# QR-Submanifolds of (p−1)

Authors: Hyang Sook Kim; Jin Suk Pak
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

The purpose of this paper is to studyn-dimensional QR-submanifolds of (p−1)QR-dimension in a quaternionic projective space QP(n+p)/4 and especially to determine such submanifolds under some curvature conditions.


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## Body

## 1. Introduction

LetM be a connected real n-dimensional submanifold of real codimension p of a quaternionic Kähler manifold M¯ with quaternionic Kähler structure {F,G,H}. If there exists an r-dimensional normal distribution ν of the normal bundle TM⊥ such that
(1)Fνx⊂νx,Gνx⊂νx,Hνx⊂νx,Fνx⊥⊂TxM,Gνx⊥⊂TxM,Hνx⊥⊂TxM
at each point x in M, then M is called a  QR-submanifold ofrQR-dimension, where ν⊥ denotes the complementary orthogonal distribution to ν in TM⊥ (cf. [1–3]). Real hypersurfaces, which are typical examples of QR-submanifold with r=0, have been investigated by many authors (cf. [2–9]) in connection with the shape operator and the induced almost contact 3-structure (for definition, see [10–13]). In their paper [2, 3], Kwon and Pak had studied QR-submanifolds of (p-1)QR-dimension isometrically immersed in a quaternionic projective space QP(n+p)/4 and proved the following theorem as a quaternionic analogy to theorems given in [14, 15], which are natural extensions of theorems proved in [6] to the case of QR-submanifolds with (p-1)QR-dimension and also extensions of theorems in [16].Theorem K-P.LetM  be ann-dimensionalQR-submanifold of(p-1)QR-dimension isometrically immersed in a quaternionic projective spaceQP(n+p)/4,and let the normal vector fieldN1  be parallel with respect to the normal connection. If the shape operatorA1  corresponding toN1  satisfies(2)A1ϕ=ϕA1,A1ψ=ψA1,A1θ=θA1,thenπ-1(M)  is locally a product ofM1×M2  where  M1  and  M2  belong to some  (4n1+3)-  and(4n2+3)-dimensional spheres (π  is the Hopf fibration  Sn+p+3(1)→QP(n+p)/4).On the other hand, whenM is a real hypersurface of QP(n+p)/4, if π-1(M) is (1) an Einstein space or (2) a locally symmetric space, then π-1(M) has a parallel second fundamental form (cf. [4, 6, 7, 9]). Projecting the quantities on π-1(M) onto M in QP(n+p)/4, we can consider QR-submanifolds of (p-1)QR-dimension with the conditions corresponding to (1) or (2). In this paper, we will study such QR-submanifolds isometrically immersed in QP(n+p)/4 and obtain Theorem 3 and other results stated in the last Section 5 as quaternionic analogies to theorems given in [16, 17] and as the extensions of theorems given in [18] by using Theorem K-P.


