# An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds

- Siraj Uddin
^{1}Email author and - Khalid Ali Khan
^{2}

**2012**:304

https://doi.org/10.1186/1029-242X-2012-304

© Uddin and Khan; licensee Springer 2012

**Received: **3 July 2012

**Accepted: **26 November 2012

**Published: **18 December 2012

## Abstract

Recently, Chen (Monatshefte Math. 133:177-195, 2001) established general sharp inequalities for CR-warped products in a Kaehler manifold. Afterward, Mihai obtained (Geom. Dedic. 109:165-173, 2004) the same inequalities for contact CR-warped product submanifolds of Sasakian space forms and derived some applications. In this paper, we obtain an inequality for the length of the second fundamental form of the warped product submanifold of a nearly cosymplectic manifold in terms of the warping function. The equality case is also discussed.

**MSC:**53C40, 53C42, 53B25.

## Keywords

## 1 Introduction

An almost contact metric structure $(\varphi ,\xi ,\eta ,g)$ satisfying $({\overline{\mathrm{\nabla}}}_{X}\varphi )X=0$ is called a nearly cosymplectic structure. If we consider ${S}^{5}$ as a totally geodesic hypersurface of ${S}^{6}$, then it is known that ${S}^{5}$ has a non-cosymplectic nearly cosymplectic structure. Almost contact manifolds with Killing structure tensors were defined in [1] as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also [2]). Later on, Blair and Showers [3] studied nearly cosymplectic structure $(\varphi ,\xi ,\eta ,g)$ on a manifold $\overline{M}$ with *η* closed from the topological viewpoint.

On the other hand, Chen [4] has introduced the notion of CR-warped product submanifolds in a Kaehler manifold. He has established a sharp relationship between the squared norm of the second fundamental form and the warping function. Later on, Mihai [5] studied contact CR-warped products and obtained the same inequality for contact CR-warped product submanifolds isometrically immersed in Sasakian space forms. Motivated by the studies of these authors, many articles dealing with the existence or non-existence of warped products in different settings have appeared; one of them is [3]. In this paper, we obtain an inequality for the length of the second fundamental form in terms of the warping function for contact CR-warped product submanifolds in a more general setting, *i.e.*, nearly cosymplectic manifold.

## 2 Preliminaries

*almost contact structure*if on $\overline{M}$ there exist a tensor field

*ϕ*of type $(1,1)$, a vector field

*ξ*and a 1-form

*η*satisfying [6]

*g*on $\overline{M}$ satisfying the following compatibility condition:

where *X* and *Y* are vector fields on $\overline{M}$ [6].

*normal*if the almost complex structure

*J*on the product manifold $\overline{M}\times \mathbb{R}$ given by

where *f* is a smooth function on $\overline{M}\times \mathbb{R}$, has no torsion, *i.e.*, *J* is integrable, the condition for normality in terms of *ϕ*, *ξ* and *η* is $[\varphi ,\varphi ]+2d\eta \otimes \xi =0$ on $\overline{M}$, where $[\varphi ,\varphi ]$ is the Nijenhuis tensor of *ϕ*. Finally, the *fundamental 2-form* Φ is defined by $\mathrm{\Phi}(X,Y)=g(X,\varphi Y)$.

*cosymplectic*if it is normal and both Φ and

*η*are closed [6]. The structure is said to be

*nearly cosymplectic*if

*ϕ*is Killing,

*i.e.*, if

for any *X*, *Y* tangent to $\overline{M}$, where $\overline{\mathrm{\nabla}}$ is the Riemannian connection of the metric *g* on $\overline{M}$. Equation (2.3) is equivalent to $({\overline{\mathrm{\nabla}}}_{X}\varphi )X=0$ for each vector field *X* tangent to $\overline{M}$. The structure is said to be *closely cosymplectic* if *ϕ* is Killing and *η* is closed. It is well known that an almost contact metric manifold is *cosymplectic* if and only if $\overline{\mathrm{\nabla}}\varphi $ vanishes identically, *i.e.*, $({\overline{\mathrm{\nabla}}}_{X}\varphi )Y=0$ and ${\overline{\mathrm{\nabla}}}_{X}\xi =0$.

**Proposition 2.1** [6]

*On a nearly cosymplectic manifold*, *the vector field* *ξ* *is Killing*.

From the above proposition, we have $g({\overline{\mathrm{\nabla}}}_{X}\xi ,X)=0$ for any vector field *X* tangent to $\overline{M}$, where $\overline{M}$ is a nearly cosymplectic manifold.

*M*be a submanifold of an almost contact metric manifold $\overline{M}$ with induced metric

*g*, and let ∇ and ${\mathrm{\nabla}}^{\perp}$ be the induced connections on the tangent bundle

*TM*and the normal bundle ${T}^{\perp}M$ of

*M*, respectively. Denote by $\mathcal{F}(M)$ the algebra of smooth functions on

*M*and by $\mathrm{\Gamma}(TM)$ the $\mathcal{F}(M)$-module of smooth sections of

*TM*over

*M*. Then the Gauss and Weingarten formulas are given by

*h*and ${A}_{N}$ are the second fundamental form and the shape operator (corresponding to the normal vector field

*N*), respectively, for the immersion of

*M*into $\overline{M}$. They are related as

where *g* denotes the Riemannian metric on $\overline{M}$ as well as induced on *M*.

where *TX* is the tangential component and *FX* is the normal component of *ϕX*.

