallel curvature tensor. Further we prove totally eta-umbilical real hypersurfaces of complex space forms have xi-parallel Ricci tensor and xi-parallel structure Jacobi operator.

Special lightlike hypersurfaces of indefinite Kaehler manifolds

In this paper, we define three types of lightlike hypersurfaces of an indefinite Kaehler manifold, which are called Hopf, recurrent and Lie recurrent lightlike hypersurfaces. After that we provide several new results on such three type lightlike hypersurfaces of an indefinite Kaehler manifold or an indefinite almost complex space form.

Characterization of real hypersurface with anti-derivatives of structure Lie operator in a complex space form

Characterization of real hypersurface with anti-derivatives of structure Lie operator in a complex space form

Characterizations of real hypersurfaces with the composition of structure Lie operator in nonflat complex space form

Characterizations of real hypersurfaces with the composition of structure Lie operator in nonflat complex space form

Lorentzian isotropic Lagrangian immersions

In this note we are interested in isotropic totally real Lorentzian submanifolds of indefinite complex space forms. We show that such submanifolds are always H-umbilical warped product immersions and we determine also the warping function.

Chen Inequalities For Submanifolds Of Generalized Space Forms With a Semi-symmetric Metric Connection

We investigate sharp inequalities for submanifolds in both generalized complex space forms and generalized Sasakian space forms with a semisymmetric metric connection.

Biharmonic surfaces with parallel mean curvature in complex space forms

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2AND ℂH2WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

Abstract. In this paper the notion of Lie derivative of a tensor fieldT of type (1,1) of real hypersurfaces in complex space forms with re-spect to the generalized Tanaka-Webster connection is introduced andis called generalized Tanaka-Webster Lie derivative. Furthermore, threedimensional real hypersurfaces in non-flat complex space forms whosegeneralized Tanaka-Webster Lie derivative of 1) shape operator, 2) struc-ture Jacobi operator coincides with the covariant derivative of them withrespect to any vector field X orthogonal to ξ are studied. 1. IntroductionA complex space form is an n-dimensional Kahler manifold of constant holo-morphic sectional curvature c. A complete and simply connected complex spaceform is analytically isometric to a complex projective space CP n if c > 0, or toa complex Euclidean space C n if c = 0, or to a complex hyperbolic space CH n if c < 0. The complex projective and complex hyperbolic spaces are callednon-flat complex space forms, since c 6= 0 and the symbol M

Real Hypersurfaces in Non-Flat Complex Space form with Structure Jacobi Operator of Lie-Codazzi Type

In this paper the notion of Lie-Codazzi type of a tensor field T of type (1,1) on real hypersurfaces, which generalizes the notion of Lie parallel, is introduced. Real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator is of Lie-Codazzi type, are studied. More precisely, the non-existence of such real hypersurfaces is proved.

Tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms

The notion of Lagrangian H-umbilical submanifolds was introduced by Chen (Isr J Math 99:69–108, 1997), and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanifolds, which are defined as submanifolds such that the bitension field of the inclusion map has vanishing tangential component. The normal bundle of a round hypersphere in $$\mathbb {R}^n$$ can be immersed as a tangentially biharmonic Lagrangian H-umbilical submanifold in $$\mathbb {C}^n$$ . Motivated by this fact, we classify tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms.

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.

Naturally reductive homogeneous real hypersurfaces in a nonflat complex space form

In this paper, in the class of homogeneous real hypersurfaces in a nonflat complex space form we study naturally reductive Riemannian homogeneous spaces having nonnegative sectional curvature.

A new characterization of the Berger sphere in complex projective space

We give a complete classification of Lagrangian immersions of homogeneous 3-manifolds (the Berger spheres, the Heisenberg group Nil 3 , the universal covering of the Lie group PSL ( 2 , R ) and the Lie group Sol 3 ) in 3-dimensional complex space forms. As a corollary, we get a new characterization of the Berger sphere in complex projective space.

On a Class of Real Hypersurfaces in a Complex Space Form with Weakly $$\phi $$ ϕ -Invariant Shape Operator

In this paper, we study real hypersurfaces in a complex space form with weakly \(\phi \)-invariant shape operator, where \(\phi \) is the almost contact structure on the real hypersurfaces induced by the complex structure on its ambient space. We first construct a class of real hypersurfaces with weakly \(\phi \)-invariant shape operator in complex Euclidean spaces and complex projective spaces and then give a characterization of such a class of real hypersurfaces. With this results, we classify minimal real hypersurfaces with weakly \(\phi \)-invariant shape operator in complex Euclidean spaces and in complex projective spaces.

NOTES ON REAL HYPERSURFACES IN A COMPLEX SPACE FORM

Abstract. We characterize a homogeneous real hypersurface of type (A)or a ruled real hypersurface in a non-flat complex space form, respectively. 1. IntroductionLet (Mf n (c),J,eg) be an n-dimensional complex space form with K¨ahlerianstructure (J,eg) of constant holomorphic sectional curvature c and let M be anorientable real hypersurface in Mf n (c). Then M has an almost contact metricstructure (η,φ,ξ,g) induced from (J,eg) (see Section 1). U.-H. Ki and Y. J.Suh [13] proved that are no real hypersurfaces in a non-flat complex space formsatisfying φA+Aφ = 0. From this we see that there are no almost cosymplecticor almost Kenmotsu real hypersurfaces in a non-flat complex space form (seeProposition 4 in Section 3). We put P = φA + Aφ. Then we prove that Pis invariant along the Reeb flow, that is, £ ξ P = 0 if and only if M is locallycongruent to a homogeneoushypersurface oftype (A) in P n Cor H n C(Theorem9).In Section 4, we prove that for a real hypersurface M in a non-flat com-plex space form Mf

Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

Abstract In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.

On Strongly Minimal Kähler Surfaces in $${\mathbf{C}^3}$$ C 3 and the Equality $${scal(p) = 4 \inf sec(\pi ^r)}$$ s c a l ( p ) = 4 inf s e c ( π r )

Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds in a Riemannian manifold (1968)], an important study of the very idea of curvature was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades, B.-Y. Chen’s fundamental inequalities have been investigated by many authors in the context of various geometric structures. In this work, we start by presenting B.-Y. Chen’s fundamental inequality for Kahler submanifolds in complex space forms, and we recall why the case of Kahler surfaces in $${\mathbf{C}^3}$$ satisfying $${scal(p) = 4 \inf sec(\pi ^r)}$$ appears naturally and is important. Then we provide several characterizations of strongly minimal complex surfaces in the complex three dimensional space. We focus our study on the question of finding further examples of strongly minimal Kahler surfaces, as the question of a complete classification of these geometric objects is still open.

On Classification of Real Hypersurfaces in a Complex Space Form with $\eta$-recurrent Shape Operator

In this paper, we classify real hypersurfaces in a non-flat complex space with η-recurrent shape operator.

On characterizations of Hopf hypersurfaces in a nonflat complex space form with commuting operators

Let $M$ be a real hypersurface in a complex space form $\mn$, $c \neq 0$. In this paper we prove that if $\rx\li = \li\rx$ holds on $M$, then $M$ is a Hopf hypersurface, where $\rx$ and $\li$ denote the structure Jacobi operator and the induced operator from the Lie derivative with respect to the structure vector field $\xi$, respectively. We characterize such Hopf hypersurfaces of $\mn$.