Semi-invariant warped product submanifolds of cosymplectic manifolds

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.

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 $\bar{M}$ 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 $\bar{M}$, 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 $\bar{M}$ 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

A (2m + 1)- dimensional C^∞-manifold $\bar{M}$ is said to have an almost contact structure if there exist on $\bar{M}$ a tensor field ϕ of type (1, 1), a vector field ξ and 1-form η satisfying:

$${\varphi }^2=-I+\eta \otimes \xi ,\varphi \left(\xi \right)=0,\eta \circ \varphi =0,\eta \left(\xi \right)=1.$$

(2.1)

There always exists a Riemannian metric g on an almost contact manifold $\bar{M}$ satisfying the following conditions

$$g\left(\varphi X,\varphi Y\right)=g\left(X,Y\right)-\eta \left(X\right)\eta \left(Y\right),\eta \left(X\right)=g\left(X,\xi \right)$$

(2.2)

where X, Y are vector fields on $\bar{M}$.

An almost contact structure (ϕ, ξ, η) is said to be normal if the almost complex structure J on the product manifold $\bar{M}\times R$ is given by

$$J\left(X,f\frac{d}{dt}\right)=\left(\varphi X-f\xi ,\eta \left(X\right)\frac{d}{dt}\right)$$

where f is the C^∞ -function on $\bar{M}\times R$ 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 $\bar{M}$, where [ϕ, ϕ] is the Nijenhuis tensor of ϕ. Finally, the fundamental two-form Φ is defined by Φ(X, Y) = g(X, ϕY).

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

$$\left({\bar{\nabla }}_X\varphi \right)Y=0$$

(2.3)

for any X, Y tangent to $\bar{M}$, where $\bar{\nabla }$ denotes the Riemannian connection of the metric g on $\bar{M}$. Moreover, for cosymplectic manifold

$${\bar{\nabla }}_X\xi =0.$$

(2.4)

Let M be a submanifold of an almost contact metric manifold $\bar{M}$ 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 $ℱ\left(M\right)$ the algebra of smooth functions on M and by Γ(TM) the $ℱ\left(M\right)$-module of smooth sections of a vector bundle TM over M, then the Gauss and Weingarten formulae are given by

$${\bar{\nabla }}_{X}Y={\nabla }_{X}Y+h\left(X,Y\right)$$

(2.5)

$${\bar{\nabla }}_{X}V=-A_{V}X+{\nabla }_X^{\perp }V$$

(2.6)

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 $\bar{M}$. They are related by

$$g\left(h\left(X,Y\right),V\right)=g\left(A_{V}X,Y\right),$$

(2.7)

where g denotes the Riemannian metric on $\bar{M}$ as well as on M. The mean curvature vector H on M is given by

$$H=\frac{1}{n}\sum _{i=1}^{n}h\left(e_i,e_i\right)$$

where n is the dimension of M and {e₁, e₂, . . . , e _n } is a local orthonormal frame of vector fields on M. The squared norm of the second fundamental form is defined as

$${\parallel h\parallel }^2=\sum _{i,j=1}^{n}g\left(h\left(e_i,e_j\right),h\left(e_i,e_j\right)\right).$$

For any X ∈ Γ(TM), we write

$$\varphi X=TX+FX,$$

(2.8)

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

Similarly, for any V ∈ Γ(T^⊥M), we write

$$\varphi V=tV+fV,$$

(2.9)

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

$$\left({\bar{\nabla }}_{X}T\right)Y={\nabla }_{X}TY-T{\nabla }_{X}Y$$

(2.10)

$$\left({\bar{\nabla }}_{X}F\right)Y={\nabla }_X^{\perp }FY-F{\nabla }_{X}Y$$

(2.11)

for all X, Y ∈ Γ(TM).

Let M be a Riemannian manifold isometrically immersed in an almost contact metric manifold $\bar{M}$, 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 $\bar{M}$ 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.

