We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the type M⊥×fMT is a usual Riemannian product of M⊥ and MT, where M⊥ and MT are anti-invariant and invariant submanifolds of a nearly cosymplectic manifold M¯, respectively. Thus we consider the warped product of the type MT×fM⊥ and obtain a characterization for such type of warped product.

1. Introduction

The notion of warped product manifolds was introduced by Bishop and O'Neill in 1969 as a natural generalization of the Riemannian product manifolds. Later on, the geometrical aspect of these manifolds has been studied by many researchers (cf., [1–3]). Recently, Chen [1] (see also [4]) studied warped product CR-submanifolds and showed that there exists no warped product CR-submanifolds of the form M=M⊥×fMT such that M⊥ is a totally real submanifold and MT is a holomorphic submanifold of a Kaehler manifold M̅. Therefore he considered warped product CR-submanifold in the form M=MT×fM⊥ which is called CR-warped product, where MT and M⊥ are holomorphic and totally real submanifolds of a Kaehler manifold M̅. Motivated by Chen's papers, many geometers studied CR-warped product submanifolds in almost complex as well as contact setting (see [3, 5, 6]).

Almost contact manifolds with Killing structure tensors were defined in [7] as nearly cosymplectic manifolds, and it was shown that normal nearly cosymplectic manifolds are cosymplectic (see also [8]). Later on, Blair and Showers [9] studied nearly cosymplectic structure (ϕ,ξ,η,g) on a manifold M̅ with η closed from the topological viewpoint.

In this paper, we have generalized the results of Chen' [1] in this more general setting of nearly cosymplectic manifolds and have shown that the warped product in the form M=M⊥×fMT is simply Riemannian product of M⊥ and MT where M⊥ is an anti-invariant submanifold and MT is an invariant submanifold of a nearly cosymplectic manifold M̅. Thus we consider the warped product submanifold of the type M=MT×fM⊥ by reversing the two factors M⊥ and MT and simply will be called warped product semi-invariant submanifold. Thus, we derive the integrability of the involved distributions in the warped product and obtain a characterization result.

2. Preliminaries

A (2n+1)-dimensional C∞ manifold M̅ is said to have an almost contact structure if there exist on M̅ a tensor field ϕ of type (1,1), a vector field ξ, and a 1-form η satisfying [9] ϕ2=-I+η⊗ξ,ϕξ=0,η∘ϕ=0,η(ξ)=1.
There always exists a Riemannian metric g on an almost contact manifold M̅ satisfying the following compatibility condition: η(X)=g(X,ξ),g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),
where X and Y are vector fields on M̅ [9].

An almost contact structure (ϕ,ξ,η) is said to be normal if the almost complex structure J on the product manifold M̅×ℝ given by J(X,fddt)=(ϕX-fξ,η(X)ddt),
where f is a C∞-function on M̅×ℝ, has no torsion, that is, J is integrable, and the condition for normality in terms of ϕ,ξ and η is [ϕ,ϕ]+2dη⊗ξ=0 on M̅, where [ϕ,ϕ] is the Nijenhuis tensor of ϕ. Finally the fundamental 2-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 [9]. The structure is said to be nearly cosymplectic if ϕ is Killing, that is, if (∇̅Xϕ)Y+(∇̅Yϕ)X=0,
for any X,Y∈TM̅, where TM̅ is the tangent bundle of M̅ and ∇̅ denotes the Riemannian connection of the metric g. Equation (2.4) is equivalent to (∇̅Xϕ)X=0, for each X∈TM̅. 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 ∇̅ϕ vanishes identically, that is, (∇̅Xϕ)Y=0 and ∇̅Xξ=0.

Proposition 2.1 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

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

From the above proposition we have ∇̅Xξ=0, for any vector field X tangent to M̅, where M̅ is a nearly cosymplectic manifold.

Let M be submanifold of an almost contact metric manifold M̅ 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, then, Gauss and Weingarten formulae are given by ∇̅XY=∇XY+h(X,Y),∇̅XN=-ANX+∇X⊥N,
for each X,Y∈TM and N∈T⊥M, where h and AN are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively, for the immersion of M into M̅. They are related as g(h(X,Y),N)=g(ANX,Y),
where g denotes the Riemannian metric on M̅ as well as being induced on M.

For any X∈TM, we write ϕX=TX+FX,
where TX is the tangential component and FX is the normal component of ϕX.

