Warped product semi-slant submanifolds in locally conformal Kaehler manifolds

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.

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

Proceedings of the International Geometry Center ◽

10.15673/tmgc.v11i3.1202 ◽

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}.

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

Advances in Mathematical Physics ◽

10.1155/2021/5900801 ◽

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.

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Some results on normal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500373 ◽

2019 ◽

Vol 16 (03) ◽

pp. 1950037

Author(s):

Megha ◽

Sangeet Kumar

Keyword(s):

Sufficient Conditions ◽

Characterization Theorem ◽

Kaehler Manifold ◽

Kaehler Manifolds ◽

Holomorphic Bisectional Curvature ◽

Necessary And Sufficient ◽

Nearly Kaehler Manifold ◽

Lightlike Submanifolds ◽

Lightlike Submanifold ◽

Normal Formula

The purpose of this paper is to study normal [Formula: see text]-lightlike submanifolds of indefinite nearly Kaehler manifolds. We find some necessary and sufficient conditions for an isometrically immersed [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal [Formula: see text]-lightlike submanifold. Further, we derive a characterization theorem for holomorphic bisectional curvature of a normal [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold.

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500312 ◽

2019 ◽

Vol 16 (02) ◽

pp. 1950031 ◽

Cited By ~ 7

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.

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Geometry of warped product pointwise semi-slant submanifolds of Kaehler manifolds

Filomat ◽

10.2298/fil1712771a ◽

2017 ◽

Vol 31 (12) ◽

pp. 3771-3788 ◽

Cited By ~ 9

Author(s):

Akram Ali ◽

Siraj Uddin ◽

Wan Othman

Keyword(s):

Warped Product ◽

Sufficient Conditions ◽

Second Fundamental Form ◽

Geometric Inequality ◽

Tensor Fields ◽

Space Forms ◽

Kaehler Manifolds ◽

Slant Submanifolds ◽

Necessary And Sufficient ◽

Complex Space Forms

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, we give some applications for Riemannian and compact Remannian submanifolds as well, i.e., we construct necessary and sufficient conditions for the non-existence of compact warped product pointwise semi-slant submanifold in complex space forms.

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On warped product semi-slant submanifolds of nearly Trans-Sasakian manifolds

Filomat ◽

10.2298/fil1817845a ◽

2018 ◽

Vol 32 (17) ◽

pp. 5845-5856

Author(s):

Akram Ali ◽

Ali Alkhaldi ◽

Rifaqat Ali

Keyword(s):

Warped Product ◽

Sufficient Conditions ◽

Characterization Theorem ◽

Sasakian Manifold ◽

Necessary And Sufficient Conditions ◽

Sasakian Manifolds ◽

Slant Submanifold ◽

Slant Submanifolds ◽

Necessary And Sufficient

In this paper, we study warped product semi-slant submanifold of type M = NT xf N? with slant fiber, isometrically immersed in a nearly Trans-Sasakian manifold by finding necessary and sufficient conditions in terms of Weingarten map. A characterization theorem is proved as main result.

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Hemi-slant submanifolds as warped products in a nearly Kaehler manifold

Mathematica Slovaca ◽

10.1515/ms-2017-0008 ◽

2017 ◽

Vol 67 (3) ◽

Author(s):

Viqar Azam Khan ◽

Kamran Khan

Keyword(s):

Present Article ◽

Warped Product ◽

Warped Products ◽

Kaehler Manifold ◽

Slant Submanifold ◽

Slant Submanifolds ◽

Nearly Kaehler Manifold

AbstractThe present article is devoted to the study of conditions on a hemi-slant submanifold of a nearly Kaehler manifold under which the submanifold is a warped product submanifold.

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Semi-invariant Riemannian submersions from nearly Kaehler manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501005 ◽

2020 ◽

Vol 17 (07) ◽

pp. 2050100

Author(s):

Rupali Kaushal ◽

Rashmi Sachdeva ◽

Rakesh Kumar ◽

Rakesh Kumar Nagaich

Keyword(s):

Sufficient Conditions ◽

Riemannian Submersion ◽

Horizontal Distribution ◽

Product Manifold ◽

Riemannian Submersions ◽

Kaehler Manifolds ◽

Necessary And Sufficient ◽

Nearly Kaehler Manifold ◽

The Vertical Distribution ◽

Usual Product

We study semi-invariant Riemannian submersions from a nearly Kaehler manifold to a Riemannian manifold. It is well known that the vertical distribution of a Riemannian submersion is always integrable therefore, we derive condition for the integrability of horizontal distribution of a semi-invariant Riemannian submersion and also investigate the geometry of the foliations. We discuss the existence and nonexistence of semi-invariant submersions such that the total manifold is a usual product manifold or a twisted product manifold. We establish necessary and sufficient conditions for a semi-invariant submersion to be a totally geodesic map. Finally, we study semi-invariant submersions with totally umbilical fibers.

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Integrability of distributions in CR-lightlike submanifolds

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.33.2002.209-222 ◽

2002 ◽

Vol 33 (3) ◽

pp. 209-222

Author(s):

Bayram Sahin ◽

Rifat Gunes

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Kaehler Manifold ◽

Kaehler Manifolds ◽

Necessary And Sufficient ◽

Lightlike Submanifolds ◽

Lightlike Submanifold

In this paper, we study CR-lighlike submanifolds of an indefinite Kaehler manifold. Integrability of distributions on CR-lightlike submanifold investigated. We give some necessary and sufficient conditions on integrability of distibutions on CR-lightlike submanifolds in an indefinite Kaehler manifolds.

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Some Result on Contact Pseudo-Slant Submanifolds of a Sasakian Manifold

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v17i0.8532 ◽

2019 ◽

Vol 17 ◽

pp. 414-425

Author(s):

Suleyman Dirik

Keyword(s):

Sufficient Conditions ◽

Sasakian Manifold ◽

Necessary And Sufficient Conditions ◽

Slant Submanifold ◽

Slant Submanifolds ◽

Integrability Conditions ◽

Definition Of ◽

Necessary And Sufficient

In this paper, we study the geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. We derive the integrability conditions of distributions in the definition of a contact pseudo-slant submanifold. The notions contact pseudo-slant product is defined, and the necessary and sufficient conditions for a submanifold to contact pseudo-slant product is given. Also, a non-trivial example is used to demonstrate that the method presented in this paper is effective.

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