Let M be a semi-invariant submanifold of an almost contact metric manifold $\bar{M}$. Then, FT _x M is a subspace of $T_x^{\perp }M$ such that

$$T_x^{\perp }M=F{T}_{x}M\oplus {\nu }_x$$

(2.12)

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

Let M be a proper semi-invariant submanifold of an almost contact metric manifold $\bar{M}$, then for any X ∈ Γ(TM), we have

$$X=P_{1}X+P_{2}X+\eta \left(X\right)\xi ,$$

(2.13)

where P₁ and P₂ are the orthogonal projections from TM to D and D^⊥, respectively. It follows immediately that

$$\left(a\right)T{P}_2=0,\left(b\right)F{P}_1=0,\left(c\right)t\left(T^{\perp }M\right)=D^{\perp },\left(d\right)f{T}^{\perp }M\subseteq \nu .$$

(2.14)

From (2.3), (2.5), (2.6), (2.8), and (2.9), we have

$$\left({\bar{\nabla }}_{X}T\right)Y=A_{FY}X+th\left(X,Y\right)$$

(2.15)

$$\left({\bar{\nabla }}_{X}F\right)Y=fh\left(X,Y\right)-h\left(X,TY\right)$$

(2.16)

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 (N₁, g₁) and (N₂, g₂) be two Riemannian manifolds with Riemannian metrics g₁ and g₂, respectively, and λ be a positive differentiable function on N₁. Then the warped product of N₁ and N₂ is the Riemannian manifold (N₁ × N₂, g), where

$$g=g_1+{\lambda }^2{g}_2.$$

The warped product manifold (N₁ × N₂, g) is denoted by N₁ ×λ N₂. If U is any vector field tangent to M = N₁ × _λ N₂ at (p, q), then

$$\parallel U{\parallel }^2=\parallel d{\pi }_{1}U{\parallel }^2+{\lambda }^2\left(p\right)\parallel d{\pi }_{2}U{\parallel }^2,$$

where π₁ and π₂ are the canonical projections of M onto N₁ and N₂, respectively.

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

Theorem 2.1 Let M = N₁ × _λ N₂ be a warped product manifold. If X, Y ∈ Γ(TN₁) and Z, W ∈ Γ(TN₂), then

• (i) ∇ _X Y ∈ Γ(TN₁)

• (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 N₂ and ∇λ is the gradient of the function λ and is defined as

$$g\left(\nabla \lambda ,U\right)=U\lambda ,$$

(2.17)

for each U ∈ Γ(TM).

Corollary 2. 1 On a warped product manifold M = N₁ × _λ N₂, we have

• (i) N₁ is totally geodesic in M,

• (ii) N₂ 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

$$g\left(h\left(X,\varphi Y\right),\varphi Z\right)=g\left(h\left(\varphi X,Y\right),\varphi Z\right)$$

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 $\bar{M}$ is integrable, then its leaves are totally geodesic in M if and only if

$$h\left(U,Y\right)\in \Gamma \left(\nu \right),$$

for each U ∈ Γ(TM) and Y ∈ Γ(D).

Proof. From (2.16), we obtain

$$F{\nabla }_{U}Y=fh\left(U,Y\right)-h\left(U,TY\right),$$

for any U ∈ Γ(TM) and Y ∈ Γ(D). Taking the inner product with ϕZ for any Z ∈ Γ(D^⊥), we get

$$g\left(F{\nabla }_{U}Y,\varphi Z\right)=-g\left(h\left(U,TY\right),\varphi Z\right).$$

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 $\bar{M}$ is integrable and its leaves are totally geodesic in M if and only if

$$\left({\bar{\nabla }}_{X}T\right)Y=0.$$

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 $\bar{M}$, the leaf N_⊥ of D^⊥ is totally geodesic in M if and only if

$$g\left(h\left(X,Z\right),\varphi W\right)=0,$$

for any X ∈ Γ(D) and Z, W ∈ Γ(D^⊥).

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

$$g\left({\nabla }_{Z}W,\varphi X\right)=g\left(h\left(X,Z\right),\varphi W\right).$$

Thus, the result follows from the above equation. ■

Theorem 3.1 A semi-invariant submanifold M of a cosymplectic manifold $\bar{M}$ is locally a semi-invariant product if and only if

$$\left({\bar{\nabla }}_{U}T\right)V=0,$$

for any U, V ∈ Γ(TM).

Proof. If T is parallel then by (2.15), we have

$$A_{FV}U=-th\left(U,V\right)$$

(3.1)

for any U, V tangent to M. In particular, if X ∈ Γ(D), then (3.1) gives, th(U, X) = 0, that is,

$$A_{FZ}X=0.$$

(3.2)

