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# Semi-invariant warped product submanifolds of cosymplectic manifolds

- Meraj Ali Khan
^{1}Email author, - Siraj Uddin
^{2}and - Rashmi Sachdeva
^{3}

**2012**:19

https://doi.org/10.1186/1029-242X-2012-19

© Khan et al.; licensee Springer. 2012

**Received:**4 October 2011**Accepted:**31 January 2012**Published:**31 January 2012

## Abstract

In this article, we obtain the necessary and sufficient conditions that the semi-invariant submanifold to be a locally warped product submanifold of invariant and anti-invariant submanifolds of a cosymplectic manifold in terms of canonical structures *T* and *F*. The inequality and equality cases are also discussed for the squared norm of the second fundamental form in terms of the warping function.

**2000 AMS Mathematics Subject Classification:** 53C25; 53C40; 53C42; 53D15.

## Keywords

- warped product
- semi-invariant submanifold
- canonical structure
- cosymplectic manifold

## 1 Introduction

Bishop and O'Neill [1] introduced the notion of warped product manifolds in order to construct a large variety of manifolds of negative curvature. Later on, the geometrical aspects of these manifolds have been studied by many researchers (c.f., [2–5]). The idea of warped product submanifolds was introduced by Chen [6]. He studied warped product CR-submanifolds of the form *M* = *M*_{⊥} ×_{
λ
} *M*_{
T
} such that *M*_{⊥} is a totally real submanifold and *M*_{
T
} is a holomorphic submanifold of a Kaehler manifold $\stackrel{\u0304}{M}$ and proved that warped product CR-submanifolds are simply CR-products. Therefore, he considered the warped product CR-submanifolds in the form of *M* = *M*_{
T
} ×_{
λ
} *M*_{⊥} which are known as CR-warped products where *M*_{
T
} and *M*_{⊥} are holomorphic and totally real submanifolds of a Kaehler manifold $\stackrel{\u0304}{M}$, respectively.

The warped product submanifolds of cosypmlectic manifolds was studied by Khan et.al [7]. Recently, Atçeken studied warped product CR-submanifolds of cosymplectic space form and obtained an inequality for the squared norm of the second fundamental form [2]. In this article, we obtain some basic results of semi-invariant submanifolds of cosymplectic manifolds and prove that a semi-invariant submanifold *M* of a cosymplectic manifold $\stackrel{\u0304}{M}$ is locally a Riemannian product if and only if the canonical structure *T* is parallel. The semi-invariant warped product submanifolds are the generalization of locally Riemannian product submanifolds, so it will be worthwhile to study warped product submanifolds in terms of canonical structures *T* and *F*, to this end we obtain some characterization results on the warped product semi-invariant submanifolds in terms of the canonical structures *T* and *F*.

## 2 Preliminaries

*m*+ 1)

*-*dimensional

*C*

^{∞}-manifold $\stackrel{\u0304}{M}$ is said to have an

*almost contact structure*if there exist on $\stackrel{\u0304}{M}$ a tensor field

*ϕ*of type (1, 1), a vector field

*ξ*and 1-form

*η*satisfying:

*g*on an almost contact manifold $\stackrel{\u0304}{M}$ satisfying the following conditions

where *X, Y* are vector fields on $\stackrel{\u0304}{M}$.

*ϕ, ξ, η*) is said to be

*normal*if the almost complex structure

*J*on the product manifold $\stackrel{\u0304}{M}\times R$ is given by

where *f* is the *C*^{∞} -function on $\stackrel{\u0304}{M}\times R$ has no torsion i.e., *J* is integrable. The condition for normality in terms of *ϕ, ξ*, and *η* is [*ϕ, ϕ*] + 2*dη* ⊗ *ξ* = 0 on $\stackrel{\u0304}{M}$, where [*ϕ, ϕ*] is the Nijenhuis tensor of *ϕ*. Finally, the fundamental two-form Φ is defined by Φ(*X, Y)* = *g*(*X, ϕY*).

