Warped product pointwise bi-slant submanifolds in metallic Riemannian manifolds

Geometry of pointwise pseudo-slant warped product submanifolds in a Kähler manifold

The purpose of this paper is to study pointwise pseudo-slant warped product submanifolds of a Kähler manifold $\widetilde{M}$. We derive the conditions of integrability and totally geodesic foliation for the distributions allied to the characterization of a pointwise pseudo-slant submanifold of $\widetilde{M}$. The necessary and sufficient conditions for isometrically immersed pointwise pseudo-slant submanfold of $\widetilde{M}$ to be a pointwise pseudo-slant warped product and a locally Riemannian product are obtained.

Kaehler流形的Warped product Quasi Bi-slant Submanifolds

Warped product Quasi Bi-slant Submanifolds of Kaehler Manifolds

本文引入Kaehler流形中的warped product quasi bi-slant submanifolds概念，证明其必为Riemannian product或warped product quasi hemi slant submanifold，并给出实例。

An optimal inequality on warped product semi-slant submanifolds of nearly Kaehler manifolds

Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in [17], it is interesting to find their existence. In this paper, we prove the existence of warped product semi-slant submanifolds of nearly Kaehler manifolds by a characterization. To this end we obtain an inequality for the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also discussed.

$\mathcal{P}\mathcal{R}$-anti-slant warped product submanifold of a nearly paracosymplectic manifold

In this paper, we study $\mathcal{P}\mathcal{R}$-anti-slant warped product submanifold of a nearly paracosymplectic manifold $\widetilde{M}$. The necessary and sufficient condition is obtained for the distributions allied to the characterization of a $\mathcal{P}\mathcal{R}$-anti-slant submanifold being integrable and totally geodesic foliation. In addition, we have defined $\mathcal{P}\mathcal{R}$-anti-slant warped product submanifold of $\widetilde{M}$ and gave some illustrations. Finally, we extracted the constraints for a submanifold of $\widetilde{M}$ to be a $\mathcal{P}\mathcal{R}$-anti…

Warped product pointwise bi-slant submanifolds of Kenmotsu manifolds

The present paper deals with the study of warped product pointwise bi-slant submanifolds of Kenmotsu manifolds with an example. The characterization for such submanifold is also discussed. An inequality of such submanifold is obtained and its equality case is also considered.

Warped Product Pointwise Semi-slant Submanifolds of Kaehler Manifolds

It is known that there exist no warped product semi-slant submanifolds in Kaehler manifolds \cite{Sahin}. Recently, Chen and Garay studied pointwise-slant submanifolds of almost Hermitian manifolds in \cite{CG} and obtained many new results for such submanifolds. In this paper, we first introduce pointwise semi-slant submanifolds of Kaehler manifolds and then we show that there exists non-trivial warped product pointwise semi-slant submanifolds of Kaehler manifold by giving an example, contrary to the semi-slant case. We present a characterization theorem and establish an inequality for the sq…

Pointwise almost h-semi-slant submanifolds

We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distributions, the conditions for such distributions to be totally geodesic foliations, the mean curvature vector fields on totally umbilic submanifolds, the properties of h-slant functions and h-semi-slant functions, the properties of non-trivial warped product proper pointwise h-semi-…

Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds

In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also considered in this paper. Finally, we provide some applications of our derived results.

Sequential warped product submanifolds having factors as holomorphic, totally real and pointwise slant

We introduce sequential warped product submanifolds of Kaehler manifolds, provide examples and establish Chen's inequality for such submanifolds. The equality case is also studied. Moreover, by inspiring Lawson and Simons's integral currrent's theorem on a submanifold, we find a similar pinching inequality for a sequential warped product submanifold and obtain geometric results when the equality case is satisfied.