Similarly for any N∈T⊥M, we write ϕN=BN+CN,
where BN is the tangential component and CN is the normal component of ϕN. The covariant derivatives of the tensor fields P and F are defined as (∇XT)Y=∇XTY-T∇XY,(∇̅XF)Y=∇X⊥FY-F∇XY
for all X,Y∈TM.

Let M be a Riemannian manifold isometrically immersed in an almost contact metric manifold M̅. then for every x∈M there exists a maximal invariant subspace denoted by 𝒟x of the tangent space TxM of M. If the dimension of 𝒟x is the same for all values of x∈M, then 𝒟x gives an invariant distribution 𝒟 on M.

A submanifold M of an almost contact metric manifold M̅ is called semi-invariant submanifold if there exists on M a differentiable invariant distribution 𝒟 whose orthogonal complementary distribution 𝒟⊥ is anti-invariant, that is,

TM=𝒟⊕𝒟⊥⊕〈ξ〉,

ϕ(𝒟x)⊆Dx,

ϕ(𝒟x⊥)⊂Tx⊥M

for any x∈M, where Tx⊥M denotes the orthogonal space of TxM in TxM̅. A semi-invariant submanifold is called anti-invariant if 𝒟x={0} and invariant if 𝒟x⊥={0}, respectively, for any x∈M. It is called the proper semi-invariant submanifold if neither 𝒟x={0} nor 𝒟x⊥={0}, for every x∈M.

Let M be a semi-invariant submanifold of an almost contact metric manifold M̅. Then, F(TxM) is a subspace of Tx⊥M. Then for every x∈M, there exists an invariant subspace νx of TxM̅ such that Tx⊥M=F(TxM)⊕νx.

A semi-invariant submanifold M of an almost contact metric manifold M̅ is called Riemannian product if the invariant distribution 𝒟 and anti-invariant distribution 𝒟⊥ are totally geodesic distributions in M.

Let (M1,g1) and (M2,g2) be two Riemannian manifolds, and let f be a positive differentiable function on M1. The warped product of M1 and M2 is the product manifold M1×fM2=(M1×M2,g), where g=g1+f2g2,
where f is called the warping function of the warped product. The warped product N1×fN2 is said to be trivial or simply Riemannian product if the warping function f is constant. This means that the Riemannian product is a special case of warped product.

We recall the following general results obtained by Bishop and O'Neill [10] for warped product manifolds.

Lemma 2.2.

Let M=M1×fM2 be a warped product manifold with the warping function f. Then

∇XY∈TM1, for each X,Y∈TM1,

∇XZ=∇ZX=(Xlnf)Z, for each X∈TM1 and Z∈TM2,

∇ZW=∇ZM2W-(g(Z,W)/f)gradf,

where ∇ and ∇M2 denote the Levi-Civita connections on M and M2, respectively.

In the above lemma gradf is the gradient of the function f defined by g(gradf,U)=Uf, for each U∈TM. From the Lemma 2.2, we have that on a warped product manifold M=M1×fM2

M1 is totally geodesic in M;

M2 is totally umbilical in M.

Now, we denote by 𝒫XY and 𝒬XY the tangential and normal parts of (∇̅Xϕ)Y, that is, (∇̅Xϕ)Y=PXY+QXY
for all X,Y∈TM. Making use of (2.5), (2.6), and (2.8)–(2.11), the following relations may easily be obtained PXY=(∇XT)Y-AFYX-Bh(X,Y),QXY=(∇̅XF)Y+h(X,TY)-Ch(X,Y).

It is straightforward to verify the following properties of 𝒫 and 𝒬, which we enlist here for later use:

(i) 𝒫X+YW=𝒫XW+𝒫YW, (ii) 𝒬X+YW=𝒬XW+𝒬YW,

(i) 𝒫X(Y+W)=𝒫XY+𝒫XW, (ii) 𝒬X(Y+W)=𝒬XY+𝒬XW,

g(𝒫XY,W)=-g(Y,𝒫XW)

for all X,Y,W∈TM.

On a submanifold M of a nearly cosymplectic manifold M̅, we obtain from (2.4) and (2.14) that (i) PXY+PYX=0,(ii) QXY+QYX=0
for any X,Y∈TM.