## 2. Preliminaries

LetM¯ be a real (n+p)-dimensional quaternionic Kähler manifold. Then, by definition, there is a 3-dimensional vector bundle V consisting of tensor fields of type (1,1) over M¯ satisfying the following conditions (a), (b), and (c).(a)
In any coordinate neighborhood𝒰¯, there is a local basis {F, G, H} of V such that
(3)F2=-I,G2=-I,H2=-I,FG=-GF=H,GH=-HG=F,HF=-FH=G.(b)
There is a Riemannian metricg which is Hermite with respect to all of F, G, and H.(c)
For the Riemannian connection∇¯ with respect to g,
(4)(∇¯F∇¯G∇¯H)=(0r-q-r0pq-p0)(FGH),wherep, q, and r are local 1-forms defined in 𝒰¯. Such a local basis {F,G,H} is called a  canonical local basis of the bundle V in 𝒰¯ (cf. [10, 19, 20]).For canonical local bases{F,G,H} and {F′,G′,H′} of V in coordinate neighborhoods 𝒰¯ and 𝒰¯′, it follows that in 𝒰¯∩𝒰¯′(5)(F′G′H′)=(sxy)(FGH)(x,y=1,2,3),
where sxy are local differentiable functions with (sxy)∈SO(3) as a consequence of (3). As is well known (cf. [19]), every quaternionic Kähler manifold is orientable.Now letM be an n-dimensional QR-submanifold of (p-1)QR-dimension isometrically immersed in M¯. Then by definition, there is a unit normal vector field N such that νx⊥=Span{N} at each point x in M. We set
(6)U=-FN,V=-GN,W=-HN.
Denoting by 𝒟x the maximal quaternionic invariant subspace TxM∩FTxM∩GTxM∩HTxM of TxM, we have 𝒟x⊥=Span{U,V,W}, where 𝒟x⊥ means the complementary orthogonal subspace to 𝒟x in TxM (cf. [1–3]). Thus, we have
(7)TxM=𝒟x⊕Span{U,V,W},x∈M,
which together with (3) and (6) implies
(8)FTxM,GTxM,HTxM⊂TxM⊕Span{N}.
Therefore, for any tangent vector field X and for a local orthonormal basis {Nα}α=1,…,p(N1∶=N) of normal vectors to M, we have
(9)FX=ϕX+u(X)N,GX=ψX+v(X)N,HX=θX+w(X)N,(10)FNα=-Uα+P1Nα,GNα=-Vα+P2Nα,HNα=-Wα+P3Nα,(α=1,…,p). Then it is easily seen that {ϕ,ψ,θ} and {P1,P2,P3} are skew-symmetric endomorphisms acting on TxM and TxM⊥, respectively. Moreover, the Hermitian property of {F,G,H} implies
(11)g(X,ϕUα)=-u(X)g(N1,P1Nα),g(X,ψVα)=-v(X)g(N1,P2Nα),α=1,…,p,g(X,θWα)=-w(X)g(N1,P3Nα),(12)g(Uα,Uβ)=δαβ-g(P1Nα,P1Nβ),g(Vα,Vβ)=δαβ-g(P2Nα,P2Nβ),α,β=1,…,p,g(Wα,Wβ)=δαβ-g(P3Nα,P3Nβ).
Also, from the hermitian properties g(FX,Nα)=-g(X,FNα), g(GX,Nα)=-g(X,GNα), and g(HX,Nα)=-g(X,HNα), it follows that
(13)g(X,Uα)=u(X)δ1α,g(X,Vα)=v(X)δ1α,g(X,Wα)=w(X)δ1α,
and hence,
(14)g(U1,X)=u(X),g(V1,X)=v(X),g(W1,X)=w(X),Uα=0,Vα=0,Wα=0,α=2,…,p.
On the other hand, comparing (6) and (10) with α=1, we have U1=U, V1=V, and  W1=W, which together with (6) and (14) implies
(15)g(U,X)=u(X),g(V,X)=v(X)g(W,X)=w(X),u(U)=1,v(V)=1,w(W)=1.
In the sequel, we will use the notations U, V, and W instead of U1, V1, and  W1.Next, applyingF to the first equation of (9) and using (10), (14), and (15), we have
(16)ϕ2X=-X+u(X)U,u(X)P1N=-u(ϕX)N.
Similarly, we have
(17)ϕ2X=-X+u(X)U,ψ2X=-X+v(X)V,θ2X=-X+w(X)W,(18)u(X)P1N=-u(ϕX)N,v(X)P2N=-v(ψX)N,w(X)P3N=-w(θX)N,
from which, taking account of the skew symmetry of P1, P2, and P3 and using (11), we also have
