# Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds

Authors: Siraj Uddin; S. H. Kon; M. A. Khan; Khushwant Singh
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the typeM⊥×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.


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## Body

## 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] (2.1)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: (2.2)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 (2.3)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-formΦ 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 (2.4)(∇̅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 [9]).
On a nearly cosymplectic manifold, the vector fieldξ 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.LetM 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 (2.5)∇̅XY=∇XY+h(X,Y),(2.6)∇̅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 (2.7)g(h(X,Y),N)=g(ANX,Y),
where g denotes the Riemannian metric on M̅ as well as being induced on M.For anyX∈TM, we write (2.8)ϕX=TX+FX,
where TX is the tangential component and FX is the normal component of ϕX.Similarly for anyN∈T⊥M, we write (2.9)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 (2.10)(∇XT)Y=∇XTY-T∇XY,(2.11)(∇̅XF)Y=∇X⊥FY-F∇XY
for all X,Y∈TM.LetM 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 submanifoldM 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,  (i)
TM=𝒟⊕𝒟⊥⊕〈ξ〉,(ii)
ϕ(𝒟x)⊆Dx,(iii)
ϕ(𝒟x⊥)⊂Tx⊥Mfor anyx∈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.LetM 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 (2.12)Tx⊥M=F(TxM)⊕νx.A semi-invariant submanifoldM 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 (2.13)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.
LetM=M1×fM2 be a warped product manifold with the warping function  f. Then (i)
∇XY∈TM1,  for each X,Y∈TM1,(ii)
∇XZ=∇ZX=(Xlnf)Z, for each  X∈TM1 and Z∈TM2,(iii)
∇ZW=∇ZM2W-(g(Z,W)/f)gradf,where  ∇ and  ∇M2 denote the Levi-Civita connections on  M and  M2, respectively.In the above lemmagradf 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(i)
M1 is totally geodesic in M;(ii)
M2 is totally umbilical in M.Now, we denote by𝒫XY and 𝒬XY the tangential and normal parts of (∇̅Xϕ)Y, that is, (2.14)(∇̅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 (2.15)PXY=(∇XT)Y-AFYX-Bh(X,Y),(2.16)QXY=(∇̅XF)Y+h(X,TY)-Ch(X,Y).It is straightforward to verify the following properties of𝒫 and 𝒬, which we enlist here for later use: (p1)
(i)𝒫X+YW=𝒫XW+𝒫YW, (ii) 𝒬X+YW=𝒬XW+𝒬YW,(p2)
(i)𝒫X(Y+W)=𝒫XY+𝒫XW, (ii) 𝒬X(Y+W)=𝒬XY+𝒬XW,(p3)
g(𝒫XY,W)=-g(Y,𝒫XW)for allX,Y,W∈TM.On a submanifoldM of a nearly cosymplectic manifold M̅, we obtain from (2.4) and (2.14) that (2.17)(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 submanifoldM 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 submanifoldM=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 anyX∈TM1 and ξ tangent to M2, we have
(3.1)∇̅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
(3.2)0=(Xln⁡f)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 productM=M1×fM2 when ξ∈TM1, where M1 and M2 are the Riemannian submanifolds of M̅. For any X∈TM2, we have (3.3)∇̅Xξ=∇Xξ+h(X,ξ).
By Proposition 2.1, and Lemma 2.2(ii), we obtain (3.4)(i)ξln⁡f=0,(ii)h(X,ξ)=0.
Thus, we consider the warped product semi-invariant submanifolds of a nearly cosymplectic manifold M̅ of the types:  (i)
M=M⊥×fMT,(ii)
M=MT×fM⊥,whereMT 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 submanifoldM=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.
Whenξ∈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
(3.5)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
(3.6)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
(3.7)g(h(X,ϕX),FZ)=-(Zln⁡f)‖X‖2.
Interchanging X by ϕX in (3.7) and using the fact that ξ∈TM⊥, we obtain
(3.8)g(h(X,ϕX),FZ)=(Zln⁡f)‖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 typeM=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.
LetM=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 anyX,Y∈𝒟, we have