A submanifold *M* tangent to the structure vector field *ξ* is said to be *invariant* (resp. *anti-invariant*) if $\varphi ({T}_{x}M)\subset {T}_{x}M$, $\mathrm{\forall}x\in M$ (resp. $\varphi ({T}_{x}M)\subset {T}_{x}^{\perp}M$, $\mathrm{\forall}x\in M$).

*M*tangent to the structure vector field

*ξ*of an almost contact metric manifold $\overline{M}$ is called a

*contact CR-submanifold*(or

*semi-invariant submanifold*) if there exists a pair of orthogonal differentiable distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp}$ on

*M*such that

- (i)
$TM=\mathcal{D}\oplus {\mathcal{D}}^{\perp}\oplus \u3008\xi \u3009$, where $\u3008\xi \u3009$ is the one-dimensional distribution spanned by

*ξ*; - (ii)
$\mathcal{D}$ is invariant under

*ϕ*,*i.e.*, $\varphi ({\mathcal{D}}_{x})\subseteq {\mathcal{D}}_{x}$, $\mathrm{\forall}x\in M$; - (iii)
${\mathcal{D}}^{\perp}$ is anti-invariant under

*ϕ*,*i.e.*, $\varphi ({\mathcal{D}}_{x}^{\perp})\subset {T}_{x}^{\perp}M$, $\mathrm{\forall}x\in M$.

*invariant*if ${\mathcal{D}}^{\perp}=\{0\}$ and

*anti-invariant*if $\mathcal{D}=\{0\}$, respectively. It is called a

*proper contact CR-submanifold*if neither $\mathcal{D}=\{0\}$ nor ${\mathcal{D}}^{\perp}=\{0\}$. Moreover, if

*μ*is the

*ϕ*-invariant subspace in the normal bundle ${T}^{\perp}M$, then in the case of a contact CR-submanifold, the normal bundle ${T}^{\perp}M$ can be decomposed as

The function *f* is called the *warping function* of the warped product [7]. A warped product ${N}_{1}{\times}_{f}{N}_{2}$ is said to be *trivial* if the warping function *f* is constant.

We recall the following general result on warped product manifolds for later use.

**Lemma 2.1**

*Let*$M={N}_{1}{\times}_{f}{N}_{2}$

*be a warped product manifold with the warping function*

*f*,

*then*

- (i)
${\mathrm{\nabla}}_{X}Y\in \mathrm{\Gamma}(T{N}_{1})$

*is the lift of*${\mathrm{\nabla}}_{X}Y$*on*${N}_{1}$, - (ii)
${\mathrm{\nabla}}_{X}Z={\mathrm{\nabla}}_{Z}X=(Xlnf)Z$,

- (iii)
${\mathrm{\nabla}}_{Z}W={\mathrm{\nabla}}_{Z}^{{N}_{2}}W-g(Z,W)\mathrm{\nabla}lnf$

*for each* $X,Y\in \mathrm{\Gamma}(T{N}_{1})$ *and* $Z,W\in \mathrm{\Gamma}(T{N}_{2})$, *where* $\mathrm{\nabla}lnf$ *is the gradient of the function* ln*f* *and* ∇ *and* ${\mathrm{\nabla}}^{{N}_{2}}$ *denote the Levi*-*Civita connections on* *M* *and* ${N}_{2}$, *respectively*.

## 3 Contact CR-warped product submanifolds

In this section, we consider the warped product submanifolds $M={N}_{1}{\times}_{f}{N}_{2}$ of a nearly cosymplectic manifold $\overline{M}$, where ${N}_{1}$ and ${N}_{2}$ are Riemannian submanifolds of $\overline{M}$. In the above product, if we assume ${N}_{1}={N}_{T}$ and ${N}_{2}={N}_{\perp}$, then the warped product of ${N}_{1}$ and ${N}_{2}$ becomes a contact CR-warped product. In this section, we discuss the contact CR-warped products and obtain an inequality for the squared norm of the second fundamental form. For the general case of warped product submanifolds of a nearly cosymplectic manifold, we have the following result.

**Theorem 3.1** [8]

*A warped product submanifold* $M={N}_{1}{\times}_{f}{N}_{2}$ *of a nearly cosymplectic manifold* $\overline{M}$ *is a usual Riemannian product if the structure vector field* *ξ* *is tangent to* ${N}_{2}$, *where* ${N}_{1}$ *and* ${N}_{2}$ *are the Riemannian submanifolds of* $\overline{M}$.

