arXiv subjects

Siraj Uddin

Publications and source records attributed to Siraj Uddin.

17 recordsLinked to original sources

arXiv2021

Warped Product Pointwise Semi-slant Submanifolds of Almost Contact Manifolds

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds. We prove that there exist no proper pointwise semi-slant warped product submanifold other than contact CR-warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds in cosypmlectic manifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds.

math.DG↗

arXiv2020

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 $\widetilde M$, respectively, and $\nabla(\ln f_1)$ and $\nabla(\ln f_2)$ denote the gradients of $\ln f_{1}$ and $\ln f_{2}$, respectively. Several applications of this inequality are given. At the end, we provide a non-trivial example of bi-warped products satisfying the equality case.

math.DG↗

arXiv2020

Geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds

In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped products are also proved in this paper.

math.DG↗

arXiv2020

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.

math.DG↗

arXiv2020

B.-Y. Chen's inequality for $CR$-warped products in a locally conformal Kaehler space form

n this paper, we obtain a geometric inequality between the length of the second fundamental form and the length of Lee form in terms of the warping function for a CR-warped product submanifold in a locally conformal Kaehler space form. The equality case is also investigated. Furthermore, the inequality is discussed for the important subclass of locally conformal Kaehler manifolds i.e., Vaisman manifold.

math.DG↗

arXiv2020

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 $c$.

math.DG↗

arXiv2019

Bi-warped product submanifolds of nearly Kaehler manifolds

We study bi-warped product submanifolds of nearly Kaehler manifolds which are the natural extension of warped products. We prove that every bi-warped product submanifold of the form $M=M_T\times_{f_1}\! M_\perp\times_{f_2}\! M_θ$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$\|h\|^2\geq 2p\|\nabla (\ln f_1)\|^2+4q\left(1+{\small \frac{10}{9}}\cot^2θ\right)\|\nabla(\ln f_2)\|^2,$$ where $p=\dim M_\perp$, $q=\frac{1}{2}\dim M_θ$, and $f_1,\,f_2$ are smooth positive functions on $M_T$. We also investigate the equality case of this inequality. Further, some applications of this inequality are also given.

math.DG↗

arXiv2018

Slant submanifolds of Golden Riemannian manifolds

In this paper, we study slant submanifolds of Riemannian manifolds with Golden structure. A Riemannian manifold $(\tilde{M},\tilde{g},φ)$ is called a Golden Riemannian manifold if the $(1,1)$ tensor field $φ$ on $\tilde{M}$ is a golden structure, that is $φ^{2}=φ+I$ and the metric $\tilde{g}$ is $φ-$ compatible. First, we get some new results for submanifolds of a Riemannian manifold with Golden structure. Later we characterize slant submanifolds of a Riemannian manifold with Golden structure and provide some non-trivial examples of slant submanifolds of Golden Riemannian manifolds.

math.DG↗

arXiv2017

Warped Product Pointwise Bi-slant Submanifolds of Kaehler Manifolds

Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds from extrinsic point of view was initiated by the first author around the beginning of this century in [7, 8]. Since then the study of warped product submanifolds has been investigated by many geometers. The notion of slant submanifolds of almost Hermitian manifolds was introduced in [5]. Bi-slant submanifolds in almost contact metric manifolds were defined in [4] by J. L. Cabrerizo et al. In [26], we studied bi-slant submanifolds and warped product bi-slant submanifolds in Kaehler manifolds. In this article, we investigate warped product pointwise bi-slant submanifolds of Kaehler manifolds. Our main results extend several important results on warped product slant submanifolds obtained in [7, 21-23, 27].

math.DG↗

arXiv2017

Invariant submanifolds of generalized Sasakian-space-forms

The present paper deals with the study of invariant submanifolds of generalized Sasakian-space-forms with respect to Levi-Civita connection as well as semi-symmetric metric connection. We provide some examples of such submanifolds and obtain many new results including, the necessary and sufficient conditions under which the submanifolds are totally geodesic. The Ricci solitons of such submanifolds are also studied.

math.DG↗

arXiv2017

A general inequality for contact CR-warped product submanifolds in cosymplectic space forms

B.-Y. Chen initiated the study of warped product submanifolds in his fundamental seminal papers \cite{C1,C2,C2.1}. In this paper, we study contact CR-warped product submanifolds of cosymplectic space forms and prove an optimal inequality by using Gauss and Codazzi equations. In addition, we obtain two geometric inequalities for contact CR-warped product submanifolds with a compact invariant factor.

math.DG↗

arXiv2017

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.

math.DG↗

arXiv2014

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 extrinsic invariants. The equality case is also considered.

math.DG↗

arXiv2014

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 the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.

math.DG↗

arXiv2014

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.

math.DG↗

arXiv2014

Classification of totally umbilical slant submanifolds of a Kenmotsu manifold

The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold $M$ of a Kenmotsu manifold $\bar M$ is either invariant or anti-invariant or $dim M=1$ or the mean curvature vector $H$ of $M$ lies in the invariant normal subbundle. Moreover, we find with an example that every totally umbilical proper slant submanifold is totally geodesic.

math.DG↗

arXiv2012

Slant submanifolds of lorentzian almost contact manifolds

In this paper we study slant submanifolds of Lorentzian almost contact manifolds. We have taken the submanifold as a space like and then defined the slant angle on a submanifold and thus we extended the results of A. Lotta (Slant submanifolds in contact geometry [8]) and M. A. Khan et. al. (Slant submanifolds of Lorentzian Paracontact manifolds [7]) in this new setting.

math.DG↗