Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber

2019 ◽  
Vol 16 (02) ◽  
pp. 1950031 ◽  
Author(s):  
Akram Ali ◽  
Jae Won Lee ◽  
Ali H. Alkhaldi
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Riemannian Product ◽  
Slant Angle ◽  
Kaehler Manifolds ◽  
Nearly Kaehler Manifold ◽  
Totally Real

There are two types of warped product pseudo-slant submanifolds, [Formula: see text] and [Formula: see text], in a nearly Kaehler manifold. We derive an optimization for an extrinsic invariant, the squared norm of second fundamental form, on a nontrivial warped product pseudo-slant submanifold [Formula: see text] in a nearly Kaehler manifold in terms of a warping function and a slant angle when the fiber [Formula: see text] is a slant submanifold. Moreover, the equality is verified for depending on what [Formula: see text] and [Formula: see text] are, and also we show that if the equality holds, then [Formula: see text] is a simply Riemannian product. As applications, we prove that the warped product pseudo-slant submanifold has the finite Kinetic energy if and only if [Formula: see text] is a totally real warped product submanifold.

scholarly journals Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Warped Product ◽  
Space Form ◽  
Sufficient Conditions ◽  
Hermitian Manifold ◽  
Second Fundamental Form ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Locally Conformal

In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}. In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}. Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form. In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold. Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}. In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form. Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}. Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$ satisfy some special equations, see Theorem~\ref{th5.2}.

scholarly journals Some characterizing results for hemi-slant warped product submanifolds of a Kaehler manifold

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3615-3625
Author(s):  
Kamran Khan ◽  
Viqar Khan
Keyword(s):  
Warped Product ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Riemannian Product ◽  
Covariant Derivatives ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real ◽  
Derivatives Of ◽  
Structure Tensors

Many differential geometric properties of submanifolds of a Kaehler manifold are looked into via canonical structure tensors P and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product (i.e. locally a Riemannian product of a holomorphic and a totally real submanifold) if and only if the canonical tensor P is parallel on the submanifold. Since, warped product manifolds are generalized version of Riemannian product of manifolds, in this article, we consider the covariant derivatives of the structure tensors on a hemi-slant submanifold of a Kaehler manifold. Our investigations have led us to characterize hemi-slant warped product submanifolds.

scholarly journals Warped product semi-slant submanifolds in locally conformal Kaehler manifolds

2017 ◽  
Vol 10 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Warped Product ◽  
Sufficient Conditions ◽  
Hermitian Manifold ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Nearly Kaehler Manifold

In 1994, in [13], N. Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of CR- and slant-submanifolds. In particular, he considered this submanifold in Kaehlerian manifolds, [13]. Then, in 2007, V. A. Khan and M. A. Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, [11]. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and gave a necessary and sufficient conditions for two distributions (holomorphic and slant) to be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form, [4]. In this paper, we define 2-kind warped product semi-slant submanifolds in a locally conformal Kaehler manifold and consider some properties of these submanifolds.

scholarly journals Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Yanlin Li ◽  
Ali H. Alkhaldi ◽  
Akram Ali
Keyword(s):  
Warped Product ◽  
Complex Space ◽  
Slant Submanifold ◽  
Codazzi Equation ◽  
Space Forms ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Generalized Complex ◽  
Complex Space Forms ◽  
Nearly Kaehler Manifold

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

scholarly journals On warped product bi-slant submanifolds of Kenmotsu manifolds

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Siraj Uddin ◽  
Ion Mihai ◽  
Adela Mihai
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Equality Case ◽  
General Inequality ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
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.

scholarly journals Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 244
Author(s):  
Ali H. Alkhaldi ◽  
Pişcoran Laurian-Ioan ◽  
Izhar Ahmad ◽  
Akram Ali
Keyword(s):  
Fundamental Form ◽  
Complex Projective Space ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Euclidean Sphere ◽  
Slant Submanifold ◽  
The Second Fundamental Form ◽  
Complex Projective Spaces ◽  
Complex Projective

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

scholarly journals Chen’s improved inequality for pointwise hemi-slant warped products in Kaehler manifolds

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 807-814
Author(s):  
Monia Naghi ◽  
Mica Stankovic ◽  
Fatimah Alghamdi
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Warped Products ◽  
Equality Case ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
The Second Fundamental Form

Recently, B.-Y. Chen discovered a technique to find the relation between second fundamental form and the warping function of warped product submanifolds. In this paper, we extend our further study of [24] by giving non-trivial examples of warped product pointwise hemi-slant submanifolds. Finally, we establish a sharp estimation for the squared norm of the second fundamental form ||h||2 in terms of the warping function f. The equality case is also investigated.

scholarly journals The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 162 ◽  
Author(s):  
Rifaqat Ali ◽  
Ali Alkhaldi ◽  
Akram Ali ◽  
Wan Othman
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Symmetric Bilinear Form ◽  
Warping Function ◽  
First Eigenvalue ◽  
Slant Submanifold ◽  
Zero Eigenvalue ◽  
Eigenvalue Estimates ◽  
The Second Fundamental Form ◽  
Nearly Cosymplectic

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type M = M ⊥ × f M θ , in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact.

Biwarped product submanifolds of complex space forms

2019 ◽  
Vol 16 (05) ◽  
pp. 1950072 ◽  
Author(s):  
Meraj Ali Khan ◽  
Kamran Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Complex Space ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Space Forms ◽  
Kaehler Manifolds ◽  
The Second Fundamental Form ◽  
Complex Space Forms ◽  
Special Case

The class of biwarped product manifolds is a generalized class of product manifolds and a special case of multiply warped product manifolds. In this paper, biwarped product submanifolds of the type [Formula: see text] embedded in the complex space forms are studied. Some characterizing inequalities for the existence of such type of submanifolds are derived. Moreover, we also estimate the squared norm of the second fundamental form in terms of the warping function and the slant function. This inequality generalizes the result obtained by Chen in [B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math. 133 (2001) 177–195]. By the application of derived inequality, we compute the Dirichlet energies of the warping functions involved. A nontrivial example of these warped product submanifolds is provided.

scholarly journals Hemi-Slant Warped Product Submanifolds of Nearly Kaehler Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Falleh R. Al-Solamy ◽  
Meraj Ali Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Kaehler Manifold ◽  
Equality Case ◽  
Kaehler Manifolds ◽  
Ambient Manifold

Hemi-slant warped product submanifolds of nearly Kaehler manifolds are studied and some interesting results are obtained. Moreover, an inequality is established for squared norm of second fundamental form and equality case is also discussed. The results obtained are also true if ambient manifold is replaced by a Kaehler manifold. These results generalize several known results in the literature.

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