Now, we consider the warped product contact CR-submanifolds of the types $M={N}_{\perp}{\times}_{f}{N}_{T}$ and $M={N}_{T}{\times}_{f}{N}_{\perp}$ of a nearly cosymplectic manifold $\overline{M}$. In [8], the present author has proved that the warped product contact CR-submanifolds of the first type are usual Riemannian products of ${N}_{\perp}$ and ${N}_{T}$, where ${N}_{\perp}$ and ${N}_{T}$ are anti-invariant and invariant submanifolds of $\overline{M}$, respectively. In the following, we consider the contact CR-warped product submanifolds $M={N}_{T}{\times}_{f}{N}_{\perp}$ and obtain a general inequality. First, we have the following preparatory result for later use.

**Lemma 3.1** [8]

*Let*$M={N}_{T}{\times}_{f}{N}_{\perp}$

*be a contact CR*-

*warped product submanifold of a nearly cosymplectic manifold*$\overline{M}$.

*If*$X,Y\in \mathrm{\Gamma}(T{N}_{T})$

*and*$Z,W\in \mathrm{\Gamma}(T{N}_{\perp})$,

*then*

- (i)
$g(h(X,Y),FZ)=0$,

- (ii)
$g(h(\varphi X,Z),FZ)=(Xlnf){\parallel Z\parallel}^{2}$.

*X*by

*ϕX*in (ii) of Lemma 3.1, then we get

*m*and a smooth function

*f*on

*M*, we recall

- (i)∇
*f*, the gradient of*f*, is defined by$g(\mathrm{\nabla}f,X)=X(f),\phantom{\rule{1em}{0ex}}\mathrm{\forall}X\in \mathrm{\Gamma}(TM).$(3.3) - (ii)Δ
*f*, the Laplacian of*f*, is defined by$\mathrm{\Delta}f=\sum _{i=1}^{m}\{({\mathrm{\nabla}}_{{e}_{i}}{e}_{i})f-{e}_{i}{e}_{i}(f)\}=-div\mathrm{\nabla}f,$(3.4)

where ∇ is the Levi-Civita connection on *M* and $\{{e}_{1},\dots ,{e}_{m}\}$ is an orthonormal frame on *M*.

Now, we prove the main result of this section using the above results.

**Theorem 3.2**

*Let*$M={N}_{T}{\times}_{f}{N}_{\perp}$

*be a contact CR*-

*warped product submanifold of a nearly cosymplectic manifold*$\overline{M}$.

*Then we have*

- (i)
*The length of the second fundamental form of**M**satisfies the inequality*${\parallel h\parallel}^{2}\ge 2q{\parallel \mathrm{\nabla}lnf\parallel}^{2},$(3.6)

*where*

*q*

*is the dimension of*${N}_{\perp}$

*and*$\mathrm{\nabla}lnf$

*is the gradient of*ln

*f*.

- (ii)
*If the equality sign of*(3.6)*holds identically*,*then*${N}_{T}$*is a totally geodesic submanifold and*${N}_{\perp}$*is a totally umbilical submanifold of*$\overline{M}$.*Moreover*,*M**is a minimal submanifold of*$\overline{M}$.

*Proof*Let $\overline{M}$ be a $(2m+1)$-dimensional nearly cosymplectic manifold and $M={N}_{T}{\times}_{f}{N}_{\perp}$ be an

*n*-dimensional contact CR-warped product submanifolds of $\overline{M}$. Let us consider the $dim{N}_{T}=2p+1$ and $dim{N}_{\perp}=q$, then $n=2p+1+q$. Let $\{{e}_{1},\dots ,{e}_{p},\varphi {e}_{1}={e}_{p+1},\dots ,\varphi {e}_{p}={e}_{2p},{e}_{2p+1}=\xi \}$ and $\{{e}_{(2p+1)+1},\dots ,{e}_{n}\}$ be the local orthonormal frames on ${N}_{T}$ and ${N}_{\perp}$, respectively. Then the orthonormal frames in the normal bundle ${T}^{\perp}M$ of $F{\mathcal{D}}^{\perp}$ and

*μ*are $\{F{e}_{(2p+1)+1},\dots ,F{e}_{n}\}$ and $\{{e}_{n+q+1},\dots ,{e}_{2m+1}\}$, respectively. Then the length of the second fundamental form

*h*is defined as

*μ*-component. Here, we equate the $F{\mathcal{D}}^{\perp}$-component, then we have

*ξ*is tangent to ${N}_{T}$ and $\xi lnf=0$, the above equation can be written for the given frame of the distribution $\mathcal{D}$ as

*M*, then by (2.4), we have

As ${N}_{T}$ is a totally geodesic submanifold in *M* (by Lemma 2.1(i)), using this fact with the first part of (3.14), we get ${N}_{T}$ is totally geodesic in $\overline{M}$. On the other hand, the second condition of (3.14) with (3.13) implies that ${N}_{\perp}$ is totally umbilical in $\overline{M}$. Moreover, from (3.14), we get *M* is a minimal submanifold of $\overline{M}$. This proves the theorem completely. □

## Declarations

## Authors’ Affiliations

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