*ϕ, ξ, η, g*) is said to be

*cosymplectic*, if it is normal and both Φ and

*η*are closed [8], and the structure equation of a cosymplectic manifold is given by

*X, Y*tangent to $\stackrel{\u0304}{M}$, where $\stackrel{\u0304}{\nabla}$ denotes the Riemannian connection of the metric

*g*on $\stackrel{\u0304}{M}$. Moreover, for cosymplectic manifold

*M*be a submanifold of an almost contact metric manifold $\stackrel{\u0304}{M}$ with induced metric

*g*and if ∇ and ∇

^{⊥}are the induced connections on the tangent bundle

*TM*and the normal bundle

*T*

^{⊥}

*M*of

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

*M*and by Γ(

*TM*) the $\mathcal{F}\left(M\right)$-module of smooth sections of a vector bundle

*TM*over

*M*, then the Gauss and Weingarten formulae are given by

*X, Y*∈ Γ(

*TM*) and

*V*∈ Γ(

*T*

^{⊥}

*M*), where

*h*and

*A*

_{ V }are the second fundamental form and the shape operator (corresponding to the normal vector field

*V*) respectively, for the immersion of

*M*into $\stackrel{\u0304}{M}$. They are related by

*g*denotes the Riemannian metric on $\stackrel{\u0304}{M}$ as well as on

*M*. The mean curvature vector

*H*on

*M*is given by

*n*is the dimension of

*M*and {

*e*

_{1},

*e*

_{2}, . . . ,

*e*

_{ n }} is a local orthonormal frame of vector fields on

*M*. The squared norm of the second fundamental form is defined as

*X*∈ Γ(

*TM*), we write

where *TX* and *FX* are the tangential and normal components of *ϕX*, respectively.

*V*∈ Γ(

*T*

^{⊥}

*M*), we write

*tV*is the tangential component and

*fV*is the normal component of

*ϕV*. The covariant derivatives of the tensors

*T*and

*F*are defined as

for all *X, Y* ∈ Γ(*TM*).

Let *M* be a Riemannian manifold isometrically immersed in an almost contact metric manifold $\stackrel{\u0304}{M}$, then for every *x* ∈ *M* there exist a maximal invariant subspace denoted by *D*_{
x
} of the tangent space *T*_{
x
}*M* of *M*. If the dimension of *D*_{
x
} is same for all values of *x* ∈ *M*, then *D*_{
x
} gives an invariant distribution *D* on *M*.

*M*of an almost contact metric manifold $\stackrel{\u0304}{M}$ is called a

*semi-invariant submanifold*if there exist on

*M*a differentiable distribution

*D*whose orthogonal complementary distribution

*D*

^{⊥}is anti-invariant, i.e.,

- (i)
*TM*=*D*⊕*D*^{⊥}⊕ 〈*ξ*〉 - (ii)
*D*is an invariant distribution - (iii)
*D*^{⊥}is an anti-invariant distribution i.e.,*ϕD*^{⊥}⊆*T*^{⊥}*M*.

A semi-invariant submanifold is *anti-invariant* if *D*_{
x
} = {0} and *invariant* if ${D}_{x}^{\perp}=\left\{0\right\}$ respectively, for every *x* ∈ *M*. It is a *proper semi-invariant submanifold* if neither *D*_{
x
} = {0} nor ${D}_{x}^{\perp}=\left\{0\right\}$, for each *x* ∈ *M*.

*M*be a semi-invariant submanifold of an almost contact metric manifold $\stackrel{\u0304}{M}$. Then,

*FT*

_{ x }

*M*is a subspace of ${T}_{x}^{\perp}M$ such that

where *ν* is the invariant subspace of *T*^{⊥}*M* under *ϕ*.

*M*be a proper semi-invariant submanifold of an almost contact metric manifold $\stackrel{\u0304}{M}$, then for any

*X*∈ Γ(

*TM*), we have

*P*

_{1}and

*P*

_{2}are the orthogonal projections from

*TM*to

*D*and

*D*

^{⊥}, respectively. It follows immediately that

for any *X, Y* ∈ Γ(*TM*).

**Definition 2.1** *A semi-invariant submanifold M is said to be a locally semi-invariant product submanifold if M is locally a Riemannian product of the leaves of distributions D, D*^{⊥}*, and* 〈*ξ*〉.

