Submanifolds of F structure manifold satisfying FK+(−)K+1F=0

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(Received 2 May 2000)

Abstract.The purpose of this paper is to study invariant submanifolds of an n -dimensional manifoldMendowed with anF-structure satisfyingFK+()K+1F=0 and FW+()W+1F0 for 1< W < K, whereK is a fixed positive integer greater than 2. The case whenK is odd(≥3)has been considered in this paper. We show that an in-variant submanifold ˜M, embedded in anF-structure manifoldMin such a way that the complementary distributionDmis never tangential to the invariant submanifoldΨ(M)˜ , is an almost complex manifold with the induced ˜F-structure. Some theorems regarding the integrability conditions of induced ˜F-structure are proved.

2000 Mathematics Subject Classification. 53C15, 53C40, 53D10.

1. Introduction. Invariant submanifolds have been studied by Blair et al. [1], Kubo [4], Yano and Okumura [7,8], and among others. Yano and Ishihara [6] have studied and shown that any invariant submanifold of codimension 2 in a contact Riemannian manifold is also a contact Riemannian manifold. We consider anF-structure manifold

M and study its invariant submanifolds. LetF be a nonzero tensor field of the type

(1,1)and of classC∞on ann-dimensional manifoldMsuch that (see [3])

FK+()K+1F=0, FW+()W+1F0, for 1< W < K, (1.1)

whereKis a fixed positive integer greater than 2. Such a structure onM is called an

F-structure of rankrand of degreeK. If the rank ofF is constant andr=r (F), then

Mis called anF-structure manifold of degreeK(≥3). Let the operator onMbe defined as follows (see [3])

=(−)KFK−1, m=I+()K+1FK−1, (1.2)

whereI denotes the identity operator onM. For the operators defined by (1.2), we have

+m=I, 2=; m2=m. (1.3)

ForFsatisfying (1.1), there exist complementary distributionDandDm correspond-ing to the projection operators and m, respectively. If rank(F)=constant onM, then dimD=r and dimDm=(n−r ). We have the following results (see [3]).

F=F=F, Fm=mF=0, (1.4a)

FK−1=()K, FK−1= −, FK−1m=0. (1.4b)


2. Invariant submanifolds ofF-structure manifold. Let ˜Mbe a differentiable man-ifold embedded differentially as a submanman-ifold in ann-dimensionalC∞Riemannian

manifoldMwith anF-structure and we denote its embedding byΨ: ˜M→M. Denote byB :T (M)˜ →T (M)the differential mapping of Ψ, where =B is the Jacobson map of Ψ. T (M)˜ and T (M)are tangent bundles of ˜M and M, respectively. We call

T (M,M)˜ as the set of all vectors tangent to the submanifoldΨ(M)˜ . It is known that

B:T (M)˜ →T (M,M)˜ is an isomorphism (see [5]).

Let ˜Xand ˜Y be twoC∞vector fields defined alongΨ(M)˜ and tangent toΨ(M)˜ . Let

XandY be the local extensions of ˜Xand ˜Y. The restriction of[X,Y ]M˜is determined

independently of the choice of these local extensionsX and Y. Therefore, we can



X,Y˜=[X,Y ]M˜. (2.1)

SinceBis an isomorphism, it is easy to see that[BX,B˜ Y ]˜ =B[X,˜ Y ]˜ for all ˜X,Y˜∈T (M)˜ . We denote byGthe Riemannian metric tensor ofMand put


gX,˜Y˜=gBX,B˜ Y˜ ∀X,˜Y˜inT (M),˜ (2.2)

wheregis the Riemannian metric inMand ˜gis the induced metric of ˜M. Definition2.1. We say that ˜Mis an invariant submanifold ofMif

(i) the tangent spaceTp(Ψ(M))˜ of the submanifoldΨ(M)˜ is invariant by the linear mappingF at each pointpofΨ(M)˜ ,

(ii) for each ˜X∈T (M)˜ , we have

F(K−1)/2BX˜=BX˜. (2.3)

Definition2.2. Let ˜Fbe a(1,1)-tensor field defined in ˜Msuch that ˜F(X)˜ =X˜ and

Mis an invariant submanifold, then we have

FBX˜=BF˜X˜, (2.4a)

F(K−1)/2BX˜=BF˜(K−1)/2X˜. (2.4b)

We see that there are two cases for any invariant submanifold ˜M. We assume the following cases.