3. Warped Product Semi-Invariant Submanifolds

Throughout the section we consider the submanifold M of a nearly cosymplectic manifold M̅ such that the structure vector field ξ is tangent to M. First, we prove that the warped product M=M1×fM2 is trivial when ξ is tangent to M2, where M1 and M2 are Riemannian submanifolds of a nearly cosymplectic manifold M̅. Thus, we consider the warped product M=M1×fM2, when ξ is tangent to the submanifold M1. We have the following nonexistence theorem.

Theorem 3.1.

A warped product submanifold M=M1×fM2 of a nearly cosymplectic manifold M̅ is a usual Riemannian product if the structure vector field ξ is tangent to M2, where M1 and M2 are the Riemannian submanifolds of M̅.

Proof.

For any X∈TM1 and ξ tangent to M2, we have
∇̅Xξ=∇Xξ+h(X,ξ).
Using the fact that ξ is Killing on a nearly cosymplectic manifold (see Proposition 2.1) and Lemma 2.2(ii), we get
0=(Xlnf)ξ+h(X,ξ).
Equating the tangential component of (3.2), we obtain Xlnf=0, for all X∈TM1, that is, f is constant function on M1. Thus, M is Riemannian product. This proves the theorem.

Now, the other case of warped product M=M1×fM2 when ξ∈TM1, where M1 and M2 are the Riemannian submanifolds of M̅. For any X∈TM2, we have ∇̅Xξ=∇Xξ+h(X,ξ).
By Proposition 2.1, and Lemma 2.2(ii), we obtain (i) ξlnf=0,(ii) h(X,ξ)=0.
Thus, we consider the warped product semi-invariant submanifolds of a nearly cosymplectic manifold M̅ of the types:

M=M⊥×fMT,

M=MT×fM⊥,

where MT and M⊥ are invariant and anti-invariant submanifolds of M̅, respectively. In the following theorem we prove that the warped product semi-invariant submanifold of the type (i) is CR-product.

Theorem 3.2.

The warped product semi-invariant submanifold M=M⊥×fMT of a nearly cosymplectic manifold M̅ is a usual Riemannian product of M⊥ and MT, where M⊥ and MT are anti-invariant and invariant submanifolds of M̅, respectively.

Proof.

When ξ∈TMT, then by Theorem 3.1, M is a Riemannian product. Thus, we consider ξ∈TM⊥. For any X∈TMT and Z∈TM⊥, we have
g(h(X,ϕX),FZ)=g(h(X,ϕX),ϕZ)=g(∇̅XϕX,ϕZ)=g(ϕ∇̅XX,ϕZ)+g((∇̅Xϕ)X,ϕZ).
From the structure equation of nearly cosymplectic, the second term of right hand side vanishes identically. Thus from (2.2), we derive
g(h(X,ϕX),FZ)=g(∇̅XX,Z)-η(Z)g(∇̅XX,ξ)=-g(X,∇̅XZ)+η(Z)g(X,∇̅Xξ).
Then from (2.5), Lemma 2.2(ii), and Proposition 2.1, we obtain
g(h(X,ϕX),FZ)=-(Zlnf)‖X‖2.
Interchanging X by ϕX in (3.7) and using the fact that ξ∈TM⊥, we obtain
g(h(X,ϕX),FZ)=(Zlnf)‖X‖2.
It follows from (3.7) and (3.8) that Zlnf=0, for all Z∈TM⊥. Also, from (3.4) we have ξlnf=0. Thus, the warping function f is constant. This completes the proof of the theorem.

From the above theorem we have seen that the warped product of the type M=M⊥×fMT is a usual Riemannian product of an anti-invariant submanifold M⊥ and an invariant submanifold MT of a nearly cosymplectic manifold M̅. Since both M⊥ and MT are totally geodesic in M, then M is CR-product. Now, we study the warped product semi-invariant submanifold M=MT×fM⊥ of a nearly cosymplectic manifold M̅.

Theorem 3.3.

Let M=MT×fM⊥ be a warped product semi-invariant submanifold of a nearly cosymplectic manifold M̅. Then the invariant distribution 𝒟 and the anti-invariant distribution 𝒟⊥ are always integrable.

Proof.