$k$-slant distributions

Inspired by the concepts of slant distribution and slant submanifold, with their variants of hemi-slant, semi-slant, bi-slant, or almost bi-slant, we introduce the more general concepts of $k$-slant distribution and $k$-slant submanifold in the settings of an almost Hermitian, an almost product Riemannian, an almost contact metric, and an almost paracontact metric manifold and study some of their properties. We prove that, for any proper $k$-slant distribution in the tangent bundle of a Riemannian manifold, there exists another one in its orthogonal complement, and we establish basic relations…

A geometric inequality for warped product semi-slant submanifolds of nearly cosymplectic manifolds

Recently, we have shown that there do not exist the warped product semi-slant submanifolds of cosymplectic manifolds [10]. As nearly cosymplectic structure generalizes cosymplectic ones same as nearly Kaehler generalizes Kaehler structure in almost Hermitian setting. It is interesting that the warped product semi-slant submanifolds exist in nearly cosymplectic case while in case of cosymplectic do not exist. In the beginning, we prove some preparatory results and finally we obtain an inequality such as $\|h\|^2 \geq 4q\csc^2θ\{1+\frac{1}{9}\cos^2θ\}\|\nabla \ln f\|^2$ in terms of intrinsic and…

Kähler流形上逐点$k$-slant子流形的若干性质

Some properties of pointwise $k$-slant submanifolds of Kähler manifolds

该研究聚焦近Hermite流形尤其是Kähler流形上的逐点$k$-slant子流形，刻画了其分量分布的可积性，并探讨了全测地情形，丰富了相关子流形的性质研究。

Existence and uniqueness theorems for pointwise slant immersions in complex space forms

An isometric immersion $f: M^{n} \rightarrow \tilde M^{m}$ from an $n$-dimensional Riemannian manifold $M^{n}$ into an almost Hermitian manifold $\tilde M^{m}$ of complex dimension $m$ is called pointwise slant if its Wirtinger angles define a function defined on $M$. In this paper we establish the existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds $M^{n}$ into a complex space form $\tilde M^{n}(c)$ of constant holomorphic sectional curvature $c$.

关于斜子流形的新见解

New insights on slant submanifolds

该研究在微分几何领域，给出了逐点斜子流形关于反交换近埃尔米特结构生成族为斜的充要条件，证明了逐点斜性质在一类子流形上的传递性，并辅以示例说明。

Some inequalities on hemi-slant product submanifolds in a cosymplectic manifold

Recently, M. Atcken studied Contact CR-warped prod- uct submanifolds in cosymplectic space forms and established gen- eral sharp inequalities for CR-warped products in a cosymplectic manifold [1]. In the present paper, we obtain an inequality for the squared norm of the second fundamental form in terms of constant ?$ϕ$-sectional curvature for hemi-slant products in cosymplectic mani- folds. An inequality for hemi-slant warped products in a cosymplectc manifold is also given. The equality case is considered.

Characterization of warped product submanifolds of Lorentzian concircular structure Manifolds

Recently Hui et al. (\cite{HAP}, \cite{HAN}) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a $(LCS)_n$-manifold $\bar{M}$. In this paper we have studied the characterization for both these classes of warped product submanifolds. It is also shown that there do not exists any proper warped product bi-slant submanifold of a $(LCS)_n$-manifold. Although we constructed an example of a bi-slant submanifold of $(LCS)_n$-manifold.

On warped product submanifolds of Kenmotsu manifolds

In the present paper, we study warped product semi-slant submanifolds of Kenmotsu manifolds. We have obtained results on the existence of warped product semi-slant submanifolds of Kenmotsu manifolds in term of the canonical structure .

Generalized inequalities of warped product submanifolds of nearly Kenmotsu $f$-manifolds

In the present paper, we discuss the non-trivial warped product pseudo slant submanifolds of type $M_{\bot }\times _{f}M_{θ}$ and $M_θ\times _{f}M_{\bot }$ of nearly Kenmotsu $f$-manifold $\overline{M}$. Firstly, we get some basic properties of these type warped product submanifolds. Then, we establish the general sharp inequalities for squared norm of second fundamental form for mixed totally geodesic warped product pseudo slant submanifolds of both cases, in terms of the warping function and the slant angle. Also the equality cases are verified. We show that some previous results are trivial…

从或到Kähler流形的共形逐点斜Riemann映射

Conformal pointwise slant Riemannian maps from or to Kähler manifolds

本文研究从或到Kähler流形的共形逐点斜Riemann映射，提供非平凡例子，推导重要结果，讨论分布的可积性与全测地性，并研究其同伦性、调和性条件及不等式。

金属黎曼流形中的扭曲积子流形

Warped product submanifolds in metallic Riemannian manifolds

该研究聚焦金属（黄金）黎曼流形中真扭曲积子流形的存在性，讨论三类扭曲积子流形的相关性质，并给出欧氏空间中的对应示例，丰富了黎曼几何子流形理论的研究内容。

关于 para-Kähler 流形中广义类 $\mathcal{P}\mathcal{R}$-warped 积子流形的研究

On the Generalized Class of $\mathcal{P}\mathcal{R}$-warped product submanifolds in para-Kähler Manifolds

