Siraj Uddin

King Abdulaziz University · Department of Mathematics

This research investigates k-Almost Newton-Ricci solitons (k-ANRS) embedded in a metallic Riemannian manifold Mn having the potential function ψ. Furthermore, we prove geodesic and minimal conditions for hypersurfaces of metallic Riemannian manifolds. Beside this, we have explained some applications of metallic Riemannian manifold admitting k-Almos...

In this paper, we first define the notion of Lagrangian statistical submersion from a K\"ahler-like statistical manifold onto a statistical manifold. Then we prove that the horizontal distribution of a Lagrangian statistical submersion is integrable. Next, we establish Chen-Ricci inequality for Lagrangian statistical submersions from K\"ahler-like...

In this paper, we prove that every hemi-slant warped product submanifold of the form N θ × f N ⊥ in a nearly trans-Sasakian manifold M͠ satisfies the following inequality: ∥ h ∥ ² ≥ n 2 cot ² θ (∥∇̂(ln f )∥ ² – β ² ), whereas the warped product by reversing these two factors, i.e., N ⊥ × f N θ satisfying the inequality: $\begin{array}{} \displaysty...

In this paper, we first define the notion of Lagrangian statistical submersion from a Kahler-like statistical manifold onto a statistical manifold. Then we prove that the horizontal distribution of a Lagrangian statistical submersion is integrable. Next, we establish Chen-Ricci inequality for Lagrangian statistical submersions from Kahler-like stat...

Let \((M, g, {\nabla }^{(\alpha )})\) be a statistical manifold and \(g^\flat : TM \rightarrow T^*M\) be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the \(\alpha \)-vertical and \(\alpha \)-horizontal lifts on the tangent bundle of the statistical manifold M, we construct the g-\(\alpha \)-vertical and g-\(\alpha...

In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such submanifolds.

In this chapter, we survey important results on CR-products, CR-warped products, bi-slant warped products, hemi-slant warped products, semi-slant warped products, and CR-slant warped products in Kaehler and nearly Kaehler manifolds. In the last two sections, we present related results on slant submanifolds in generalized complex space forms and in...

The aim of this paper is to find some important classes of Einstein manifolds using conformal [Formula: see text]-Ricci solitons and conformal [Formula: see text]-Ricci almost solitons. We prove that a Kenmotsu metric as conformal [Formula: see text]-Ricci soliton is Einstein if it is [Formula: see text]-Einstein or the potential vector field [Form...

A bi-warped product of the form: $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_\theta$ in a contact metric manifold is called a CRS bi-warped product, where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_θ$ are invariant, anti-invariant and proper pointwise slant submanifolds, respectively. First, we prove that there are no proper CRS bi-warpe...

In this paper, we study slant helix using modified orthogonal frame in Minkowski space E31 with timelike, lightlike and spacelike axes. We also study a general slant helix with the Killing vector field axis. Furthermore, we give a non-trivial example and find the relations for curvature and torsion of f-biharmonic slant helix.

In this paper, we prove that every pointwise semi-slant warped product submanifold M = NT xf N? in a nearly Kenmotsu manifold ?M satisfies the following inequality: ||h||2 ? 2n2 (1 + 10/9 cot2?)(|| ??(lnf)||2-1), where n2 = dimN?, ??(ln f) is the gradient of ln f and ||h|| is the length of the second fundamental form of M. The equality and special...

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...

An isometric immersion f : Mn ? ?Mm from an n-dimensional Riemannian manifold Mn into an almost Hermitian manifold ?Mm of complex dimension m is called pointwise slant if its Wirtinger angles define a function defined on Mn. In this paper we establish the Existence and Uniqueness Theorems for pointwise slant immersions of Riemannian manifolds Mn in...

In this paper, we study the biharmonic submanifolds of Riemannian manifolds endowed with metallic and complex metallic structures. In case of both the structures, we obtain the necessary and sufficient conditions for a submanifold to be biharmonic. Particularly, we find the estimates for mean curvature of Lagrangian and complex surfaces.

