MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation23037410.1155/2011/230374230374Research ArticleWarped Product Semi-Invariant Submanifolds of Nearly Cosymplectic ManifoldsUddinSiraj1KonS. H.1KhanM. A.2SinghKhushwant3Elias-ZunigaAlex1Institute of Mathematical SciencesFaculty of ScienceUniversity of Malaya50603 Kuala LumpurMalaysiaum.edu.my2Department of MathematicsUniversity of TabukTabukSaudi Arabiathapar.edu3School of Mathematics and Computer ApplicationsThapar UniversityPatiala 147 004Indiathapar.edu201104092011201120042011160620112011Copyright © 2011 Siraj Uddin et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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., ). Recently, Chen  (see also ) 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  as nearly cosymplectic manifolds, and it was shown that normal nearly cosymplectic manifolds are cosymplectic (see also ). Later on, Blair and Showers  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'  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  ϕ2=-I+ηξ,ϕξ=0,ηϕ=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 .

An almost contact structure (ϕ,ξ,η) is said to be normal if the almost complex structure J on the product manifold M̅× given by J(X,fddt)=(ϕX-fξ,  η(X)ddt), where f is a C-function on M̅×, has no torsion, that is, J is integrable, and the condition for normality in terms of ϕ,  ξ and η is [ϕ,ϕ]+2dηξ=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 . The structure is said to be nearly cosymplectic if ϕ is Killing, that is, if (̅Xϕ)Y+(̅Yϕ)X=0, for any X,YTM̅, 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 XTM̅. 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 TM of M, respectively, then, Gauss and Weingarten formulae are given by ̅XY=XY+h(X,Y),̅XN=-ANX+XN, for each X,YTM and NTM, 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 XTM, we write ϕX=TX+FX, where TX is the tangential component and FX is the normal component of ϕX.

Similarly for any NTM, 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-TXY,(̅XF)Y=XFY-FXY for all X,YTM.

Let M be a Riemannian manifold isometrically immersed in an almost contact metric manifold M̅. then for every xM 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 xM, 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=𝒟𝒟ξ,

ϕ(𝒟x)Dx,

ϕ(𝒟x)TxM

for any xM, where TxM 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 xM. It is called the proper semi-invariant submanifold if neither 𝒟x={0} nor 𝒟x={0}, for every xM.

Let M be a semi-invariant submanifold of an almost contact metric manifold M̅. Then, F(TxM) is a subspace of TxM. Then for every xM, there exists an invariant subspace νx of TxM̅ such that TxM=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  for warped product manifolds.

Lemma 2.2.

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

XYTM1,  for each X,YTM1,

XZ=ZX=(Xlnf)Z, for each XTM1 and ZTM2,

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 UTM. 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,YTM. 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,WTM.

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,YTM.

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 XTM1 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 XTM1, 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 ξTM1, where M1 and M2 are the Riemannian submanifolds of M̅. For any XTM2, 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 ξTMT, then by Theorem 3.1, M is a Riemannian product. Thus, we consider ξTM. For any XTMT and ZTM, we have g(h(X,ϕX),FZ)=g(h(X,ϕX),ϕZ)=g(̅XϕX,ϕZ)=g(ϕ̅XX,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)X2. Interchanging X by ϕX in (3.7) and using the fact that ξTM, we obtain g(h(X,ϕX),FZ)=(Zlnf)X2. It follows from (3.7) and (3.8) that Zlnf=0, for all ZTM. 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,Y𝒟, we have F[X,Y]=FXY-FYX. 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,YD, we have h(X,TY)+XTY=̅XTY=̅XϕY. Using the covariant derivative property of ̅ϕ, we obtain h(X,TY)+XTY=(̅Xϕ)Y+ϕ̅XY. 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+TXY+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,YD. This proves the integrability of D. Now, for the integrability of D, we consider any XD and Z,WD, 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,W𝒟. 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,YTMT and Z,WTM, then

g(𝒫XY,Z)=g(h(X,Y),FZ)=0,

g(𝒫XZ,W)=g(h(X,Z),FW)-g(h(X,W),FZ)=-(ϕXlnf)g(Z,W)-g(h(X,Z),FW),

g(h(ϕX,Z),FZ)=(Xlnf)Z2.