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 $\left({\bar{\nabla }}_{U}T\right)Y=0$. Similarly, for any Z ∈ Γ(D^⊥) and U ∈ Γ(TM), we obtain ∇ _U Z ∈ Γ(D^⊥) and then by (2.10), we get $\left({\bar{\nabla }}_{U}T\right)Z=0$ and it is easy to see that $\left({\bar{\nabla }}_{U}T\right)\xi =0$. By these observations we find that $\left({\bar{\nabla }}_{U}T\right)V=0$, 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 $\bar{M}$, respectively. The warped product semi-invariant submanifolds of a cosymplectic manifold $\bar{M}$ 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 $\bar{M}$. Then

$$\left({\bar{\nabla }}_{Z}T\right)X=\left(TX\text{ ln }\lambda \right)Z$$

$$\left({\bar{\nabla }}_{U}T\right)Z=g\left(P_{2}U,Z\right)T\left(\nabla \text{ ln }\lambda \right).$$

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 $\bar{M}$, then by Theorem 2.1 (ii), we have

$${\nabla }_{X}Z={\nabla }_{Z}X=\left(X ln \lambda \right)Z$$

(4.1)

for X ∈ Γ(TN _T ) and Z ∈ Γ(TN_⊥). Then, from (2.10) and (4.1), we get

$$\left({\bar{\nabla }}_{Z}T\right)X=\left(TX ln \lambda \right)Z,$$

which proves the first part of the lemma. Now, for any U ∈ Γ(TM), we have TU ∈ Γ(TN _T ), therefore $\left({\bar{\nabla }}_{U}T\right)Z\in \Gamma \left(T{N}_T\right)$ for any U ∈ Γ(TM). Furthermore, for any X ∈ Γ(TN _T ), we obtain

$$g\left(\left({\bar{\nabla }}_{U}T\right)Z,X\right)=-g\left(Z,{\nabla }_{U}TX\right).$$

Using (2.13), the above equation reduced to

$$\begin{array}{cc}\hfill g\left(\left({\bar{\nabla }}_{U}T\right)Z,X\right)& =-g\left(Z,{\nabla }_{P_{1}U+P_{2}U+\eta \left(U\right)\xi }TX\right).\hfill \\ =-g\left(Z,{\nabla }_{P_{2}U}TX\right)+\eta \left(U\right)g\left({\nabla }_{\xi }Z,TX\right).\hfill \end{array}$$

Using (4.1), the second term of right hand side is identically zero, then the above equation takes the form

$$\begin{array}{cc}\hfill g\left(\left({\bar{\nabla }}_{U}T\right)Z,X\right)& =-g\left(Z,{\nabla }_{P_{2}U}TX\right)\hfill \\ =-\left(TX ln \lambda \right)g\left(Z,P_{2}U\right).\hfill \end{array}$$

Using (2.17), we obtain

$$g\left(\left({\bar{\nabla }}_{U}T\right)Z,X\right)=g\left(T\nabla ln \lambda ,X\right)g\left(Z,P_{2}U\right).$$

That is,

$$\left({\bar{\nabla }}_{U}T\right)Z=T\left(\nabla ln \lambda \right)g\left(Z,P_{2}U\right).$$

This proves the lemma completely. ■

Theorem 4.1 A proper semi-invariant submanifold of a cosymplectic manifold $\bar{M}$ is locally a warped product semi-invariant submanifold if and only if

$$\left({\bar{\nabla }}_{U}T\right)V=\left(TV\mu \right)P_{2}U+g\left(P_{2}U,P_{2}V\right)\varphi \nabla \mu ,$$

(4.2)

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 $\bar{M}$, then from (2.10) and (2.13), we have

$$\left({\bar{\nabla }}_{U}T\right)V=\left({\bar{\nabla }}_{U}T\right)P_{1}V+\left({\bar{\nabla }}_{U}T\right)P_{2}V+\eta \left(U\right)\left({\bar{\nabla }}_{U}T\right)\xi .$$

Again using (2.10) and (2.13), the above equation takes the form

$$\left({\bar{\nabla }}_{U}T\right)V=\left({\bar{\nabla }}_{P_{1}U}T\right)P_{1}V+\left({\bar{\nabla }}_{P_{2}U}T\right)P_{1}V+\left({\bar{\nabla }}_{U}T\right)P_{2}V.$$

(4.3)

Now, from Lemma 4.1, we have

$$\left({\bar{\nabla }}_{P_{2}U}T\right)P_{1}V=\left(TV ln \lambda \right)P_{2}U$$

and

$$\left({\bar{\nabla }}_{U}T\right)P_{2}V=g\left(P_{2}U,P_{2}V\right)T\left(\nabla ln \lambda \right).$$

Substituting these values in (4.3), we obtain

$$\left({\bar{\nabla }}_{U}T\right)V=\left(TV\mu \right)P_{2}U+g\left(P_{2}U,P_{2}V\right)\varphi \nabla \mu .$$