(19)u(ϕX)=0,v(ψX)=0,w(θX)=0,ϕU=0,ψV=0,θW=0,P1N=0,P2N=0,P3N=0.
So (10) can be rewritten in the form
(20)FN=-U,GN=-V,HN=-W,FNα=P1Nα,GNα=P2Nα,HNα=P3Nα(α=2,…,p). Applying G and H to the first equation of (9) and using (3), (9), and (20), we have
(21)θX+w(X)N=-ψ(ϕX)-v(ϕX)N+u(X)V,ψX+v(X)N=θ(ϕX)+w(ϕX)N-u(X)W,
and consequently,
(22)ψ(ϕX)=-θX+u(X)V,v(ϕX)=-w(X),θ(ϕX)=ψX+u(X)W,w(ϕX)=v(X).
Similarly, the other equations of (9) yield
(23)ϕ(ψX)=θX+v(X)U,u(ψX)=w(X),θ(ψX)=-ϕX+v(X)W,w(ψX)=-u(X),ϕ(θX)=-ψX+w(X)U,u(θX)=-v(X),ψ(θX)=ϕX+w(X)V,v(θX)=u(X).
From the first three equations of (20), we also have
(24)ψU=-W,v(U)=0,θU=V,w(U)=0,ϕV=W,u(V)=0,θV=-U,w(V)=0,ϕW=-V,u(W)=0,ψW=U,v(W)=0.Equations (14)–(17), (19), and (22)–(24) tell us that M admits the so-called almost contact 3-structure and consequently n=4m+3 for some integer m (cf. [12]).Now let∇ be the Levi-Civita connection on M, and let ∇⊥ be the normal connection induced from ∇¯ in the normal bundle of M. Then Gauss and Weingarten formulae are given by
(25)∇¯XY=∇XY+h(X,Y),(26)∇¯XNα=-AαX+∇X⊥Nα,α=1,…,p,
for X, Y tangent to M. Here h denotes the second fundamental form and Aα the shape operator corresponding to Nα. They are related by h(X,Y)=∑α=1p‍g(AαX,Y)Nα. Furthermore, put
(27)∇X⊥Nα=∑β=1p‍sαβ(X)Nβ,
where (sαβ) is the skew-symmetric matrix of connection forms of ∇⊥.Differentiating the first equation of (9) covariantly and using (4), (9), (10), (14) (25), and (26), we have
(28)(∇Yϕ)X=r(Y)ψX-q(Y)θX+u(X)A1Y-g(A1Y,X)U,(∇Yu)X=r(Y)v(X)-q(Y)w(X)+g(ϕA1Y,X).
From the other equations of (9), we also have
(29)(∇Yψ)X=-r(Y)ϕX+p(Y)θX+v(X)A1Y-g(A1Y,X)V,(∇Yv)X=-r(Y)u(X)+p(Y)w(X)+g(ψA1Y,X),(∇Yθ)X=q(Y)ϕX-p(Y)ψX+w(X)A1Y-g(A1Y,X)W,(∇Yw)X=q(Y)u(X)-p(Y)v(X)+g(θA1Y,X).Next, differentiating the first equation of (20) covariantly and comparing the tangential and normal parts, we have
(30)∇YU=r(Y)V-q(Y)W+ϕA1Y,g(AαU,Y)=-∑β=2p‍s1β(Y)P1βα,α=2,…,p.
From the other equations of (20), we have similarly
(31)∇YV=-r(Y)U+p(Y)W+ψA1Y,g(AαV,Y)=-∑β=2p‍s1β(Y)P2βα,α=2,…,p,∇YW=q(Y)U-p(Y)V+θA1Y,g(AαW,Y)=-∑β=2p‍s1β(Y)P3βα,α=2,…,p.Finally the equation of Gauss is given as follows (cf. [21]):
(32)g(R¯(X,Y)Z,W)=g(R(X,Y)Z,W)+∑α‍{g(AαX,Z)g(AαY,W)-g(AαY,Z)g(AαX,W)},
for X,Y, and Z tangent to M, where R¯ and R denote the Riemannian curvature tensor of M¯ and M, respectively.In the rest of this paperwe assume that the distinguished normal vector fieldN1:=N  is parallel with respect to the normal connection∇⊥. Then it follows from (27) that s1β=0, and consequently, (30)-(31) imply
(33)AαU=0,AαV=0,AαW=0,α=2,…,p.On the other hand, since the curvature tensorR¯ of QP(n+p)/4 is of the form
(34)R¯(X¯,Y¯)Z¯=g(Y¯,Z¯)X¯-g(X¯,Z¯)Y¯+g(FY¯,Z¯)FX¯-g(FX¯,Z¯)FY¯-2g(FX¯,Y¯)FZ¯+g(GY¯,Z¯)GX¯-g(GX¯,Z¯)GY¯-2g(GX¯,Y¯)GZ¯+g(HY¯,Z¯)HX¯-g(HX¯,Z¯)HY¯-2g(HX¯,Y¯)HZ¯
for X¯,Y¯, and Z¯ tangent to QP(n+p)/4, (32) reduces to
(35)R(X,Y)Z=g(Y,Z)X-g(X,Z)Y+g(ϕY,Z)ϕX-g(ϕX,Z)ϕY-2g(ϕX,Y)ϕZ+g(ψY,Z)ψX-g(ψX,Z)ψY-2g(ψX,Y)ψZ+g(θY,Z)θX-g(θX,Z)θY-2g(θX,Y)θZ+∑α‍{g(AαY,Z)AαX-g(AαX,Z)AαY}.