Being motivated by a well-known Nash’s embedding theorem, Chen introduced a method to discover the relationship for the extrinsic invariants controlled by the intrinsic one. In this paper, we extend Chen’s inequality for the intrinsic and extrinsic invariants for pointwise bi-slant warped products in locally conformal Kaehler space forms with quart...

Chen–Ricci inequality is derived for CR-warped products in complex space forms, Theorem 4.1, involving an intrinsic invariant (Ricci curvature) controlled by extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, this inequality is applied to derive a necessary condition for the immersed subma...

In this paper, we establish optimal inequalities involving generalized δ-Casorati curvature δC (k; s − 1) for the bi-slant submanifolds of generalized Sasakian space forms endowed with a quarter-symmetric connection. The equality case is discussed for ideal submanifolds. Furthermore, the special case of derived inequality is given for C-totally rea...

In the present paper, we introduce left-invariant (almost) Kenmotsu structures on Hom-Lie groups (or, almost Kenmotsu Hom-Lie algebras). Also, we present examples of such structures. It is proved that if the Ricci tensor of Kenmotsu Hom-Lie algebras is η-parallel, then the scalar curvature is constant. We describe η-Einstein Kenmotsu Hom-Lie algebr...

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 manifolds. We give non-trivial examples of such submanifolds and obtain several fundament...

In this paper, we establish B.-Y. Chen?s optimal inequalities for statistical submanifolds involving Casorati curvature and the normalized scalar curavture in a statistical manifold of quasi constant curvature. The equality cases of these inequalities are also considered. Further, we provide some applications of our results. Moreover, as a new exam...

The geometry of slant submanifolds was initiated by Chen [15, 16] as a natural generalization of both holomorphic and totally real submanifolds. Since then, many geometers studied these submanifolds. A. Lotta defined and studied slant submanifolds in contact geometry [31, 32]. Papaghiuc [35] introduced semi-slant submanifolds. Cabrerizo et al. stud...

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...

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. Th...

We call a submanifold M of a Kaehler manifold $\tilde M$ a pointwise CR-Slant warped product if it is a warped product, B x_f N_θ of a CR-product B = N_T x N_⊥ and a proper pointwise slant submanifold N_θ with slant function θ, where N_T and N_⊥ are complex and totally real submanifolds of $\tilde M$. We prove that if a pointwise CR-Slant warped pr...

An isometric immersion f : M^n →M^m from an n-dimensional Riemannian manifold M^n into an almost Hermitian manifold M^m of complex dimension m is called pointwise slant if its Wirtinger angles define a function defined on M^n. In this paper we establish the existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds M^n...

Recently, B.-Y. Chen and O. J. Garay introduced the notion of pointwise slant submanifolds of almost Hermitian manifolds. In this paper, we introduce pointwise semi-slant submanifolds of locally product Riemannian manifolds. Using this notion, we investigate the geometry of warped product pointwise semi-slant submanifolds. We provide some non-trivi...

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 $M=M_T\times_{f_1} M_\perp\times_{f_2} M_\theta$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$||h|^2 ≥ 2p||grad(ln f_1)||^2+4q\left(1+{\smal...

In 2008, Chen and Dillen obtained a sharp estimation for the squared norm of the second fundamental form of multiply warped CR-submanifold \(M=M_1\times _{f_2}M_2\times \ldots \times _{f_k}M_k\) in an arbitrary Kähler manifold \({\tilde{M}}\) such that \(M_1\) is a holomorphic submanifold and \(M_\perp =_{f_2}M_2\times \cdots \times _{f_k}M_k\) is...

We show in this paper that many well-known theorems about the geometry of warped product submanifolds of Kaehler manifolds and itself nearly Kaehler manifolds can be generalized to CR-slant warped products in nearly Kaehler manifolds.

In this paper, we consider Mθ, a pointwise slant submanifold and prove that every bi-warped product M⊥×f1MT×f2Mθ in a locally product Riemannian manifold satisfies a general inequality: ‖σ‖2≥n2‖∇→T(lnf1)‖2+n3cos2θ‖∇→θ(lnf2)‖2,where n2=dim(MT),n3=dim(Mθ) and σ is the second fundamental form and ∇T(lnf1) and ∇θ(lnf2) are the gradient components along...