Conversely, suppose that M is a semi-invariant submanifold of a cosymplectic manifold $\bar{M}$ and (4.2) holds, then $\left({\bar{\nabla }}_{X}T\right)Y=0$, 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

$$g\left(\left({\bar{\nabla }}_{Z}T\right)X,W\right)=\left(TX\mu \right)g\left(Z,W\right).$$

for X ∈ Γ(D) and Z, W ∈ Γ(D^⊥). Using (2.3), (2.8), and (2.10), we obtain

$$g\left(\varphi {\bar{\nabla }}_{Z}X,W\right)=\left(TX\mu \right)g\left(Z,W\right).$$

That is,

$$g\left({\bar{\nabla }}_{Z}X,\varphi W\right)=-\left(TX\mu \right)g\left(Z,W\right).$$

Using cosymplectic character and (2.5), we derive

$$g\left({\nabla }_{Z}W,\varphi X\right)=-\left(TX\mu \right)g\left(Z,W\right).$$

By (2.17), the above equation takes the form

$$g\left({\nabla }_{Z}W,\varphi X\right)=g\left(T\nabla \mu ,X\right)g\left(Z,W\right).$$

(4.4)

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

$$g\left(h^{\prime }\left(Z,W\right),X\right)=g\left({\nabla }_{Z}W,X\right).$$

Using (4.4), we get

$$g\left(h^{\prime }\left(Z,W\right),\varphi X\right)=-g\left(\nabla \mu ,\varphi X\right)g\left(Z,W\right)$$

or,

$$h^{\prime }\left(Z,W\right)=-g\left(Z,W\right)\nabla \mu .$$

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 $\bar{\nabla }T$, when the submanifold is a warped product semi-invariant submanifold.

Theorem 4.2 A semi-invariant submanifold M of a cosymplectic manifold $\bar{M}$ is locally a warped product semi-invariant submanifold if and only if

$$g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)=-\left(P_{1}V\mu \right)g\left(U,W\right),$$

(4.5)

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 $\bar{M}$, then N _T and N_⊥ are totally geodesic and totally umbilical in M, respectively. Moreover, we have

$${\nabla }_{X}Z={\nabla }_{Z}X=\left(X ln \text{λ}\right)Z.$$

(4.5.1)

for any X ∈ Γ(D) and Z ∈ Γ(D^⊥). Now, by (2.13), we have

$$\begin{array}{cc}\hfill \left({\bar{\nabla }}_{U}F\right)V& =\left({\bar{\nabla }}_{P_{1}U+P_{2}U+\eta \left(U\right)\xi }F\right)V\hfill \\ =\left({\bar{\nabla }}_{P_{1}U}F\right)V+\left({\bar{\nabla }}_{P_{2}U}F\right)V+\eta \left(U\right)\left({\bar{\nabla }}_{\xi }F\right)V.\hfill \end{array}$$

Again, using (2.13), the above equation takes the form

$$\begin{array}{c}\left({\bar{\nabla }}_{U}F\right)V=\left({\bar{\nabla }}_{P_{1}U}F\right)P_{1}V+\left({\bar{\nabla }}_{P_{1}U}F\right)P_{2}V+\eta \left(V\right)\left({\bar{\nabla }}_{P_{1}U}F\right)\xi +\left({\bar{\nabla }}_{P_{2}U}F\right)P_{1}V\hfill \\ +\left({\bar{\nabla }}_{P_{2}U}F\right)P_{2}V+\eta \left(V\right)\left({\bar{\nabla }}_{P_{2}U}F\right)\xi +\eta \left(U\right)\left({\bar{\nabla }}_{\xi }F\right)P_{1}V\hfill \\ +\eta \left(U\right)\left({\bar{\nabla }}_{\xi }F\right)P_{2}V+\eta \left(U\right)\eta \left(V\right)\left({\bar{\nabla }}_{\xi }F\right)\xi .\hfill \end{array}$$

In view of (2.4), (2.5), and (2.16), the above equation reduced to

$$\left({\bar{\nabla }}_{U}F\right)V=\left({\bar{\nabla }}_{P_{1}U}F\right)P_{1}V+\left({\bar{\nabla }}_{P_{1}U}F\right)P_{2}V+\left({\bar{\nabla }}_{P_{2}U}F\right)P_{1}V+\left({\bar{\nabla }}_{P_{2}U}F\right)P_{2}V.$$