该研究在 para-Kähler 流形中提出并刻画 $\mathcal{P}\mathcal{R}$-伪斜 warped 积子流形，证明其在真斜因子条件下的存在与不存在性，并提供基本示例。

逐点斜黎曼 submersion 沿垂直与水平分布的 Casorati 不等式

Casorati inequalities along vertical and horizontal distributions of pointwise slant riemannian submersions

该研究针对逐点斜黎曼 submersion 的垂直、水平分布建立最优 Casorati 不等式，界定其归一化标量曲率，特殊情形涵盖多种空间形式并给出几何特征，验证了结果的严格性。

Warped product submanifolds of Lorentzian paracosymplectic manifolds

In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\times_{f}M_{\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to $M$ is an usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifo…

Geometric inequalities on bi-warped product submanifolds of locally conformal almost cosymplectic manifolds

In this paper we present not only some properties related to bi-warped product submanifolds of locally conformal almost cosymplectic manifolds, but also we show how the squared norm of the second fundamental form and the bi-warped product's warping functions are related when the bi-warped product submanifold has a proper slant submanifold as a base or fiber.

Geometry of contact skew CR-warped product submanifolds of Sasakian manifolds

In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products. We establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. The equality case in the statement of the inequality is investigated and some applications of derived inequality are given. Furthermore, we provide a non-trivial example of such submanifolds.

Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds

In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type $N_θ\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}N_θ}$. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16, 18, 19, 24], while by a new technique, the second inequality is constructed to express th…

On a class of immersions between almost para-Hermitian manifolds

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an analogue of the idea of slant submanifold in almost Hermitian geometry. There are classified pointwise slant surfaces of four dimensional almost para-Hermitian manifold.

Another class of warped product skew CR-submanifolds of Kenmotsu manifolds

Recently, Naghi et al. \cite{NAGHI} studied warped product skew CR-submanifold of the form $M_1\times_fM_\bot$ of order $1$ of a Kenmotsu manifold $\bar{M}$ such that $M_1=M_T\times M_θ$, where $M_T$, $M_\bot$ and $M_θ$ are invariant, anti-invariant and proper slant submanifolds of $\bar{M}$. The present paper deals with the study of warped product submanifolds by interchanging the two factors $M_T$ and $M_\bot$, i.e, the warped products of the form $M_2\times_fM_T$ such that $M_2=M_\bot\times M_θ$. The existence of such warped product is ensured by an example and then we characterize such war…

Warped Product Manifolds 中的双调和曲线 $I\times_{f}M^{n}\left( c\right) $

Biharmonic Curves in Warped Product Manifolds $I\times_{f}M^{n}\left( c\right) $

研究双调和曲线在Warped Product Manifolds中的几何性质及构造方法

Geometry of $CRS$ bi-warped product submanifolds in Sasakian and cosymplectic manifolds

In this paper, we prove that there are no proper $CRS$ bi-warped product submanifolds other than contact CR-biwarped products in Sasakian manifolds. On the other hand, we prove that if $M$ is a $CRS$ bi-warped product of the form $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_θ$ in a cosymplectic manifold $\widetilde M$, then its second fundamental form $h$ satisfies the inequality: $$\|h\|^2\geq 2n_1\|\nabla(\ln f_1)\|^2+2n_2(1+2\cot^2θ)\|\nabla(\ln f_2)\|^2,$$ where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_θ$ are invariant, anti-invariant and proper pointwise slant submanifolds of $…

Sasakian流形的部分全实子流形

Partially Totally Real Submanifolds of Sasakian Manifolds

本文针对切于Reeb向量场的Sasakian流形子流形，发展了部分全实（PTR）子流形的对应框架，证明了几类特殊子流形属于该框架，研究了相关几何性质并给出了模型说明。

Riemannian manifolds of dimension 7 whose skew-symmetric curvature operator has constant eigenvalues

A Riemannian manifold is called IP, if the eigenvalues of its skew-symmetric curvature operator are pointwise constant. It was previously shown that for all n\ge 4, except n=7, any IP manifold either has constant curvature, or is a warped product, with some specific function, of a line and a space of constant curvature. We extend this result to the case n = 7, and also study 3-dimensional IP manifolds.