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Semiinvariant warped product submanifolds of cosymplectic manifolds
Journal of Inequalities and Applications volume 2012, Article number: 19 (2012)
Abstract
In this article, we obtain the necessary and sufficient conditions that the semiinvariant submanifold to be a locally warped product submanifold of invariant and antiinvariant 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.
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 CRsubmanifolds 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 CRsubmanifolds are simply CRproducts. Therefore, he considered the warped product CRsubmanifolds in the form of M = M_{ T } ×_{ λ } M_{⊥} which are known as CRwarped 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 CRsubmanifolds 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 semiinvariant submanifolds of cosymplectic manifolds and prove that a semiinvariant 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 semiinvariant 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 semiinvariant submanifolds in terms of the canonical structures T and F.
2 Preliminaries
A (2m + 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 1form η satisfying:
There always exists a Riemannian metric g on an almost contact manifold $\stackrel{\u0304}{M}$ satisfying the following conditions
where X, Y are vector fields on $\stackrel{\u0304}{M}$.
An almost contact structure (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] + 2dη ⊗ ξ = 0 on $\stackrel{\u0304}{M}$, where [ϕ, ϕ] is the Nijenhuis tensor of ϕ. Finally, the fundamental twoform Φ is defined by Φ(X, Y) = g(X, ϕY).
An almost contact metric structure ( 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
for any 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
Let 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
for each 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
where g denotes the Riemannian metric on $\stackrel{\u0304}{M}$ as well as on M. The mean curvature vector H on M is given by
where 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
For any X ∈ Γ(TM), we write
where TX and FX are the tangential and normal components of ϕX, respectively.
Similarly, for any V ∈ Γ(T^{⊥}M), we write
where 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.
A submanifold M of an almost contact metric manifold $\stackrel{\u0304}{M}$ is called a semiinvariant submanifold if there exist on M a differentiable distribution D whose orthogonal complementary distribution D^{⊥} is antiinvariant, i.e.,