Taking the inner product with ϕW, for any W ∈ Γ(D^⊥), we obtain

$$\begin{array}{cc}\hfill g\left({\bar{\nabla }}_{U}F\right)V,\varphi W)& =g(\left({\bar{\nabla }}_{P_{1}U}F\right)P_{1}V+\left({\bar{\nabla }}_{P_{1}U}F\right)P_{2}V\hfill \\ +\left({\bar{\nabla }}_{P_{2}U}F\right)P_{1}V+\left({\bar{\nabla }}_{P_{2}U}F\right)P_{2}V,\varphi W).\hfill \end{array}$$

Using (2.14), (2.16) and the fact that P₁U ∈ Γ(D) and P₂U ∈ Γ(D^⊥), for any U ∈ Γ(TM), then the above equation becomes

$$\begin{array}{c}g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)=g\left(fh\left(P_{1}U,P_{1}V\right),\varphi W\right)-g\left(h\left(P_{1}U,T{P}_{1}V\right),\varphi W\right)\\ +g\left(fh\left(P_{1}U,P_{2}V\right),\varphi W\right)+g\left(fh\left(P_{2}U,P_{1}V\right),\varphi W\right)\\ +g\left(fh\left(P_{2}U,P_{2}V\right),\varphi W\right)-g\left(h\left(P_{2}U,T{P}_{1}V\right),\varphi W\right).\end{array}$$

From (2.2), the above equation becomes

$$g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)=-g\left(h\left(P_{1}U,T{P}_{1}V\right)+h\left(P_{2}U,T{P}_{1}V\right),\varphi W\right).$$

Using (2.5), we derive

$$g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)=-g\left({\bar{\nabla }}_{P_{1}U}\varphi P_{1}V,\varphi W\right)-g\left({\bar{\nabla }}_{P_{2}U}\varphi P_{1}V,\varphi W\right).$$

Using the covariant differentiation property of ϕ and the fact that P₁V ∈ Γ(D) and P₂V ∈ Γ(D^⊥), for any V ∈ Γ(TM), then from (2.2), we obtain

$$g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)=g\left(P_{1}V,{\bar{\nabla }}_{P_{1}U}W\right)-g\left({\bar{\nabla }}_{P_{2}U}{P}_{1}V,W\right).$$

Again using (2.5), we arrive at

$$g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)=g\left(P_{1}V,{\nabla }_{P_{1}U}W\right)-g\left({\nabla }_{P_{2}U}{P}_{1}V,W\right).$$

The first term of right-hand side is zero by (4.1) and the fact that P₁V ∈ Γ(D) and W ∈ Γ(D^⊥), thus we obtain

$$\begin{array}{cc}\hfill g\left(\left({\bar{\nabla }}_{U}F\right)V,\varphi W\right)& =-\left(P_{1}V ln \lambda \right)g\left(P_{2}U,W\right)\hfill \\ =-\left(P_{1}V ln \lambda \right)g\left(U,W\right)\hfill \\ =-\left(P_{1}V\mu \right)g\left(U,W\right).\hfill \end{array}$$

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

$$g\left(\left({\bar{\nabla }}_{X}F\right)Y,\varphi W\right)=0,$$

for each X, Y ∈ Γ(D) and W ∈ Γ(D^⊥). Thus, by (2.16) we obtain

$$g\left(h\left(X,\varphi Y\right),\varphi W\right)=0.$$

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

$$g\left(\left({\bar{\nabla }}_{Z}F\right)X,\varphi W\right)=-\left(X\mu \right)g\left(Z,W\right).$$

Using (2.16), we get

$$g\left(h\left(\varphi X,Z\right),\varphi W\right)=\left(X\mu \right)g\left(Z,W\right).$$

(4.6)

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

$${\nabla }_{Z}W={\nabla }_Z^{\prime }W+h^{\prime }\left(Z,W\right).$$

(4.7)

Now for any Z, W ∈ Γ(D^⊥) and X ∈ Γ(D), by (2.3) and (2.5), we have

$$g\left(h\left(Z,X\right),\varphi W\right)=g\left(\varphi X,{\nabla }_{Z}W\right).$$

From (4.7), we obtain

$$g\left(h\left(Z,X\right),\varphi W\right)=g\left(h^{\prime }\left(Z,W\right),\varphi X\right).$$

(4.8)

Thus, by (4.6) and (4.8), we derive

$$g\left(h^{\prime }\left(Z,W\right),X\right)=-\left(X\mu \right)g\left(Z,W\right).$$