Proof.

For a warped product manifold M=MT×fM, we have that MT is totally geodesic in M; then by (2.10), (̅XT)YTMT, for any X,YTMT, 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=-TXZ-AFZX-Bh(X,Z) for any XTMT and ZTM. Using Lemma 2.2(ii), the first term of right-hand side is zero. Thus, taking the product with WTM, 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-TZX-Bh(X,Z). Thus using Lemma 2.2(ii), we derive PZX=(TXlnf)Z-Bh(X,Z). Taking inner product with WTM 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=-TXZ-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)Z2=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}lnfZ2=-g(h(ϕX,Z),FZ). Thus by (3.4)(i), the above equation reduces to (Xlnf)Z2=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,XDξ,  ZD for some function μ on M satisfying V(μ)=0 for each V𝒟.

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(ϕ̅XY,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(TXY,Z)-g(Bh(X,Y),Z). Thus by (2.2), the above equation reduces to g(TXY,Z)=g(h(X,Y),ϕZ). Hence using (2.7) and (3.36), we get g(TXY,Z)=g(AϕZX,Y)=0, which implies XY𝒟ξ, that is, 𝒟ξ is integrable and its leaves are totally geodesic in M. Now, for any Z,W𝒟 and X𝒟ξ, we have g(ZW,ϕX)=g(̅ZW,ϕX)=-g(ϕ̅ZW,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=-TXZ-AFZX-Bh(X,Z). Taking the product with WD and using (3.36), we obtain g(PXZ,W)=-g(TXZ,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 W𝒟. 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,W𝒟, 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 V𝒟, 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 , M is locally a warped product MT×fM, where MT and M denote the integral manifolds of the distributions 𝒟ξ and 𝒟, respectively and f is the warping function. Thus the theorem is proved.

ChenB.-Y.Geometry of warped product CR-submanifolds in Kaehler manifoldsMonatshefte für Mathematik20011333177195186113610.1007/s006050170019ZBL0996.53045HasegawaI.MihaiI.Contact CR-warped product submanifolds in Sasakian manifoldsGeometriae Dedicata200310214315010.1023/B:GEOM.0000006582.29685.222026842ZBL1066.53103KhanK. A.KhanV. A.UddinS.Warped product submanifolds of cosymplectic manifoldsBalkan Journal of Geometry and its Applications200813155652395375ZBL1161.53036ChenB.-Y.Geometry of warped product CR-submanifolds in Kaehler manifolds. IIMonatshefte für Mathematik2001134210311910.1007/s0060501700021878074ZBL0996.53045AtçekenM.Warped product semi-invariant submanifolds in almost paracontact Riemannian manifoldsMathematical Problems in Engineering200920091610.1155/2009/6216256216252539715BonanzingaV.MatsumotoK.Warped product CR-submanifolds in locally conformal Kaehler manifoldsPeriodica Mathematica Hungarica2004481-2207221207769710.1023/B:MAHU.0000038976.01030.49ZBL1104.53049BlairD. E.Almost contact manifolds with Killing structure tensorsPacific Journal of Mathematics1971392852920305299ZBL0239.53031BlairD. E.YanoK.Affine almost contact manifolds and f-manifolds with affine Killing structure tensorsKōdai Mathematical Seminar Reports197123473479031242610.2996/kmj/1138846415ZBL0234.53041BlairD. E.ShowersD. K.Almost contact manifolds with Killing structure tensors. IIJournal of Differential Geometry197495775820346695BishopR. L.O'NeillB.Manifolds of negative curvatureTransactions of the American Mathematical Society1969145149025166410.1090/S0002-9947-1969-0251664-4ZBL0191.52002HiepkoS.Eine innere Kennzeichnung der verzerrten produkteMathematische Annalen1979241320921510.1007/BF01421206535555ZBL0387.53014