In this paper, we study a generalized class of warped product sub-manifolds of Sasakian manifolds. We establish B.-Y. Chen's first inequality for such warped products. The equality case is also discussed. Several non-trivial examples and applications are given. Mathematics Subject Classification (2010). 53C05;53C40; 53A40; 53C15.

In this paper, we introduce a new class of warped products, called generic warped product submanifolds in locally product Riemannian manifolds with pointwise slant fiber. We prove that every generic warped product submanifold B×fMθ in a locally product Riemannian manifold satisfies the following inequality: ‖h‖2≥s[cos2θ‖∇→⊥(lnf)‖2+2cscθ+cotθ2‖∇→T(l...

Recently, B.-Y. Chen estabished a general relationship between Ricci curvature and the mean curvature vector of a submanifold in Riemannian manifolds. Later, the same inequality was derived for other structures, but not for warped products. In this paper, we derive Chen-Ricci inequality for warped product semi-slant submanifolds in Kenmotsu space f...

Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form). In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant...

In this paper, we prove DDVV conjecture (the generalized Wintgen inequality) for statistical submanifolds of Kenmotsu statistical manifolds of constant φ-sectional curvature. Further, we give some applications of derived inequality.

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 $M=M_T\times_{f_1}\! M_\perp\times_{f_2}\! M_\theta$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$\|h\|^2\geq 2p\|\nabla (\ln f_1)\|^2+4q\le...

In this paper, we deal with the study of warped product semi-slant submanifolds isometrically immersed into a Kenmotsu manifold. We prove two characterization theorems for a warped product semi-slant submanifold in Kenmotsu manifolds in terms of the tensor fields.

In this paper, we prove DDVV conjecture (the generalized Wintgen inequality) for statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. Further, we give some applications of derived inequality.

In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products in Kenmotsu manifolds. We obtain a lower bound relationship between the squared norm of the second fundamental form and the warping function. Furthermore, the equality case is investigated and some applications of derived inequality are g...

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 in...

In this paper, we study bi-warped product submanifolds of the form: $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_\theta$ in a contact metric manifold $\widetilde M$, where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_\theta$ are invariant, anti-invariant and proper pointwise slant submanifolds of $\widetilde M$, respectively. We simply calle...

In this paper, we study bi-warped product submanifolds of the form M = M ×f1MT ×f2M⊥ in a Kenmotsu manifold. We obtain a lower bound for the squared norm of the second fundamental form of a bi-warped product submanifold such as h2 ≥ m 1csc2(1 +cos2)(|| Δ →(ln f 1) 2 - 1) + m 2cot2(|| Δ → (ln f 2) 2 - 1), where m1 =dim(MT) and m2 =dim(M⊥) and f1,f2...

In this paper, we have studied submanifolds especially, totally umbilical submanifolds of generalized \((k,\mu )\)-space-forms. We have found a necessary and sufficient condition for such submanifolds to be either invariant or anti-invariant. It is also shown that every totally umbilical submanifold of a generalized \((k,\mu )\)-space-form is a pse...

In this paper, we establish a sharp inequality for the squared norm of the second fundamental form of bi-warped product submanifolds of Kenmotsu manifolds. The equality case is also considered. We also provide a non-trivial example and some applications of derived inequality.

The article is concerned with the study of real hypersurfaces of the complex quadric Q m. We establish B. Y. Chen's inequalities for real hypersurfaces of the complex quadric Q m and by considering the equality case, we obtain some consequences. Also, we establish an inequality in terms of the warping function and the scalar curvature for a warped...

In this paper we introduce pointwise hemi 3-slant submanifolds of almost contact metric 3-structures. We characterize these submanifolds and give non-trivial examples of such submanifolds. In addition, we prove that the distribution spanned by the structure vector fields is totally geodesic and integrable. Moreover, we investigate the integrability...

In this study, we introduce a new class of pseudo f-structure, called hyperbolic f-structure. We give some classifications of this new structure. Further, we extend the notion of (κ, µ, ν)-nullity distribution to hyperbolic almost Kenmotsu f-manifolds. Finally, we construct some non-trivial examples of such manifolds.