**Definition 2.2**

*Let*(

*N*

_{1},

*g*

_{1})

*and*(

*N*

_{2},

*g*

_{2})

*be two Riemannian manifolds with Riemannian metrics g*

_{1}

*and g*

_{2}

*, respectively, and λ be a positive differentiable function on N*

_{1}.

*Then the warped product of N*

_{1}

*and N*

_{2}

*is the Riemannian manifold*(

*N*

_{1}×

*N*

_{2},

*g*),

*where*

*The warped product manifold*(

*N*

_{1}×

*N*

_{2},

*g*)

*is denoted by N*

_{1}×

*λ N*

_{2}.

*If U is any vector field tangent to M*=

*N*

_{1}×

_{ λ }

*N*

_{2}

*at*(

*p, q*)

*, then*

*where π*_{1} *and π*_{2} *are the canonical projections of M onto N*_{1} *and N*_{2}*, respectively*.

Bishop and O'Neill [1] proved the following results:

**Theorem 2.1**

*Let M*=

*N*

_{1}×

_{ λ }

*N*

_{2}

*be a warped product manifold. If X, Y*∈ Γ(

*TN*

_{1})

*and Z, W*∈ Γ(

*TN*

_{2})

*, then*

- (i)
∇

_{ X }*Y*∈ Γ(*TN*_{1}) - (ii)
${\nabla}_{X}Z={\nabla}_{Z}X=\left(\frac{X\lambda}{\lambda}\right)Z,$

- (iii)
${\nabla}_{Z}W={\nabla}_{Z}^{{N}_{2}}W-\frac{g\left(Z,W\right)}{\lambda}\nabla \lambda $.

*where*${\nabla}^{{N}_{2}}$

*is the connection on N*

_{2}

*and*∇

*λ is the gradient of the function λ and is defined as*

*for each U* ∈ Γ(*TM*).

**Corollary 2. 1**

*On a warped product manifold M*=

*N*

_{1}×

_{ λ }

*N*

_{2},

*we have*

- (i)
*N*_{1}*is totally geodesic in M*, - (ii)
*N*_{2}*is totally umbilical in M*.

## 3 Some basic results on semi-invariant submanifolds

In the following section, we discuss some basic results on semi-invariant submanifolds of a cosymplectic manifold for later use. First, we obtain the integrability conditions of involved distributions in the definition of a semi-invariant submanifold and then we will see the geometric properties of their leaves.

**Proposition 3.1** [9] *Let M be a semi-invariant submanifold of a cosymplectic manifold then the anti-invariant distribution D*^{⊥} *is integrable*.

**Proposition 3.2**

*The invariant distribution D on a semi-invariant submanifold of a cosymplectic manifold is integrable if and only if*

*for each X, Y* ∈ Γ(*D*) *and Z* ∈ Γ(*D*^{⊥}).

*Proof*. The result can be obtained by making use of (2.2), (2.3), and (2.5). ■

**Proposition 3.3**

*If the invariant distribution D on a semi-invariant submanifold M of a cosymplectic manifold*$\stackrel{\u0304}{M}$

*is integrable, then its leaves are totally geodesic in M if and only if*

*for each U* ∈ Γ(*TM*) *and Y* ∈ Γ(*D*).

*Proof*. From (2.16), we obtain

*U*∈ Γ(

*TM*) and

*Y*∈ Γ(

*D*). Taking the inner product with

*ϕZ*for any

*Z*∈ Γ(

*D*

^{⊥}), we get

The result follows from the above equation. ■

Now, we have the following corollary for later use.

**Corollary 3.1**

*The invariant distribution D on a semi-invariant submanifold M of a cosymplectic manifold*$\stackrel{\u0304}{M}$

*is integrable and its leaves are totally geodesic in M if and only if*

*for any X, Y* ∈ Γ(*D*).

*Proof*. The result follows from (2.15) and Proposition 3.3. ■

**Lemma 3.1**

*For a semi-invariant submanifold M of a cosymplectic manifold*$\stackrel{\u0304}{M}$,

*the leaf N*

_{⊥}

*of D*

^{⊥}

*is totally geodesic in M if and only if*

*for any X* ∈ Γ(*D*) *and Z, W* ∈ Γ(*D*^{⊥}).