Case1. The distributionDmis never tangential toΨ(M)˜ . Case2. The distributionDmis always tangential toΨ(M)˜ .

We will considerCase 1 and assume that no vector field of the typemX, where

X∈T (Ψ(M))˜ is tangential toΨ(M)˜ .

Theorem2.3. An invariant submanifoldM˜ is an almost complex manifold if the following two conditions are satisfied:

(i) the distributionDmis never tangential toΨ(M)˜ , and

(ii) ˜F inM˜defines an induced almost complex structure satisfyingF˜K−1=()KI. Proof. ApplyingF(K−1)/2in (2.4), we obtain


Making use of (2.4a) in (2.5), we get

FK−1BX˜=BF˜K−1X˜. (2.6)

In order to show that vector fields of the typeBX˜belong to the distributionD, we suppose thatm(BX)˜ ≠0, then using (1.2) we have

mBX˜=I+(−)K+1FK−1BX˜=BX˜+()K+1FK−1BX˜ (2.7)

which in view of (2.6) becomes

mBX˜=BX˜+(−)K+1BF˜K−1X˜=BX˜+()K+1F˜K−1X˜ (2.8)

which, contrary to our assumption, shows thatm(BX)˜ is tangential toΨ(M)˜ . Thus

m(BX)˜ =0.

Also, in view of (1.4b), (1.3), and (2.6) we obtain

BF˜K−1X˜=FK−1BX˜=()KBX˜=()K(Im)BX˜ =(−)KBX˜()KmBX,˜



SinceBis an isomorphism, we get


FK−1=()KI. (2.10)

LetᏲ(M)be the ring of real-valued differentiable functions onM, and letᐄ(M)be the module of derivatives ofᏲ(M). Thenᐄ(M)is Lie algebra over the real numbers and the elements ofᐄ(M)are called vector fields. ThenMis equipped with(1,1)-tensor fieldFwhich is a linear map such that

F:ᐄ(M) →ᐄ(M). (2.11)

LetM be of degreeK and letK be a positive odd integer greater than 2. Then we consider a positive definite Riemannian metric with respect to whichD andDmare orthogonal so that

g(X,Y )=g(HX,HY )+g(mX,Y ), (2.12)

whereH=F(K−1)/2for allX,Y(M).

Definition2.4. The induced metric ˜gdefined by (2.2) is Hermitian if the following is satisfied:


gHX,H˜ Y˜=g˜X,˜Y˜, whereH=F(K−1)/2. (2.13) Theorem2.5. IfF-structure manifold has the following two properties, that is,

(a) ˜Mis an invariant submanifold ofF-structure manifoldMsuch that distribution

Dmis never tangential toΨ(M)˜ ,


Proof. In view of (2.2) and (2.13) we obtain


gF˜(K−1)/2X,˜F˜(K−1)/2Y˜=gBF˜(K−1)/2X,B˜ F˜(K−1)/2Y˜. (2.14)

Applying (2.4) and (2.12) in (2.14), we get


gF˜(K−1)/2X,˜F˜(K−1)/2Y˜=gF(K−1)/2BX,F˜ (K−1)/2BY˜

=gBX,B˜ Y˜−gmBX,B˜ Y˜. (2.15)

Since the distributionDmis never tangential toΨ(M)˜ , on using (2.2) we get


gF˜(K−1)/2X,˜ F˜(K−1)/2Y˜=gBX,B˜ Y˜=g˜X,˜Y˜. (2.16)

Now, we consider the second case and assume that the distributionDm is always tangential toΨ(M)˜ . In view ofCase 2, we havemBX˜=BX˜, where ˜XT (M)˜ for

some ˜X∗T (M)˜ .

We define (1,1)-tensor fields ˜mand ˜in ˜Mas follows:


=(−)KF˜K−1, m˜=I˜+()K+1F˜K−1, (2.17a)


mX˜=X˜, mBX˜=Bm˜X˜. (2.17b)

Theorem2.6. We have

B˜X˜=BX˜. (2.18) Proof. In view of (2.17a), equation (2.18) assumes the following form:

B˜X˜=B(−)KF˜K−1X˜=()KBF˜K−1X˜. (2.19)

Making use of (2.6) and (2.15) in (2.19), we get

B˜X˜=(−)KF˜K−1BX˜=˜BX˜. (2.20)

Theorem2.7. For˜andm˜ satisfying (2.17a), we have


+m˜ =˜I, ˜2=,˜ m˜2=m.˜ (2.21)

Proof. From (1.3) we have+m=I, which can be written as+mBX˜=BX˜, thus we have

BX˜+mBX˜=BX˜ (2.22) which in view of (2.17b) and (2.18) becomes

B˜X˜+Bm˜X˜=B˜+m˜X˜=BX.˜ (2.23)

Therefore ˜+m˜ =˜IsinceBis an isomorphism. Proof of the other relations follows in a similar manner.