In this paper, we study ϕ-recurrent almost cosymplec-tic (κ, µ)-space and prove that it is an η-Einstein manifold with constant coefficients. Next, we show that a three-dimensional locally ϕ-recurrent almost cosymplectic (κ, µ)-space is the space of constant curvature.

In this paper, we study pseudo-slant submanifolds and their warped products in Kenmotsu manifolds. We obtain the necessary conditions that a pseudoslant submanifold is locally a warped product and establish an inequality for the squared norm of the second fundamental form in terms of the warping function. The equality case is also considered.

In this paper, we study slant submanifolds of Riemannian manifolds with Golden structure. A Riemannian manifold $(\tilde{M},\tilde{g},{\varphi})$ is called a Golden Riemannian manifold if the $(1,1)$ tensor field ${\varphi}$ on $\tilde{M}$ is a golden structure, that is ${\varphi}^{2}={\varphi}+I$ and the metric $\tilde{g}$ is ${\varphi}-$ compatib...

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 [Geometry of warped product CR-submanifolds in Kaehler manifolds, I & II, Monat. Math. 133 (2001), 177-195 & 134 (2001), 10...

In [3], it was shown that there are no warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold of a warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function and show that there exists a class of non-trivial warped product submanifolds...

Recently, we have discussed the warped product pseudo-slant submanifolds of the type M θ × f M ⊥ of Kenmotsu manifolds. In this paper, we study other type of warped product pseudo-slant submanifolds by reversing these two factors in Kenmotsu manifolds. The existence of such warped product immersions is proved by a characterization. Also, we provide...

In this paper, we introduce the notion of warped product skew CR-submanifolds in Kenmotsu manifolds. We obtain several results on such submanifolds. A characterization for skew CR-submanifolds is obtained. Furthermore, we establish an inequality for the squared norm of the second fundamental form of a warped product skew CR-submanifold M 1 × f M ⊥...

Recently, B.-Y. Chen and O.J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. In this paper, first we study pointwise slant and pointwise pseudo-slant submanifolds of almost contact metric manifolds and then using this notion, we show that there exist a non-trivial class of warped product pointwise pseudo-slant submanifold...

In this paper, we study warped product bi-slant submanifolds of cosymplectic manifolds. It is shown that there is no proper warped product bi-slant submanifold other than pseudo-slant warped product. Finally, we give an example of warped product pseudo-slant submanifolds.

Recently, we have proved that there do not exist warped product semislant submanifolds of Sasakian manifolds other than contact CR-warped products which have been studied by Hasegawa and Mihai. In this paper, we introduce another class of submanifolds, called warped product pseudo-slant submanifolds. A characterization theorem for such immersions i...

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...

In this paper, we study semi-slant submanifolds and warped product semi-slant submanifolds of Kenmotsu manifolds, which have not been considered in earlier studies. A characterization and a general sharp inequality for the lower bound of the squared norm of the second fundamental form of such immersions are obtained. The necessary condition is obta...

B.-Y. Chen initiated the study of warped product submanifolds in his fundamental seminal papers [6, 7, 8]. 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 su...

Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in Sahin (Geom Dedic 117:195–202, 2006), it is interesting to find their existence in a more general setting, e.g., nearly Kaehler manifolds. In this paper, we obtain a necessary and sufficient condition for a semi-slant submanifold of a nearly Kaehler manifold...

The bundle of the 2-forms over a 6-dimensional base manifold decomposes to three subbundles such that Λ 2 (R 6) ≡ Λ 1 2 ⊕ Λ 6 2 ⊕ Λ 8 2 with dimensions 1, 6 and 8, respectively. A duality notion for the 2-forms called Φ-duality is given by equation η = λ * Φ (η ∧ Φ) and an anti self dual SU (3) Yang–Mills theory is studied on the subbundle Λ 6 2. T...

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....

A submanifold M of an almost Hermitian manifold (N, g, J) is called slant, if for every point p in M and 0 ≠ X in T_p M, the angle between JX and T_p M is constant [see Chen in Bull Aust Math Soc 41:135–147, 1990]. Later, Carriazo [Proceedings of the ICRAMS 2000, Kharagpur, 2000] defined the notion of bi-slant immersions as an extension of slant im...