*Proof*. From (2.2), (2.3), (2.5), and (2.6), we obtain

Thus, the result follows from the above equation. ■

**Theorem 3.1**

*A semi-invariant submanifold M of a cosymplectic manifold*$\stackrel{\u0304}{M}$

*is locally a semi-invariant product if and only if*

*for any U, V* ∈ Γ(*TM*).

*Proof*. If

*T*is parallel then by (2.15), we have

*U, V*tangent to

*M*. In particular, if

*X*∈ Γ(

*D*), then (3.1) gives,

*th*(

*U, X*) = 0, that is,

for any *Z* ∈ Γ(*D*^{⊥}). Thus by Proposition 3.2 and Lemma 3.1, *D* is integrable and the leaf *N*_{⊥} of *D*^{⊥} is totally geodesic in *M*. Let *N*_{
T
} be a leaf of *D*, now for any *X, Y* ∈ Γ(*D*) and *Z* ∈ Γ(*D*^{⊥}) by (3.2), we obtain *g*(*A*_{
ϕZ
}*X, Y)* = 0 and using (2.2), (2.3), (2.5), and (2.6), we get *g*(∇_{
X
}*ϕY, Z*) = 0, which shows that leaf of *D* is totally geodesic in *M* and distribution 〈*ξ*〉 is already totally geodesic in *M* and hence *M* is locally a semi-invariant product.

Conversely, if *M* is locally a semi-invariant product then ∇_{
U
} *×* ∈ Γ(*D*) for any *X* ∈ Γ(*D*) and *U* ∈ Γ(*TM*), thus by (2.15) and the Proposition 3.3, we get $\left({\stackrel{\u0304}{\nabla}}_{U}T\right)Y=0$. Similarly, for any *Z* ∈ Γ(*D*^{⊥}) and *U* ∈ Γ(*TM*), we obtain ∇_{
U
} *Z* ∈ Γ(*D*^{⊥}) and then by (2.10), we get $\left({\stackrel{\u0304}{\nabla}}_{U}T\right)Z=0$ and it is easy to see that $\left({\stackrel{\u0304}{\nabla}}_{U}T\right)\xi =0$. By these observations we find that $\left({\stackrel{\u0304}{\nabla}}_{U}T\right)V=0$, for all *U, V* ∈ Γ(*TM*), this proves the theorem completely. ■

## 4 Semi-invariant warped product submanifolds

Throughout this section, we denote *N*_{
T
} and *N*_{⊥} the invariant and anti-invariant submanifolds of a cosymplectic manifold $\stackrel{\u0304}{M}$, respectively. The warped product semi-invariant submanifolds of a cosymplectic manifold $\stackrel{\u0304}{M}$ are denoted by *N*_{⊥} ×_{
λ
} *N*_{
T
} and *N*_{
T
} ×_{
λ
} *N*_{⊥}. The first type of warped products do not exist of a cosymplectic manifold in the sense of [5], here we discuss the second type of warped products and obtain some interesting results. First, we have the following lemma:

**Lemma 4.1**

*Let M*=

*N*

_{ T }×

_{ λ }

*N*

_{⊥}

*be a warped product semi-invariant submanifold of an almost contact metric manifold*$\stackrel{\u0304}{M}$.

*Then*

for any *X, Z*, and *U* tangent to *N*_{
T
}*, N*_{⊥}, and *M*, respectively.

*Proof*. Let

*M*=

*N*

_{ T }×

_{ λ }

*N*

_{⊥}be a warped product submanifold of invariant and anti-invariant submanifolds of an almost contact metric manifold $\stackrel{\u0304}{M}$, then by Theorem 2.1 (

*ii*), we have

*X*∈ Γ(

*TN*

_{ T }) and

*Z*∈ Γ(

*TN*

_{⊥}). Then, from (2.10) and (4.1), we get

*U*∈ Γ(

*TM*), we have

*TU*∈ Γ(

*TN*

_{ T }), therefore $\left({\stackrel{\u0304}{\nabla}}_{U}T\right)Z\in \Gamma \left(T{N}_{T}\right)$ for any

*U*∈ Γ(

*TM*). Furthermore, for any

*X*∈ Γ(

*TN*

_{ T }), we obtain

This proves the lemma completely. ■

**Theorem 4.1**

*A proper semi-invariant submanifold of a cosymplectic manifold*$\stackrel{\u0304}{M}$

*is locally a warped product semi-invariant submanifold if and only if*

*for each U, V* ∈ Γ(*TM*) *and μ, a C*^{∞} -*function on M satisfying W μ* = 0, *for each W* ∈ Γ(*D*^{⊥}).