For any X,Y∈𝒟, we have
F[X,Y]=F∇XY-F∇YX.
Using (2.11), we obtain
F[X,Y]=(∇̅XF)Y-(∇̅YF)X.
Then by (2.16), we derive
F[X,Y]=QXY-h(X,TY)+Ch(X,Y)-QYX+h(Y,TX)-Ch(X,Y).
Thus from (2.17)(ii), we get
F[X,Y]=2QXY+h(Y,TX)-h(X,TY).
Now, for any X,Y∈D, we have
h(X,TY)+∇XTY=∇̅XTY=∇̅XϕY.
Using the covariant derivative property of ∇̅ϕ, we obtain
h(X,TY)+∇XTY=(∇̅Xϕ)Y+ϕ∇̅XY.
Then by (2.5) and (2.14), we get
h(X,TY)+∇XTY=PXY+QXY+ϕ(∇XY+h(X,Y)).
Since MT is totally geodesic in M (see Lemma 2.2(i)), then using (2.8) and (2.9), we obtain
h(X,TY)+∇XTY=PXY+QXY+T∇XY+Bh(X,Y)+Ch(X,Y).
Equating the normal components of (3.16), we get
h(X,TY)=QXY+Ch(X,Y).
Similarly, we obtain
h(Y,TX)=QYX+Ch(X,Y).
Then from (3.17) and (3.18), we arrive at
h(Y,TX)-h(X,TY)=QYX-QXY.
Hence, using (2.17)(ii), we get
h(Y,TX)-h(X,TY)=-2QXY.
Thus, it follows from (3.12) and (3.20) that F[X,Y]=0, for all X,Y∈D. This proves the integrability of D. Now, for the integrability of D⊥, we consider any X∈D and Z,W∈D⊥, and we have
g([Z,W],X)=g(∇̅ZW-∇̅WZ,X).=-g(∇ZX,W)+g(∇WX,Z).
Using Lemma 2.2(ii), we obtain
g([Z,W],X)=-(Xlnf)g(Z,W)+(Xlnf)g(Z,W)=0.
Thus from (3.22), we conclude that [Z,W]∈𝒟⊥, for each Z,W∈𝒟⊥. Hence, the theorem is proved completely.

Lemma 3.4.

Let M=MT×fM⊥ be a warped product submanifold of a nearly cosymplectic manifold M̅. If X,Y∈TMT and Z,W∈TM⊥, then

For a warped product manifold M=MT×fM⊥, we have that MT is totally geodesic in M; then by (2.10), (∇̅XT)Y∈TMT, for any X,Y∈TMT, and therefore from (2.15), we get
g(PXY,Z)=-g(Bh(X,Y),Z)=g(h(X,Y),FZ).
The left-hand side of (3.23) is skew symmetric in X and Y whereas the right hand side is symmetric in X and Y, which proves (i). Now, from (2.10) and (2.15), we have
PXZ=-T∇XZ-AFZX-Bh(X,Z)
for any X∈TMT and Z∈TM⊥. Using Lemma 2.2(ii), the first term of right-hand side is zero. Thus, taking the product with W∈TM⊥, we obtain
g(PXZ,W)=-g(AFZX,W)-g(Bh(X,Z),W),
Then by (2.2) and (2.7), we get
g(PXZ,W)=-g(h(X,W),FZ)+g(h(X,Z),FW).
which proves the first equality of (ii). Again, from (2.10) and (2.15), we have
PZX=∇ZTX-T∇ZX-Bh(X,Z).
Thus using Lemma 2.2(ii), we derive
PZX=(TXlnf)Z-Bh(X,Z).
Taking inner product with W∈TM⊥ and using (2.2), we obtain
g(PZX,W)=(ϕXlnf)g(Z,W)+g(h(X,Z),FW).
Then from (2.17)(i), we get
g(PXZ,W)=-(ϕXlnf)g(Z,W)-g(h(X,Z),FW).
This is the second equality of (ii). Now, from (3.24) and (3.28), we have
PXZ+PZX=-T∇XZ-AFZX+(TXlnf)Z-2Bh(X,Z).
Left-hand side and the first term of right-hand side are zero on using (2.17)(i) and Lemma 2.2(i), respectively. Thus the above equation takes the form
(TXlnf)Z=AFZX+2Bh(X,Z).
Taking the product with Z and on using (2.2) and (2.7), we get
(ϕXlnf)‖Z‖2=g(h(X,Z),FZ)-2g(h(X,Z),FZ)=-g(h(X,Z),FZ).
Interchanging X by ϕX and using (2.1), we obtain
{-X+η(X)ξ}lnf‖Z‖2=-g(h(ϕX,Z),FZ).
Thus by (3.4)(i), the above equation reduces to
(Xlnf)‖Z‖2=g(h(ϕX,Z),FZ).
This proves the lemma completely.

Theorem 3.5.