It was shown in \cite{S1, S2} that there does not exist any warped product submanifold of a Kaehler manifold such that the spherical manifold of the warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function. We show that there exists a class of non-trivial warped product submanifolds...

In this paper, we introduce the notion of semi-invariant submanifolds of a normal almost paracontact manifold. We study their fundamental properties and the particular cases. The necessary and sufficient conditions are given for a submanifold to be invariant or anti-invariant. Also, we give some results for semi-invariant submanifolds of a normal a...

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant sub- manifolds of almost Hermitian manifolds. By using this notion, we investigate pointwise semi-slant submanifolds and their warped products in Sasakian manifolds. We give an example of such submanifolds and obtain several fundamental results, including a characterization for warped pro...

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise \phi-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some a...

In this paper, first we study pseudo-slant submanifolds of cosymplectic manifolds in detail and then we discuss their warped products. To give the answer of the question that: Is there any warped product submanifold of almost contact metric manifolds with slant factor? We study warped product pseudo-slant submanifolds of cosymplectic manifolds. We...

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 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 eve...

Recently, Chen established a relation for the squared norm second fundamental form of warped product immersion by using Codazzi equation. We establish a sharp inequality for a contact CR-warped product submanifold in a cosymplectic space form by using the Gauss equation. The equality case is also discussed.

In this paper, we study warped product pointwise semi-slant submanifolds of a Kaehler manifold. First, we prove some characterizations results in terms of the tensor fields T and F and then, we obtain a geometric inequality for the second fundamental form in terms of intrinsic invariants. Furthermore, the equality case is also discussed. Moreover,...

Non-existence of warped product semi-slant submanifolds of locally product Riemannian manifolds is proved in [3, 21]. In this paper , we study warped products of slant and anti-invariant submanifolds of a locally product Riemannian manifold and we prove the existence of such kind of warped products by a characterization. Also, we construct examples...

In this paper, first we study pseudo-slant submanifolds of cosymplectic manifolds in detail and then we discuss their warped products. To give the answer of the question that: Is there any warped product submanifold of almost contact metric manifolds with slant factor? We study warped product pseudo-slant submanifolds of cosymplectic manifolds. We...

Non-existence of warped product pseudo-slant submanifolds of nearly Kaehler manifolds was proved under some conditions in [16]. In this paper, we continue the study of such warped products for their existence. We characterise pseudo-slant submanifolds of a nearly Keahler manifold to be locally warped products. Also, we obtain a geometric inequality...

In the present paper, we show the existence of warped product semi-slant submanifolds in a Kenmotsu manifold by an example. We locally characterize the warped product semi-slant submanfiolds in a Kenmotsu manifold. Such submanifold does not exist in K¨ahler, Sasakian and cosymplectic manifolds. Further, we search some geometric properties to constr...

In the present paper, we study the extrinsic and intrinsic geometry of sub-manifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection. We deduce Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and show some applications of these equations. Finally, we give an example ve...

In [6], Cabras, Ianus and Pitis proved that in a cosymplectic manifold there does not exist any extrinsic sphere tangent to the structure vector field ξ. We consider the structure vector field ξ normal to the submanifold in the sense of Papaghiuc [12] and derive that a totally umbilical CR-submanifold of a cosymplectic manifold is either (i) totall...

In the present paper, we study the extrinsic and intrinsic geometry of submanifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection. We deduce Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and show some applications of these equations. Finally, we give an example ver...

In the present paper, we study totally umbilical submanifolds of cosym-plectic manifolds. We obtain a result on the classification of totally umbilical contact CR-submanifolds of a cosymplectic manifold.

In the present paper, we study totally umbilical submanifolds of cosym-plectic manifolds. We obtain a result on the classification of totally umbilical contact CR-submanifolds of a cosymplectic manifold.

In this paper, we prove the existence of warped product semi-slant submanifolds in a Kenmotsu manifold by its characterization. Also, we obtain an inequality for the length of second fundamental form in terms of intrinsic invariant such as the gradient of warping function and the slant angle. We discuss the equality case and our inequality generali...

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 $N_\theta\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}N_\theta}$. We establish...

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...

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...