Contact CR-Warped Product Submanifolds in Sasakian Space Forms

Iordache, Mihai

doi:10.1007/s10711-004-5459-z

Cited by 55 publications

(44 citation statements)

References 13 publications

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“…He also established general sharp inequalities for CR-warped products in Kaehler manifolds. Motivated by Chen's papers, CR-warped product submanifolds have been studied in [3], [11], [13], [14] and [15].…”

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confidence: 99%

Warped product submanifolds of Kaehler manifolds with a slant factor

2009

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Abstract. Recently, we showed that there exist no warped product semi-slant submanifolds in Kaehler manifolds. On the other hand, Carriazo introduced anti-slant submanifolds as a particular class of bi-slant submanifolds. In this paper, we study such submanifolds in detail and show that they are useful to define a new kind of warped product submanifolds of Kaehler manifolds. In this direction, we obtain the existence of warped product hemi-slant (anti-slant) submanifolds with examples. We give a characterization theorem and establish an inequality for the squared norm of the second fundamental form in terms of the warping function for such submanifolds. The equality case is also considered.1. Introduction. CR-submanifolds of Kaehler manifolds were introduced by Bejancu [1] as a generalization of totally real submanifolds and holomorphic submanifolds. Since then, many papers have appeared on these submanifolds. Recently, Chen [7] (see also [8], [9]) studied warped product CR-submanifolds and showed that there exist no warped product CRsubmanifolds of the form M ⊥ × f M T such that M ⊥ is a totally real submanifold and M T is a holomorphic submanifold of a Kaehler manifold M . Then he introduced CR-warped product submanifolds as follows: A submanifold M of a Kaehler manifold M is called a CR-warped product if it is the warped product

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confidence: 99%

Warped product submanifolds of Kaehler manifolds with a slant factor

2009

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“…In particular, we know that the sphere

S^{2 m + 1}

is a Sasakian manifold of constant sectional curvature one (see ), then Theorem can be written as: Theorem Assume that

ℓ : M^{n} = M_{T}^{p} \times_{f} M_{θ}^{q} \to {double-struckS}^{2 m + 1}

is an isometric immersion from an n ‐dimensional warped product pointwise semi‐slant submanifold

M_{T}^{p} \times_{f} M_{θ}^{q}

into a sphere

S^{2 m + 1}

. Then (i)The squared norm of the second fundamental form of

M^{n}

is defined as

{| | h | |}^{2} \geq {2 q (| | \nabla λ | |}^{2} + p - Δ λ),

where p and q are the dimensions of the invariant

M_{T}^{p}

and the pointwise slant submanifold

M_{θ}^{q}

, respectively, and

λ = ln f

. (ii)Equality holds in if and only if

M_{T}^{p}

is totally geodesic and

M_{θ}^{q}

are totally umbilical submanifolds in

{\overset{M}{true}}^{2 m + 1} false(c false)

.…”

Section: Inequality For Warped Product Pointwise Semi‐slant Submanifomentioning

confidence: 99%

“…In particular, we know that the sphere 2 +1 is a Sasakian manifold of constant sectional curvature one (see [31,34]), then Theorem 4.2 can be written as:…”

Section: Inequalit Y For Warped Product Point Wise Semi-slant Submanimentioning

confidence: 99%

Geometric classification of warped products isometrically immersed into Sasakian space forms

Ali

Laurian-Ioan

2018

Mathematische Nachrichten

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The main objective of this paper is to study the warped product pointwise semi‐slant submanifolds which are isometrically immersed into Sasakian manifolds. First, we prove some characterizations results in terms of the shape operator, under which influence a pointwise semi‐slant submanifold of a Sasakian manifold can be reduced to a warped product submanifold. Then, we determine a geometric inequality for the second fundamental form regarding to intrinsic invariant and extrinsic invariant using the Gauss equation instead of the Codazzi equation. Evenmore, we give some applications of this inequality into Sasakian space forms, and we will investigate the status of equalities in the inequality. As a particular case, we provide numerous applications of the Green lemma, the Laplacian of warped functions and some partial differential equations. Some triviality results for connected, compact warped product pointwise semi‐slant submanifolds of Sasakian space form by means of Hamiltonian and the kinetic energy of warped function involving boundary conditions are established.

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“…The work of Chen is about the characterization of CR-warped products in Kaehler manifolds, and derives the inequality for the second fundamental form. In fact, distinct classes of warped product submanifolds of the different kinds of structures were studied by several geometers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Recently, Ali et al [15], established general inequalities for warped product pseudo-slants isometrically immersed in nearly Kenmotsu manifolds for mixed, totally geodesic submanifolds.…”

Section: Introductionmentioning

confidence: 99%

The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

et al. 2019

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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.

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Contact CR-Warped Product Submanifolds in Sasakian Space Forms

Cited by 55 publications

References 13 publications

Warped product submanifolds of Kaehler manifolds with a slant factor

Warped product submanifolds of Kaehler manifolds with a slant factor

Geometric classification of warped products isometrically immersed into Sasakian space forms

The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

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