(3.9)F[X,Y]=F∇XY-F∇YX.
Using (2.11), we obtain
(3.10)F[X,Y]=(∇̅XF)Y-(∇̅YF)X.
Then by (2.16), we derive
(3.11)F[X,Y]=QXY-h(X,TY)+Ch(X,Y)-QYX+h(Y,TX)-Ch(X,Y).
Thus from (2.17)(ii), we get
(3.12)F[X,Y]=2QXY+h(Y,TX)-h(X,TY).
Now, for any X,Y∈D, we have
(3.13)h(X,TY)+∇XTY=∇̅XTY=∇̅XϕY.
Using the covariant derivative property of ∇̅ϕ, we obtain
(3.14)h(X,TY)+∇XTY=(∇̅Xϕ)Y+ϕ∇̅XY.
Then by (2.5) and (2.14), we get
(3.15)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
(3.16)h(X,TY)+∇XTY=PXY+QXY+T∇XY+Bh(X,Y)+Ch(X,Y).
Equating the normal components of (3.16), we get
(3.17)h(X,TY)=QXY+Ch(X,Y).
Similarly, we obtain
(3.18)h(Y,TX)=QYX+Ch(X,Y).
Then from (3.17) and (3.18), we arrive at
(3.19)h(Y,TX)-h(X,TY)=QYX-QXY.
Hence, using (2.17)(ii), we get
(3.20)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
(3.21)g([Z,W],X)=g(∇̅ZW-∇̅WZ,X).=-g(∇ZX,W)+g(∇WX,Z).
Using Lemma 2.2(ii), we obtain
(3.22)g([Z,W],X)=-(Xln⁡f)g(Z,W)+(Xln⁡f)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.
LetM=MT×fM⊥ be a warped product submanifold of a nearly cosymplectic manifold  M̅. If  X,Y∈TMT and  Z,W∈TM⊥, then (i)
g(𝒫XY,Z)=g(h(X,Y),FZ)=0,(ii)
g(𝒫XZ,W)=g(h(X,Z),FW)-g(h(X,W),FZ)=-(ϕXlnf)g(Z,W)-g(h(X,Z),FW),(iii)
g(h(ϕX,Z),FZ)=(Xlnf)∥Z∥2.Proof.
For a warped product manifoldM=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
(3.23)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
(3.24)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
(3.25)g(PXZ,W)=-g(AFZX,W)-g(Bh(X,Z),W),
Then by (2.2) and (2.7), we get
(3.26)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
(3.27)PZX=∇ZTX-T∇ZX-Bh(X,Z).
Thus using Lemma 2.2(ii), we derive
(3.28)PZX=(TXln⁡f)Z-Bh(X,Z).
Taking inner product with W∈TM⊥ and using (2.2), we obtain
(3.29)g(PZX,W)=(ϕXln⁡f)g(Z,W)+g(h(X,Z),FW).
Then from (2.17)(i), we get
(3.30)g(PXZ,W)=-(ϕXln⁡f)g(Z,W)-g(h(X,Z),FW).
This is the second equality of (ii). Now, from (3.24) and (3.28), we have
(3.31)PXZ+PZX=-T∇XZ-AFZX+(TXln⁡f)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
(3.32)(TXln⁡f)Z=AFZX+2Bh(X,Z).
Taking the product with Z and on using (2.2) and (2.7), we get
(3.33)(ϕXln⁡f)‖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
(3.34){-X+η(X)ξ}ln⁡f‖Z‖2=-g(h(ϕX,Z),FZ).
Thus by (3.4)(i), the above equation reduces to
(3.35)(Xln⁡f)‖Z‖2=g(h(ϕX,Z),FZ).
This proves the lemma completely.Theorem 3.5.
A proper semi-invariant submanifoldM of a nearly cosymplectic manifold  M̅ is locally a semi-invariant warped product if and only if the shape operator of  M satisfies
(3.36)AϕZX=-(ϕXμ)Z,X∈D⊕〈ξ〉,Z∈D⊥
for some function  μ on  M satisfying  V(μ)=0 for each  V∈𝒟⊥.Proof.
IfM=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, supposeM is a semi-invariant submanifold of a nearly cosymplectic manifold M̅ satisfying (3.36). Then
(3.37)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
(3.38)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
(3.39)g(∇XTY,Z)=g(PXY,Z).
Using (2.10) and (2.15), we obtain
(3.40)g(∇XTY,Z)=g(∇XTY,Z)-g(T∇XY,Z)-g(Bh(X,Y),Z).
Thus by (2.2), the above equation reduces to
(3.41)g(T∇XY,Z)=g(h(X,Y),ϕZ).
Hence using (2.7) and (3.36), we get
(3.42)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
(3.43)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
(3.44)g(∇ZW,ϕX)=g(PZW,X)+g(AϕWZ,X).
Thus from (2.7) and the property (p3), we arrive at
(3.45)g(∇ZW,ϕX)=-g(W,PZX)+g(h(Z,X),ϕW).
Again using (2.7) and (2.17)(i), we get
(3.46)g(∇ZW,ϕX)=g(PXZ,W)+g(AϕWX,Z).
On the other hand, from (2.10) and (2.15), we have
(3.47)PXZ=-T∇XZ-AFZX-Bh(X,Z).
Taking the product with W∈D⊥ and using (3.36), we obtain
(3.48)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
(3.49)g(PXZ,W)=(ϕXμ)g(Z,W)+g(AϕWX,Z).
Thus from (3.36), we derive
(3.50)g(PXZ,W)=(ϕXμ)g(Z,W)-(ϕXμ)g(Z,W)=0.
Then from (3.36), (3.46), and (3.50), we obtain
(3.51)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
(3.52)g(h⊥(Z,W),ϕX)=g(∇ZW,ϕX).
Hence, from (3.51) and (3.52), we conclude that
(3.53)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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