*Proof*. Let

*M*=

*N*

_{ T }×

_{ λ }

*N*

_{⊥}be a warped product semi-invariant submanifold of a cosymplectic manifold $\stackrel{\u0304}{M}$, then from (2.10) and (2.13), we have

*M*is a semi-invariant submanifold of a cosymplectic manifold $\stackrel{\u0304}{M}$ and (4.2) holds, then $\left({\stackrel{\u0304}{\nabla}}_{X}T\right)Y=0$, for each

*X, Y*∈ Γ(

*D*). Then by Corollary 3.1,

*D*is integrable and each leave

*N*

_{ T }of

*D*is totally geodesic in

*M*. Moreover, from (4.2), we have

*X*∈ Γ(

*D*) and

*Z, W*∈ Γ(

*D*

^{⊥}). Using (2.3), (2.8), and (2.10), we obtain

*N*

_{⊥}is a leaf of

*D*

^{⊥}and

*h*' is the second fundamental form of the immersion of

*N*

_{⊥}into

*M*, then

This means that *N*_{⊥} is totally umbilical in *M* with non vanishing mean curvature ∇*μ*. Also, as *W μ* = 0, for all *W* ∈ Γ(*D*^{⊥}), i.e., the mean curvature vector of *N*_{⊥} is parallel and the leaves of *D*^{⊥} are extrinsic spheres in *M*. Hence from a result of Hiepko [10], the submanifold *M* is locally a warped product semi-invariant submanifold of *N*_{
T
} and *N*_{⊥} with warping function *λ* = *e*^{
μ
}. ■

**Note**. Theorem 4.1 is a generalization of Theorem 3.1, and shows that what is the effect on $\stackrel{\u0304}{\nabla}T$, when the submanifold is a warped product semi-invariant submanifold.

**Theorem 4.2**

*A semi-invariant submanifold M of a cosymplectic manifold*$\stackrel{\u0304}{M}$

*is locally a warped product semi-invariant submanifold if and only if*

*for U, V* ∈ Γ(*TM*) *and W* ∈ Γ(*D*^{⊥})*, where μ is a C*^{∞} -*function on M such that Z μ* = 0, *for all Z* ∈ *D*^{⊥}.

*Proof*. If

*M*=

*N*

_{ T }×

_{ λ }

*N*

_{⊥}is a warped product semi-invariant submanifold of a cosymplectic manifold $\stackrel{\u0304}{M}$, then

*N*

_{ T }and

*N*

_{⊥}are totally geodesic and totally umbilical in

*M*, respectively. Moreover, we have

*X*∈ Γ(

*D*) and

*Z*∈ Γ(

*D*

^{⊥}). Now, by (2.13), we have

*ϕW*, for any

*W*∈ Γ(

*D*

^{⊥}), we obtain

*P*

_{1}

*U*∈ Γ(

*D*) and

*P*

_{2}

*U*∈ Γ(

*D*

^{⊥}), for any

*U*∈ Γ(

*TM*), then the above equation becomes

*ϕ*and the fact that

*P*

_{1}

*V*∈ Γ(

*D*) and

*P*

_{2}

*V*∈ Γ(

*D*

^{⊥}), for any

*V*∈ Γ(

*TM*), then from (2.2), we obtain

*P*

_{1}

*V*∈ Γ(

*D*) and

*W*∈ Γ(

*D*

^{⊥}), thus we obtain

*M*is a semi-invariant submanifold of a cosymplectic manifold satisfying (4.5), then it is easy to see that