Using (2.17), we obtain

$$h^{\prime }\left(Z,W\right)=-g\left(Z,W\right)\nabla \mu ,$$

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 $\bar{M}$. Then

• (i) $h_{\varphi D^{\perp }}\left(\varphi X,Z\right)=\left(X ln \text{λ}\right)\varphi Z$

• (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

$$h\left(\varphi X,Z\right)=\varphi {\nabla }_{Z}X+\varphi h\left(X,Z\right)-{\nabla }_Z\varphi X.$$

Using (4.1), we get

$$h\left(\varphi X,Z\right)=\left(X ln \text{λ}\right)\varphi Z+\varphi h\left(X,Z\right)-\left(\varphi X ln \text{λ}\right)Z.$$

(4.9)

Equating the tangential components of (4.9), we get

$$\left(\varphi X ln \text{λ}\right)Z=th\left(X,Z\right),$$

Taking the inner product with W ∈ Γ(D^⊥), we obtain

$$g\left(h\left(X,Z\right),\varphi W\right)=-\left(\varphi X \; ln\; \text{λ}\right)g\left(Z,W\right),$$

or equivalently

$$h_{\varphi D^{\perp }}\left(X,Z\right)=-\left(\varphi X ln \text{λ}\right)\varphi Z.$$

Replacing X by ϕX, we obtain

$$h_{\varphi D^{\perp }}\left(\varphi X,Z\right)=\left(X \; ln\; \text{λ}\right)\varphi Z,$$

which proves the part (i) of proposition. Now, for the second part comparing the normal components of (4.9), we get

$$h\left(\varphi X,Z\right)=\left(X \; ln\; \text{λ}\right)\varphi Z+\varphi h_{\nu }\left(X,Z\right),$$

or,

$$h\left(\varphi X,Z\right)-\varphi h_{\nu }\left(X,Z\right)=\left(X ln \text{λ}\right)\varphi Z,$$

Taking the inner product with ϕ h(X, Z), we derive

$$g\left(h\left(\varphi X,Z\right),\varphi h\left(X,Z\right)\right)=\parallel h_{\nu }\left(X,Z\right){\parallel }^2,$$

which completes the proof. ■

Theorem 4.3. Let M = N _T × _λ N_⊥ be a warped product semi-invariant submanifold of a cosymplectic manifold $\bar{M}$. Then

• (i) The squared norm of the second fundamental form satisfies

  $$\parallel h{\parallel }^2\ge 2q\parallel \nabla \; ln\; \text{λ}{\parallel }^2,$$

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$\bar{M}$, N_⊥ is a totally umbilical submanifold of $\bar{M}$ and M is minimal.

Proof. Let {X₁, X₂, . . . , X _p , X_p+1= ϕX₁, . . . , X_2p= ϕX _p , X_2p+1= ξ} be a local orthonormal frame of vector fields on N _T and {Z₁, Z₂, . . . , Z _q } a local orthonormal frame on N_⊥. Then by definition of squared norm of mean curvature vector

$$\begin{array}{c}\left|\right|h|{|}^2=\sum _{i,j=1}^{2p+1}g\left(h\left(X_i,X_j\right),h\left(X_i,X_j\right)\right)\hfill \\ +\sum _{i=1}^{2p+1}\sum _{r=1}^{q}g\left(h\left(X_i,Z_r\right),h\left(X_i,Z_r\right)\right)\hfill \\ +\sum _{r,s=1}^{q}g\left(h\left(Z_r,Z_s\right),h\left(Z_r,Z_s\right)\right)\hfill \end{array}$$

(4.10)

or,

$${\parallel h\parallel }^2\ge \sum _{i=1}^{2p}\sum _{r=1}^{q}g\left(h\left(X_i,Z_r\right),h\left(X_i,Z_r\right)\right).$$

In view of Proposition 4.1 (i), we get

$$\begin{array}{cc}\hfill \parallel h{\parallel }^2& \ge \sum _{i=1}^{2p}\sum _{r=1}^q{\left(\varphi X_i ln \text{λ}\right)}^{2}g\left(Z_r,Z_r\right),\hfill \\ \ge 2q\parallel \nabla ln \text{λ}{\parallel }^2.\hfill \end{array}$$

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

$$h\left(D,D\right)=0,,h\left(D^{\perp },D^{\perp }\right)=0\mathsf{\text{and}}h\left(D,D^{\perp }\right)\in \Gamma \left(\varphi D^{\perp }\right).$$

(4.11)

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