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-STRUCTURE MANIFOLD SATISFYING
LOVEJOY S. DAS
(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+1F≠0 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. , Kubo , Yano and Okumura [7,8], and among others. Yano and Ishihara  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 )
FK+(−)K+1F=0, FW+(−)W+1F≠0, 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 )
=(−)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 ).
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, wheredΨ =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 ).
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
Deﬁnition2.2. Let ˜Fbe a(1,1)-tensor ﬁeld deﬁned in ˜Msuch that ˜F(X)˜ =X˜ and
Mis an invariant submanifold, then we have
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
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
which in view of (2.6) becomes
which, contrary to our assumption, shows thatm(BX)˜ is tangential toΨ(M)˜ . Thus
Also, in view of (1.4b), (1.3), and (2.6) we obtain
SinceBis an isomorphism, we get
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)
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
=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 ˜X∗∈T (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:
Making use of (2.6) and (2.15) in (2.19), we get
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
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
which in view of (1.1) and (2.4a) yields
which shows that ˜F deﬁnes an ˜F-structure manifold which satisﬁes
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 )
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˜
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
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