A proper semi-invariant submanifold M of a nearly cosymplectic manifold M̅ is locally a semi-invariant warped product if and only if the shape operator of M satisfies
AϕZX=-(ϕXμ)Z,X∈D⊕〈ξ〉,Z∈D⊥
for some function μ on M satisfying V(μ)=0 for each V∈𝒟⊥.

Proof.

If M=MT×fM⊥ is a warped product semi-invariant submanifold, then by Lemma 3.4 (iii), we obtain (3.36). In this case μ=lnf.

Conversely, suppose M is a semi-invariant submanifold of a nearly cosymplectic manifold M̅ satisfying (3.36). Then
g(h(X,Y),ϕZ)=g(AϕZX,Y)=-(ϕXμ)g(Y,Z)=0.
Now, from (2.5) and the property of covariant derivative of ∇̅, we have
g(h(X,Y),ϕZ)=g(∇̅XY,ϕZ)=-g(ϕ∇̅XY,Z)=-g(∇̅XϕY,Z)+g((∇̅Xϕ)Y,Z).
Then from (2.5), (2.14), and (3.37), the above equation takes the form
g(∇XTY,Z)=g(PXY,Z).
Using (2.10) and (2.15), we obtain
g(∇XTY,Z)=g(∇XTY,Z)-g(T∇XY,Z)-g(Bh(X,Y),Z).
Thus by (2.2), the above equation reduces to
g(T∇XY,Z)=g(h(X,Y),ϕZ).
Hence using (2.7) and (3.36), we get
g(T∇XY,Z)=g(AϕZX,Y)=0,
which implies ∇XY∈𝒟⊕〈ξ〉, that is, 𝒟⊕〈ξ〉 is integrable and its leaves are totally geodesic in M. Now, for any Z,W∈𝒟⊥ and X∈𝒟⊕〈ξ〉, we have
g(∇ZW,ϕX)=g(∇̅ZW,ϕX)=-g(ϕ∇̅ZW,X)=g((∇̅Zϕ)W,X)-g(∇̅ZϕW,X).
Then, using (2.6) and (2.14), we obtain
g(∇ZW,ϕX)=g(PZW,X)+g(AϕWZ,X).
Thus from (2.7) and the property (p3), we arrive at
g(∇ZW,ϕX)=-g(W,PZX)+g(h(Z,X),ϕW).
Again using (2.7) and (2.17)(i), we get
g(∇ZW,ϕX)=g(PXZ,W)+g(AϕWX,Z).
On the other hand, from (2.10) and (2.15), we have
PXZ=-T∇XZ-AFZX-Bh(X,Z).
Taking the product with W∈D⊥ and using (3.36), we obtain
g(PXZ,W)=-g(T∇XZ,W)+(ϕXμ)g(Z,W)+g(h(X,Z),FW).
The first term of right-hand side of above equation is zero using the fact that TW=0, for any W∈𝒟⊥. Again using (2.7), we get
g(PXZ,W)=(ϕXμ)g(Z,W)+g(AϕWX,Z).
Thus from (3.36), we derive
g(PXZ,W)=(ϕXμ)g(Z,W)-(ϕXμ)g(Z,W)=0.
Then from (3.36), (3.46), and (3.50), we obtain
g(∇ZW,ϕX)=-(ϕXμ)g(Z,W).
Let M⊥ be a leaf of 𝒟⊥, and let h⊥ be the second fundamental form of the immersion of M⊥ into M. Then for any Z,W∈𝒟⊥, we have
g(h⊥(Z,W),ϕX)=g(∇ZW,ϕX).
Hence, from (3.51) and (3.52), we conclude that
g(h⊥(Z,W),ϕX)=-(ϕXμ)g(Z,W).
This means that integral manifold M⊥ of 𝒟⊥ is totally umbilical in M. Since the anti-invariant distribution 𝒟⊥ of a semi-invariant submanifold M is always integrable (Theorem 3.3) and V(μ)=0 for each V∈𝒟⊥, which implies that the integral manifold of 𝒟⊥ is an extrinsic sphere in M; that is, it is totally umbilical and its mean curvature vector field is nonzero and parallel along M⊥. Hence by virtue of results obtained in [11], M is locally a warped product MT×fM⊥, where MT and M⊥ denote the integral manifolds of the distributions 𝒟⊕〈ξ〉 and 𝒟⊥, respectively and f is the warping function. Thus the theorem is proved.

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