Bi-slant Submanifolds Research Articles

Generalized Wintgen inequality for submanifolds in generalized (κ, µ)-space forms

In this paper, we obtain a generalized Wintgen-type inequality for submanifolds in generalized (κ,µ)-space forms. Further, we discuss the inequality for bi-slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds and slant submanifolds in various ambient spaces. In particular, we generalize some recent results of Mihai, Tohoku Math. J. 69 (2017), and Aquib et al., Tamkang J. Math. 50 (2019) and Mathematics 7(12) (2019).

Warped product pointwise bi-slant submanifolds of kenmotsu manifolds

This 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.

An integral formula for bi-slant submanifolds in complex space forms

The present paper deals with bi-slant submanifolds in complex space forms. We obtain an integral formula of Simons' type for bi-slant submanifolds of a complex space form with positive constant holomorphic sectional curvature. Then, we apply it to prove our main result. We also discuss the same result in different submanifolds such as hemi-slant and semi-slant submanifolds.

Chen’s [formula omitted]-invariants type inequalities for bi-slant submanifolds in generalized Sasakian space forms

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 cases of derived inequality is given for C-totally real submanifolds and some other classes of submanifolds.

Almost bi-slant submanifolds of an almost contact metric manifold

In this paper we introduce and study the almost bi-slant submanifolds of an almost contact metric manifold. We give some characterization theorems for almost bi-slant submanifolds. Moreover, we obtain integrability conditions of the distributions which are involved in the definition of almost bi-slant submanifolds. We also get some results for totally geodesic and totally umbilical almost bi-slant submanifolds of cosymplectic manifolds and Sasakian manifolds.

A Note on Quasi bi-slant submanifolds of cosymplectic manifolds

The aim of the present paper is to define and study the notion of quasi bi-slant submanifolds of almost contact metric manifolds. We mainly concerned with quasi bi-slant submanifolds of cosymplectic manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant and quasi hemi-slant submanifolds. First, we give non-trivial examples in order to demostrate the method presented in this paper is effective and investigate the geometry of distributions. Moreover, We study these types of submanifolds with parallel canonical structures.

Warped product pointwise bi-slant submanifolds of trans-Sasakian manifold

The purpose of this paper is to study pointwise bi-slant submanifolds of trans-Sasakian manifold. Firstly, we obtain a non-trivial example of a pointwise bi-slant submanifolds of an almost contact metric manifold. Next we provide some fundamental results, including a characterization for warped product pointwise bi-slant submanifolds in trans-Sasakian manifold. Then we establish that there does not exist warped product pointwise bi-slant submanifold of trans-Sasakian manifold \tilde{M} under some certain considerations. Next, we consider that M is a proper pointwise bi-slant submanifold of a trans-Sasakian manifold \tilde{M} with pointwise slant distrbutions \mathcal{D}_1\oplus<\xi> and \mathcal{D}_2, then using Hiepko’s Theorem, M becomes a locally warped product submanifold of the form M_1\times_fM_2, where M_1 and M_2 are pointwise slant submanifolds with the slant angles \theta_1 and \theta_2 respectively. Later, we show that pointwise bi-slant submanifolds of trans-Sasakian manifold become Einstein manifolds admitting Ricci soliton and gradient Ricci soliton under some certain conditions..

Geometry of warped product pointwise submanifolds of Sasakian manifolds

Recently, Chen and Uddin introduced and studied warped product pointwise bi-slant submanifolds of K?hler manifolds in [13]. They have obtained many interesting results. In the present paper, we investigate warped product pointwise bi-slant submanifolds in Sasakian manifolds and we derive contact version of results obtain in [13]. We give a non-trivial example to prove the existence of these submanifolds.

The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.

Ricci curvature on warped product submanifolds of complex space forms and its applications

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

Bi-Slant Submanifolds of Para Hermitian Manifolds

In this paper, we introduce the notion of bi-slant submanifolds of a para Hermitian manifold. They naturally englobe CR, semi-slant, and hemi-slant submanifolds. We study their first properties and present a whole gallery of examples.

On warped product bi-slant submanifolds of Kenmotsu manifolds

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) of warped product bi-slant submanifolds of Kenmotsu manifolds in terms of the warping function (an intrinsic invariant) and bi-slant angles. The equality case is also considered. Some applications of derived inequality are given.

Optimal casorati inequalities on bi-slant submanifolds of generalized sasakian space forms

In this paper, we use T Oprea's optimization method to establish some optimal Casorati inequalities, which involve the normalized scalar curvature for bi-slant submanifolds of generalized Sasakian space forms. In the continuation, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical.

Geometric inequalities for warped product bi-slant submanifolds with a warping function

In this paper, we prove that the squared norm of the second fundamental form for bi-slant submanifolds with any codimension of nearly trans-Sasakian manifolds is bounded below by the gradient of a warping function and also find the conditions on which the equality holds. Some related examples are also provided.

Generalized Wintgen inequality for submanifolds in Kenmotsu space forms

In this paper, we obtain the generalized Wintgen inequality for Legendrian submanifolds in Kenmotsu space forms and discuss the equality case of the inequality. Further, we discuss the inequality for bi-slant submanifold in the same ambient space and derive its application in various slant cases.

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].

Lower extremities for generalized normalized δ-Casorati curvatures of bi-slant submanifolds in generalized complex space forms

In this paper, we obtain the inequalities for the generalized normalized δ-Casorati curvature and the normalized scalar curvature for different submanifolds in generalized complex space form, which is based on an optimization procedure involving a quadratic polynomial in the components of the second fundamental form and characterizes the submanifolds on which equalities hold. We also develop, the same inequalities for semi-slant, hemi-slant, CR, slant, invariant and anti-invariant submanifolds in the same target space and consider the equality case. Moreover, we obtain a geometric inequality involving Casorati curvature for warped product bi-slant submanifolds in same ambient and obtain an obstruction result.

Bounds for generalized normalized δ-Casorati curvatures for Bi-slant submanifolds in T-space forms

In this paper, we prove the inequality between the generalized normalized ?-Casorati curvatures and the normalized scalar curvature for the bi-slant submanifolds in T-space forms and consider the equality case of the inequality. We also develop same results for semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds, slant submanifolds, invariant and anti-invariant submanifolds in T-space forms.

Warped Product Bi-slant Immersions in Kaehler Manifolds

A submanifold M of an almost Hermitian manifold \((\widetilde{M},g,J)\) is called slant, if for each point \(p\in M\) and \(0\ne X\in T_p M\), the angle between JX and \(T_p M\) is constant (see Chen in Bull Aust Math Soc 41:135–147, 1990). Later, Carriazo (in: Proceedings of the ICRAMS 2000, Kharagpur, 2000) defined the notion of bi-slant immersions as an extension of slant immersions. In this paper, we study warped product bi-slant submanifolds in Kaehler manifolds and provide some examples of warped product bi-slant submanifolds in some complex Euclidean spaces. Our main theorem states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product or a warped product hemi-slant submanifold.

Warped product bi-slant submanifolds of cosymplectic manifolds

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.