*X, Y*∈ Γ(

*D*) and

*W*∈ Γ(

*D*

^{⊥}). Thus, by (2.16) we obtain

*D*is integrable and its leaves are totally geodesic in

*M*. Now for any

*Z*∈ Γ(

*D*

^{⊥}), by (4.5), we have

*N*

_{⊥}be a leaf of

*D*

^{⊥}and

*h*' be the second fundamental form of the immersion of

*N*

_{⊥}into

*M*and ∇' is the induced connection on

*N*

_{⊥}, then by Gauss formula, we have

*Z, W*∈ Γ(

*D*

^{⊥}) and

*X*∈ Γ(

*D*), by (2.3) and (2.5), we have

which implies that *N*_{⊥} is totally umbilical in *M* with non vanishing mean curvature vector ∇*μ*. Moreover, as *Z μ* = 0 for all *Z* ∈ Γ(*D*^{⊥}) that is, the mean curvature is parallel on *N*^{⊥}, this show that *N*_{⊥} is extrinsic sphere. Hence, from a result of [10], *M* is locally a warped product submanifold. ■

**Proposition 4.1**.

*Let M*=

*N*

_{ T }×

_{ λ }

*N*

_{⊥}

*be a warped product semi-invariant submanifold of a cosymplectic manifold of*$\stackrel{\u0304}{M}$.

*Then*

- (i)
${h}_{\varphi {D}^{\perp}}\left(\varphi X,Z\right)=\left(Xln\text{\lambda}\right)\varphi Z$

- (ii)
*g*(*h*(*ϕX, Z*),*ϕh*(*X, Z*)) = ||*h*_{ ν }(*X, Z*)||^{2}

*for any ×* ∈ Γ(*D*) *and Z* ∈ Γ(*D*^{⊥}).

*Proof*. For any

*X*∈ Γ(

*D*) and

*Z*∈ Γ(

*D*

^{⊥}), by Gauss formula, we have

*W*∈ Γ(

*D*

^{⊥}), we obtain

*X*by

*ϕX*, we obtain

*ϕ h*(

*X, Z*), we derive

which completes the proof. ■

**Theorem 4.3**.

*Let M*=

*N*

_{ T }×

_{ λ }

*N*

_{⊥}

*be a warped product semi-invariant submanifold of a cosymplectic manifold*$\stackrel{\u0304}{M}$.

*Then*

- (i)
*The squared norm of the second fundamental form satisfies*$\parallel h{\parallel}^{2}\phantom{\rule{2.77695pt}{0ex}}\ge 2q\parallel \nabla ln\text{\lambda}{\parallel}^{2},$

*where*∇ ln λ

*is the gradient of the function*ln

*λ and q is the dimension of N*

_{⊥}.

- (ii)
*If the equality holds identically, then N*_{ T }*is a totally geodesic submanifold of*$\stackrel{\u0304}{M}$*, N*_{⊥}*is a totally umbilical submanifold of*$\stackrel{\u0304}{M}$ and*M is minimal*.

*Proof*. Let {

*X*

_{1},

*X*

_{2}, . . . ,

*X*

_{ p }

*, X*

_{p+1}=

*ϕX*

_{1}, . . . ,

*X*

_{2p}=

*ϕX*

_{ p }

*, X*

_{2p+1}=

*ξ*} be a local orthonormal frame of vector fields on

*N*

_{ T }and {

*Z*

_{1},

*Z*

_{2}, . . . ,

*Z*

_{ q }} a local orthonormal frame on

*N*

_{⊥}. Then by definition of squared norm of mean curvature vector

*i*). If the equality sign holds, then from (4.10) and Proposition 4.1 (i), we get

As *N*_{
T
} is a totally geodesic submanifold of *M*, the first condition of (4.11) implies that *N*_{
T
} is totally geodesic in $\stackrel{\u0304}{M}$. Moreover, *N*_{⊥} is totally umbilical in *M*, the second condition of (4.11) implies that *N*_{⊥} is totally umbilical in $\stackrel{\u0304}{M}$, and also it follows from (4.11) that *M* is minimal in $\stackrel{\u0304}{M}$. ■

## Declarations

### Acknowledgements

MAK was supported by the Research Grant 0136-1432-S, Deanship of Scientific research, (University of Tabuk, K.S.A.) and SU was supported by the Grant RG117/10AFR (University of Malaya, Malaysia).

## Authors’ Affiliations

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