Complex Space Form Research Articles

GTW parallel structure Jacobi operator of real hypersurfaces in nonflat complex space forms

We classify real hypersurfaces in nonflat complex space forms of complex dimension two whose structure Jacobi operator is parallel with respect to the generalized Tanaka-Webster connection. We classify these hypersurfaces when the dimension is more than two under the Hopf assumption.

Codazzi type $h$-operators on real hypersurfaces in nonflat complex space forms

Codazzi type $h$-operators on real hypersurfaces in nonflat complex space forms

Gradient pseudo‐Ricci solitons of real hypersurfaces

AbstractLet M be a real hypersurface of a complex space form , . Suppose that the structure vector field ξ of M is an eigen vector field of the Ricci tensor S, , β being a function. We study on M, a gradient pseudo‐Ricci soliton () that is an extended concept of gradient Ricci soliton, closely related to pseudo‐Einstein real hypersurfaces. When , we show that M is a Hopf hypersurface.

Parallel mean curvature biconservative surfaces in complex space forms

AbstractIn this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non‐flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons‐type formula for a well‐chosen vector field constructed from the mean curvature vector field and use it to prove a rigidity result for CMC biconservative surfaces in two‐dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non‐flat complex space forms. We conclude by constructing examples of CMC non‐PMC biconservative submanifolds from the Segre embedding and discuss when they are proper‐biharmonic.

Estimation of Ricci Curvature for Hemi-Slant Warped Product Submanifolds of Generalized Complex Space Forms and Their Applications

In this paper, we estimate Ricci curvature inequalities for a hemi-slant warped product submanifold immersed isometrically in a generalized complex space form with a nearly Kaehler structure, and the equality cases are also discussed. Moreover, we also gave the equivalent version of these inequalities. In a later study, we will exhibit the application of differential equations to the acquired results. In fact, we prove that the base manifold is isometric to Euclidean space under a specific condition.

Ricci Soliton of CR-Warped Product Manifolds and Their Classifications

In this article, we derived an equality for CR-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR-warped product submanifolds in Ka¨hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka¨hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.

Ricci and scalar curvature inequalities for semi-slant Riemannian submersions in complex space forms

This paper is devoted to the obtain Chen inequalities containing the Ricci and scalar inequalities on the vertical and the horizontal distributions for semi-slant Riemannian submersions from complex space forms onto Riemannian manifolds. The equality case of the obtained inequalities is discussed. At the end, some geometric consequences are obtained. Moreover, many examples are constructed.

Triharmonic curves along Riemannian submersions

The purpose of this paper is to study triharmonic curves along Riemannian submersions from Riemannian manifolds onto Riemannian manifolds. We obtain necessary and sufficient conditions for a triharmonic curve on the total manifoldof Riemannian submersion from a space form ( respectively, a complex space form) to a Riemannian manifold to be triharmonic curve on the base manifold. The above research problem is also studied in the complex setting of the manifoldon which the Riemannian submersion is defined. In addition, we give several results involving curvature conditions for a triharmonic curves along Riemannian submersions.

Real hypersurfaces in complex space forms with special almost contact structures

<abstract><p>In this paper, we prove that an almost contact metric structure of a real hypersurface in a complex space form is quasi-contact if and only if it is contact. We also classify real hypersurfaces whose associated almost contact metric structures are nearly Kenmotsu or cosymplectic, which gives several extensions of some earlier results in this field.</p></abstract>

Pointwise semi-slant Riemannian (PSSR) maps from almost Hermitian manifolds

In this paper, as a generalization of pointwise slant submanifolds [B-Y. Chen and O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk J Math 36, (2012), 630-640.], pointwise slant submersions [J.W.Lee and B.S. ahin, Pointwise slant submersions, Bulletin of the Korean Mathematical Sosiety, 51(4), (2014), 115-1126.] and pointwise slant Riemannian maps [Y. G?nd?zalp and M. A. Akyol, Pointwise slant Riemannian maps from Kaehler manifolds, Journal of Geometry and Physics, 179, (2002), 104589.], we introduce pointwise semi-slant Riemannian maps (briefly, PSSR maps) from almost Hermitian manifolds to Riemannian manifolds, present examples and characterizations. We also investigate the harmonicity of such maps. Moreover, we give Chen-Ricci inequality for a PSSR map. Finally, we study some curvature relations in complex space forms, involving Casorati curvatures for PSSR maps.

Umehara algebra and complex submanifolds of indefinite complex space forms

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, the symmetrized polydisc, etc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.

Optimal Inequalities for Hemi-Slant Riemannian Submersions

In the present paper, we establish some basic inequalities involving the Ricci and scalar curvature of the vertical and the horizontal distributions for hemi-slant submersions having the total space a complex space form. We also discuss the equality case of the obtained inequalities and provide illustrative examples.

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n≥4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

Homogeneous Lagrangian foliations on complex space forms

We classify holomorphic isometric actions on complex space forms all of whose orbits are Lagrangian submanifolds, up to orbit equivalence. The only examples are Lagrangian affine subspace foliations of complex Euclidean spaces, and Lagrangian horocycle foliations of complex hyperbolic spaces.

Generic lightlike submanifolds of an indefinite Kaehler manifold with an $(\ell, m)$-type metric connection

We study generic lightlike submanifolds $M$ of an indefinite Kaehler manifold $\bar{M}$ or an indefinite complex space form $\bar{M}(c)$ with an $(\ell, m)$-type metric connection subject such that the characteristic vector field $\zeta$ of $\bar{M}$ belongs to our screen distribution $S(TM)$ of $M$.

Real hypersurfaces in a nonflat complex space form whose certain tensor is recurrent

Real hypersurfaces in a nonflat complex space form whose certain tensor is recurrent

Weakly Einstein real hypersurfaces in [formula omitted] and [formula omitted

We classify real hypersurfaces in a nonflat complex space form of complex dimension two which are weakly or strongly Einstein. Specifically, first we prove that a Hopf hypersurface of dimension three is weakly Einstein if and only if it is locally congruent to a special geodesic sphere in the complex hyperbolic plane. We prove that if a strongly Einstein real hypersurface of dimension three is non-Hopf at each point, then it is locally congruent to a strongly 2-Hopf hypersurface.

Foliated Hopf hypersurfaces in complex hyperbolic quadrics

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kähler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kähler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kähler manifolds.

Special Slant Surfaces with Non-constant Mean Curvature in 2-Dimensional Complex Space Forms

In the late 1990s, B. Y. Chen introduced the notion of special slant surfaces in K\"{a}hler surfaces and classified non-minimal proper special slant surfaces with constant mean curvature in $2$-dimensional complex space forms. In this paper, we completely classify proper special slant surfaces with non-constant mean curvature in $2$-dimensional complex space forms.

Characterization of Whitney spheres among Lagrangian submanifolds with conformal Maslov form

In this paper, we give new characterizations of Whitney spheres among n-dimensional compact and non-minimal Lagrangian submanifolds immersed in the complex space form Nn(4c) with constant holomorphic sectional curvature 4c. We first show that an n-dimensional compact and non-minimal Lagrangian submanifolds satisfying a point-wise pinching condition in Nn(4c) is a Whitney sphere. Then we prove that there is a positive constant C(n) depending only on n such that, for a compact and non-minimal submanifold Mn immersed in Nn(4c) with c≥0, if the modified second fundamental form B satisfies ∫M|B|ndμ<C(n), then B≡0 and Mn is a Whitney sphere.