Theorem2.8. IfF-structure manifold has the following property, that is,M˜ is an invariant submanifold ofF-structure manifoldMsuch that distributionDmis always tangential toΨ(M)˜ . Then there exists an inducedF˜-structure manifold which admits a similar Riemannian metricg˜satisfying


gX,˜ Y˜=g˜H˜X,˜H˜Y˜+g˜m˜X˜Y˜. (2.24) Proof. From (2.4b) we get

BF˜(K−1)/2X˜=F(K−1)/2BX˜. (2.25)


BF˜KX˜=FKBX˜ (2.26)

which in view of (1.1) and (2.4a) yields

BF˜KX˜=B()K+1F˜X˜ (2.27)

which shows that ˜F defines an ˜F-structure manifold which satisfies


FK+()K+1F˜=0. (2.28)

In consequence of (2.2), (2.4b), and (2.12) we obtain


gH,˜ X,˜H˜Y˜+g˜m˜X,˜Y˜=gBH˜X,B˜ H˜Y˜+gBm˜X,B˜ Y˜ =gHBX,HB˜ Y˜+gmBX,B˜ Y˜

=gBX,B˜ Y˜, where ˜H=F˜(K−1)/2


which in view of the fact thatBis an isomorphism gives


gH,˜ X,˜H˜Y˜+g˜m˜X,˜Y˜=g˜X,˜Y˜. (2.30)

3. Integrability conditions. The Nijenhuis tensorNof the type (1.2) ofFsatisfying (1.1) inMis given by (see [2])

N(X,Y )=[FX,FY ]−F[FX,Y ]−F[X,F,Y ]+F2[X,Y ], (3.1)

and the Nijenhuis tensor ˜Nof ˜Fsatisfying (2.28) in ˜Mis given by

NX,˜Y˜=F˜X,˜ F˜Y˜−F˜F˜X,˜Y˜−F˜X˜F˜Y˜+F˜2X,˜Y˜. (3.2)

Theorem3.1. The Nijenhuis tensorsNandN˜ofMandM˜given by (3.1) and (3.2) satisfy the following relation:

NBX,B˜ Y˜=BN˜X,˜Y˜. (3.3)

Proof. We have


which in view of (2.4a) becomes

NBX,B˜ Y˜=BF˜X,˜F˜Y˜−FBF˜X˜,BY˜−FBX,B˜ F˜Y˜+F2BX,B˜ Y˜ =BF˜X,˜F˜Y˜−FB[F˜X,˜ Y˜−FBX,˜F˜Y˜+BF2X,˜Y˜

=BF˜X,˜F˜Y˜−BF˜F,˜X,˜Y˜−BF˜X,˜ F˜Y˜+BF˜2X,˜Y˜=BN˜X,˜Y˜.


Theorem3.2. The following identities hold:

BN˜˜X,˜ ˜Y˜=NB˜ X,˜B˜ Y˜, BN˜m˜X,˜m˜Y˜=NmB˜ X,˜ mB˜ Y˜,

Bm˜n˜X,˜Y˜=mNBX,B˜ Y˜. (3.6)

Proof. The proof of (3.6) follows by virtue ofTheorem 3.1, equations (1.4a), (2.4a), (2.17a), (2.17b), and (3.3).

For ˜F satisfying (2.28), there exists complementary distributionD˜andDm˜

corre-sponding to the projection operators ˜and ˜min ˜Mgiven by (2.17a). Then in view of the integrability conditions of ˜F structure we state the following theorems.

Theorem 3.3. If D is integrable inM, then D˜is also integrable inM˜. IfDm is

integrable inM, thenDm˜ is also integrable inM˜.

Theorem3.4. IfD andDm are both integrable inM, thenD˜andDm˜ are also

integrable inM˜.

Theorem3.5. IfF-structure is integrable inM, then the induced structureF˜is also integrable inM˜.


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Lovejoy S. Das: Department of Mathematics and Computer Science, Kent State University, Tuscarawas Campus, New Philadelphia, OH44663, USA