(i)
TM = D ⊕ D^{⊥} ⊕ 〈ξ〉

(ii)
D is an invariant distribution

(iii)
D^{⊥} is an antiinvariant distribution i.e., ϕD^{⊥} ⊆ T^{⊥}M.
A semiinvariant submanifold is antiinvariant if D_{ x } = {0} and invariant if ${D}_{x}^{\perp}=\left\{0\right\}$ respectively, for every x ∈ M. It is a proper semiinvariant submanifold if neither D_{ x } = {0} nor ${D}_{x}^{\perp}=\left\{0\right\}$, for each x ∈ M.
Let M be a semiinvariant 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.
Let M be a proper semiinvariant submanifold of an almost contact metric manifold $\stackrel{\u0304}{M}$, then for any X ∈ Γ(TM), we have
where P_{1} and P_{2} are the orthogonal projections from TM to D and D^{⊥}, respectively. It follows immediately that
From (2.3), (2.5), (2.6), (2.8), and (2.9), we have
for any X, Y ∈ Γ(TM).
Definition 2.1 A semiinvariant submanifold M is said to be a locally semiinvariant 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 semiinvariant submanifolds
In the following section, we discuss some basic results on semiinvariant submanifolds of a cosymplectic manifold for later use. First, we obtain the integrability conditions of involved distributions in the definition of a semiinvariant submanifold and then we will see the geometric properties of their leaves.
Proposition 3.1 [9] Let M be a semiinvariant submanifold of a cosymplectic manifold then the antiinvariant distribution D^{⊥} is integrable.
Proposition 3.2 The invariant distribution D on a semiinvariant 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 semiinvariant 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
for any 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 semiinvariant 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 semiinvariant 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 semiinvariant submanifold M of a cosymplectic manifold $\stackrel{\u0304}{M}$ is locally a semiinvariant product if and only if
for any U, V ∈ Γ(TM).
Proof. If T is parallel then by (2.15), we have
for any 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 semiinvariant product.
Conversely, if M is locally a semiinvariant 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 Semiinvariant warped product submanifolds
Throughout this section, we denote N_{ T } and N_{⊥} the invariant and antiinvariant submanifolds of a cosymplectic manifold $\stackrel{\u0304}{M}$, respectively. The warped product semiinvariant 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 semiinvariant 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 antiinvariant submanifolds of an almost contact metric manifold $\stackrel{\u0304}{M}$, then by Theorem 2.1 (ii), we have
for X ∈ Γ(TN_{ T }) and Z ∈ Γ(TN_{⊥}). Then, from (2.10) and (4.1), we get
which proves the first part of the lemma. Now, for any 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
Using (2.13), the above equation reduced to
Using (4.1), the second term of right hand side is identically zero, then the above equation takes the form
Using (2.17), we obtain
That is,
This proves the lemma completely. ■
Theorem 4.1 A proper semiinvariant submanifold of a cosymplectic manifold $\stackrel{\u0304}{M}$ is locally a warped product semiinvariant 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 semiinvariant submanifold of a cosymplectic manifold $\stackrel{\u0304}{M}$, then from (2.10) and (2.13), we have
Again using (2.10) and (2.13), the above equation takes the form
Now, from Lemma 4.1, we have
and
Substituting these values in (4.3), we obtain
Conversely, suppose that M is a semiinvariant 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
for X ∈ Γ(D) and Z, W ∈ Γ(D^{⊥}). Using (2.3), (2.8), and (2.10), we obtain
That is,
Using cosymplectic character and (2.5), we derive
By (2.17), the above equation takes the form
Let us assume that N_{⊥} is a leaf of D^{⊥} and h' is the second fundamental form of the immersion of N_{⊥} into M, then
Using (4.4), we get
or,
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 semiinvariant 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 semiinvariant submanifold.
Theorem 4.2 A semiinvariant submanifold M of a cosymplectic manifold $\stackrel{\u0304}{M}$ is locally a warped product semiinvariant 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 semiinvariant 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
for any X ∈ Γ(D) and Z ∈ Γ(D^{⊥}). Now, by (2.13), we have
Again, using (2.13), the above equation takes the form
In view of (2.4), (2.5), and (2.16), the above equation reduced to
Taking the inner product with ϕW, for any W ∈ Γ(D^{⊥}), we obtain
Using (2.14), (2.16) and the fact that P_{1}U ∈ Γ(D) and P_{2}U ∈ Γ(D^{⊥}), for any U ∈ Γ(TM), then the above equation becomes
From (2.2), the above equation becomes
Using (2.5), we derive
Using the covariant differentiation property of 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
Again using (2.5), we arrive at
The first term of righthand side is zero by (4.1) and the fact that P_{1}V ∈ Γ(D) and W ∈ Γ(D^{⊥}), thus we obtain
Conversely, suppose that M is a semiinvariant submanifold of a cosymplectic manifold satisfying (4.5), then it is easy to see that
for each X, Y ∈ Γ(D) and W ∈ Γ(D^{⊥}). Thus, by (2.16) we obtain
Therefore by Propositions 3.2 and 3.3, the distribution D is integrable and its leaves are totally geodesic in M. Now for any Z ∈ Γ(D^{⊥}), by (4.5), we have
Using (2.16), we get
Let 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
Now for any Z, W ∈ Γ(D^{⊥}) and X ∈ Γ(D), by (2.3) and (2.5), we have
From (4.7), we obtain
Thus, by (4.6) and (4.8), we derive
Using (2.17), we obtain
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 semiinvariant 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
Using (4.1), we get
Equating the tangential components of (4.9), we get
Taking the inner product with W ∈ Γ(D^{⊥}), we obtain
or equivalently
Replacing X by ϕX, we obtain
which proves the part (i) of proposition. Now, for the second part comparing the normal components of (4.9), we get
or,
Taking the inner product with ϕ h(X, Z), we derive
which completes the proof. ■
Theorem 4.3. Let M = N_{ T } ×_{ λ } N_{⊥} be a warped product semiinvariant 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
or,
In view of Proposition 4.1 (i), we get
This verifies the assertion (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 $\stackrel{\u0304}{M}$. ■
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Acknowledgements
MAK was supported by the Research Grant 01361432S, Deanship of Scientific research, (University of Tabuk, K.S.A.) and SU was supported by the Grant RG117/10AFR (University of Malaya, Malaysia).
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Authors' contributions
MAK carried out the geometric properties of the leaves of the involved distributions and participated to find out the geometric properties of warped products. SU participated in the study of warped products and drafted the manuscript. RS participated in the proof reading of the manuscript. All authors read and approved the final manuscript.
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Keywords
 warped product
 semiinvariant submanifold
 canonical structure
 cosymplectic manifold