Hypersurfaces In Complex Space Forms Research Articles

Ruled Real Hypersurfaces in the Indefinite Complex Projective Space

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 \({\mathbb {C}}P^n_p\). In a previous work, Kimura and the second author introduced Hopf real hypersurfaces in \({\mathbb {C}}P^n_p\). 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.

A characterization of ruled hypersurfaces in complex space forms in terms of the Lie derivative of shape operator

<abstract><p>In this paper, it is proved that if a non-Hopf real hypersurface in a nonflat complex space form of complex dimension two satisfies Ki and Suh's condition (J. Korean Math. Soc., 32 (1995), 161–170), then it is locally congruent to a ruled hypersurface or a strongly $ 2 $-Hopf hypersurface. This extends Ki and Suh's theorem to real hypersurfaces of dimension greater than or equal to three.</p></abstract>

Quasi-Einstein Hypersurfaces of Complex Space Forms

Based on a well-known fact that there are no Einstein hypersurfaces in a nonflat complex space form, in this article, we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hypersurface of a nonflat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of Cho and Kimura.

Ricci operator in a Hopf real Hypersurfaces of complex space form

We initiate the study of real hypersurface of a complex space form and it is proved that Lie parallelism of the Ricci operator in the direction of ξ, locally symmetric Ricci operator in a real hypersurface of a complex space form reduce the hypersurface to Hopf hypersurface.

On the Minkowski Formula for Hypersurfaces in Complex Space Forms

Abstract In this paper, we discuss various Minkowski-type formulas for real hypersurfaces in complex space forms. In particular, we investigate the formulas suggested by the natural splitting of the tangent space. In this direction, our main result concerns a new kind of 2nd Minkowski formula.

Generalized Sasakian Space Forms Which Are Realized as Real Hypersurfaces in Complex Space Forms

We prove a classification theorem of the generalized Sasakian space forms which are realized as real hypersurfaces in complex space forms.

Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2

Let [Formula: see text] be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that [Formula: see text] is [Formula: see text]-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in [Formula: see text] or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in [Formula: see text]. Moreover, [Formula: see text]-parallel real hypersurfaces in [Formula: see text] and [Formula: see text] under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with [Formula: see text]-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

Higher-order Minkowski formula in complex space forms

In this paper, we prove higher-order Minkowski-type formulas for closed real hypersurfaces in complex space forms. As applications, we obtain several integral inequalities.

Ricci Curvature of Real Hypersurfaces in Non-flat Complex Space Forms

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.

Certain types of real hypersurfaces in complex space forms

A nice characterization of real hypersurfaces of a nonflat complex space form is very much familiar when its structure vector field $$\xi $$ is Killing. This motivates us to classify a real hypersurface of a nonflat complex space form when the structure vector field $$\xi $$ is 2-Killing (i.e., $$\pounds _{\xi }\pounds _{\xi }g = 0)$$ , where $$\pounds _{\xi }$$ denotes the Lie derivative along the vector field $$\xi $$ . Further, we classify real hypersurfaces of a nonflat complex space form when the tensor field $$T (= \pounds _{\xi }g)$$ is (i) weakly Lie $$\xi $$ -parallel, and (ii) weakly parallel.

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.

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.

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 characterizations of real hypersurface in complex space form with Codazzi type structure Lie operator

In this paper, we prove that if \((\nabla _{X} L_{\xi })Y= (\nabla _{Y} L_{\xi })X\) holds on \(M\), then \(M\) is a Hopf hypersurface, where \(L_\xi \) denote the induced operator from the Lie derivative with respect to the structure vector field \(\xi \). We characterize such Hopf hypersurfaces of \(M_n(c)\).

A study on hypersurface of complex space form

We show that quasi-umbilical, generalized quasi-umbilical, super quasi-umbilical hypersurfaces of a complex space form are quasi-Einstein, mixed generalized quasi-Einstein and mixed super quasi-Einstein manifolds, respectively. We also prove that a Bochner at space of generalized complex space form is an Einstein manifold.

Ruled real hypersurfaces of complex space forms

Ruled real hypersurfaces of complex space forms

Ricci solitons and real hypersurfaces in a complex space form

We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.

Levi-parallel hypersurfaces in a complex space form

In this paper, we classify a Hopf hypersurface in a nonflat complex space form whose Levi-form is parallel with respect to the generalized Tanaka-Webster connection.

PSEUDO-PARALLEL REAL HYPERSURFACES IN COMPLEX SPACE FORMS

Pseudo-parallel real hypersurfaces in complex space forms can be defined as an extrinsic analogues of pseudo-symmetric real hypersurfaces, that generalize the notion of semi-symmetric real hypersurface. In this paper a classification of the pseudo-parallel real hypersurfaces in a non-flat complex space forms is obtained.

NONEXISTENCE OF RICCI-PARALLEL REAL HYPERSURFACES IN P2C OR H2C

Niebergall and Ryan posed many open problems on real hypersurfaces in complex space forms. One of them is Are there any Ricci-parallel real hypersurfaces in complex projective space or complex hyperbolic space \ulcorner The purpose of present paper is to prove the nonexistence of such hypersurfaces.