Complex Space Form Research Articles

Ricci flow of Kaehlerian slant submanifolds in complex space forms and its applications

AbstractThe normalized Ricci flow converges to a constant curvature metric for a connected Kaehlerian slant submanifold in a complex space form if the squared norm of the second fundamental form satisfies certain upper bounds. These bounds include the constant sectional curvature, the slant angle, and the squared norm of the mean curvature vector. Additionally, we demonstrate that the submanifold is diffeomorphic to the sphere $$\mathbb {S}^{n_1}$$ S n 1 under some restriction on the mean curvature. We claim that some of our previous results are rare cases.

$$\mathcal {D}$$-Parallel structure Lie operators on real hypersurfaces in nonflat complex space forms

$$\mathcal {D}$$-Parallel structure Lie operators on real hypersurfaces in nonflat complex space forms

Pointwise hemi-slant Riemannian maps ($\mathcal{PHSRM}$) from almost Hermitian manifolds

In 2022, the notion of pointwise slant Riemannian maps were introduced by Y. G\"{u}nd\"{u}zalp and M. A. Akyol [Journal of Geometry and Physics, 179, 104589, 2022] as a natural generalization of slant Riemannian maps \cite{s3}, slant Riemannian submersions \cite{lee}, slant submanifolds \cite{c1}. As a generalization of pointwise slant Riemannian maps and many subclasses notions (see also: \cite{ga2}), we introduce {\textit pointwise hemi-slant Riemannian maps (briefly, $\mathcal{PHSRM}$)} from almost Hermitian manifolds to Riemannian manifolds, giving non-trivial (proper) examples and investigate some properties of the map, we deal with their properties: the J-pluriharmonicity, the J-invariant, and the totally geodesicness of the map. Finally, we study some curvature relations in complex space form, involving Chen inequalities and Casorati curvatures for pointwise hemi-slant Riemannian maps, respectively.

Homology groups in CR-warped products of complex space forms

In the present paper, by using the result of Lawson-Simons [15], we adopt a different technique to show that there are no stable integral q-currents and a non-trivial homology group in a compact oriented CR-warped product submanifold Nn in a complex space forms space Qcm(4c) by imposing some restrictions on the squared norm of gradient and the Laplacian of warped function. Finally, we show that the same approach can be given by using Dirichlet energy, positive eigenvalue, and Hamiltonian.

Tube formulas for valuations in complex space forms

AbstractGiven an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in Riemannian manifolds.

Chen–Ricci inequalities for quasi bi-slant Riemannian submersions from complex space forms

Chen–Ricci inequalities for quasi bi-slant Riemannian submersions from complex space forms

New characterization of Hopf hypersurfaces in nonflat complex space forms

New characterization of Hopf hypersurfaces in nonflat complex space forms

Semi-Symmetric Metric Connections and Homology of CR-Warped Product Submanifolds in a Complex Space Form Admitting a Concurrent Vector Field

In this paper, we conduct a thorough study of CR-warped product submanifolds in a Kaehler manifold, utilizing a semi-symmetric metric connection within the framework of warped product geometry. Our analysis yields fundamental and noteworthy results that illuminate the characteristics of these submanifolds. Additionally, we investigate the implications of our findings on the homology of these submanifolds, offering insights into their topological properties. Notably, we present a compelling proof demonstrating that, under a specific condition, stable currents cannot exist for these warped product submanifolds. Our research outcomes contribute significant knowledge concerning the stability and behavior of CR-warped product submanifolds equipped with a semi-symmetric metric connection. Furthermore, this work establishes a robust groundwork for future explorations and advancements in this particular field of study.

Statistical Killing vector fields on the Hopf hypersurfaces in the complex space forms

Statistical Killing vector field is introduced and Hopf hypersurfaces of complex space forms with the condition of being structural statistical Killing vector field are studied. It is shown that these hypersurfaces have at most three distinct constant principal curvatures.

Rigidity results on totally real submanifolds in complex space forms

Rigidity results on totally real submanifolds in complex space forms

Complex submanifolds of indefinite complex space forms

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.

Quasi Yamabe Solitons on CR Submanifolds of Maximal CR Dimension in Kähler Manifolds

In this paper we give necessary and sufficient conditions for a CR submanifold of maximal CR dimension in arbitrary Kähler manifold to admit (quasi-)Yamabe structure, with naturally chosen soliton vector field. When the ambient manifold is a non-flat complex space form, we give a complete classification of such solitons, under certain conditions.

Polyharmonic hypersurfaces into complex space forms

Polyharmonic hypersurfaces into complex space forms

Some inequalities for bi-slant Riemannian submersions in complex space forms

The goal of this paper is to analyze sharp-type inequalities including the scalar and Ricci curvatures of bi-slant Riemannian submersions in complex space forms. Then, for bi-slant Riemannian submersion between a complex space form and a Riemannian manifold, we give inequalities involving the Casorati curvature of the space ker [Formula: see text]. Also, we mention some examples.

Geometric inequalities of bi-warped product submanifold in generalized complex space form

In this article, we obtain some inequalities for the bi-warped product submanifold of nearly-Kähler manifold which is expressed in terms of Laplacian, gradient, slant coefficient, and second fundamental form. Our inequality generalizes the inequalities of [1,20]. We also compute the Dirichlet energy and first eigenvalue of the Laplacian of warping functions.

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

<abstract><p>In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.</p></abstract>

Real hypersurfaces of nonflat complex space forms with weakly transversal Killing operators

Wang and Zhang in (2022) [20] and (2023) [21] characterized type (A) real hypersurfaces of the nonflat complex space forms as having transversal Killing structure Lie operator Lξ or contact Lie operator Lξϕ. In this note, we extend the above results by showing that the class of real hypersurfaces of type (A), (B) and the ruled real hypersurfaces in the nonflat complex space forms are locally characterized by having weakly transversal Killing operator Lξ or Lξϕ.

Pointwise Hemislant Submanifolds in a Complex Space Form

In this paper, pointwise hemislant submanifolds were introduced in a Kahler manifold. The integrability conditions for the distributions which are involved in the definition of a pointwise hemislant submanifold were investigated. In addition, the necessary and sufficient conditions were given for a pointwise hemislant submanifold to be a pointwise hemislant product.

Real Hypersurfaces in Nonflat Complex Space Forms with Reeb Recurrence Condition

In this paper we prove that the structure Lie operator of a real hypersurface in a nonflat complex space form is Reeb recurrent if and only if it is Lie Reeb recurrent, and this is equivalent to that the hypersurface is of type (A).

Characterization of simply connected complex space forms in terms of Bergman functions

It is well known that the Bergman functions of simply connected complex space forms (M,ω) of complex dimension n are constants ∏j=1n(m+λj)(λ≤0orλ=1k) on M for all m>m0. Conversely, this paper, under certain conditions, obtain that simply connected complete Kähler manifolds whose Bergman functions are constants ∏j=1n(m+λj)(λ≤0orλ=1) on M for all m>m0 must be holomorphically isometric to the complex space forms.