## 3. Fibrations and Immersions

From now onn-dimensional QR-submanifolds of (p-1)QR-dimension isometrically immersed in QP(n+p)/4 only will be considered. Moreover, we will use the assumption and the notations as in Section 2.LetSn+p+3(a) be the hypersphere of radius a (>0) in Q(n+p+4)/4 the quaternionic space of quaternionic dimension (n+p+4)/4, which is identified with the Euclidean (n+p+4)-space ℝn+p+4. The unit sphere Sn+p+3(1) will be briefly denoted by Sn+p+3. Let π~:Sn+p+3→QP(n+p)/4 be the natural projection of Sn+p+3 onto QP(n+p)/4 defined by the Hopf fibration S3→Sn+p+3→QP(n+p)/4. As is well known (cf. [10, 11, 20]), Sn+p+3 admits a Sasakian 3-structure whereby ξ~,η~,andζ~ are mutually orthogonal unit Killing vector fields. Thus it follows that
(36)∇~ξ~ξ~=0,∇~η~η~=0,∇~ζ~ζ~=0,∇~ζ~η~=-∇~η~ζ~=ξ~,∇~ξ~ζ~=-∇~ζ~ξ~=η~,∇~η~ξ~=-∇~ξ~η~=ζ~,
where ∇~ denotes the Riemannian connection with respect to the canonical metric g~ on Sn+p+3 (cf. [6, 9–13]). Moreover, each fibre π~-1(x) of x in QP(n+p)/4 is a maximal integral submanifold of the distribution spanned by ξ~,η~, and ζ~. Thus the base space QP(n+p)/4 admits the induced quaternionic Kähler structure of constant Q-sectional curvature 4 (cf. [10, 11]). We have especially a fibration π:π-1(M)→M which is compatible with the Hopf fibration π~. More precisely speaking, π:π-1(M)→M is a fibration with totally geodesic fibers such that the following diagram is commutative:
(37)
where i′:π-1(M)→Sn+p+3 and i:M→QP(n+p)/4 are isometric immersions.Now, letξ,η, and ζ be the unit vector fields tangent to the fibers of π-1(M) such that i*′ξ=ξ~, i*′η=η~, and i*′ζ=ζ~. (In what follows, we will again delete the i′ and i*′ in our notation.) Furthermore, we denote by X* the horizontal lift of a vector field X tangent to M. Then, the horizontal lifts Nα* (α=1,…,p) of the normal vectors Nα to M form an orthonormal basis of normal vectors to π-1(M) in Sn+p+3. Let Aα′ and sαβ′ be the corresponding shape operators and normal connection forms, respectively. Then, as shown in [3, 9, 22], the fundamental equations for the submersion π are given by
(38)∇′X*Y*=(∇XY)*+g′((ϕX)*,Y*)ξ+g′((ψX)*,Y*)η+g′((θX)*,Y*)ζ,(39)[X*,Y*]=[X,Y]*+2g′((ϕX)*,Y*)ξ+2g′((ψX)*,Y*)η+2g′((θX)*,Y*)ζ,(40)∇′X*ξ=∇′ξX*=-(ϕX)*,∇′X*η=∇′ηX*=-(ψX)*,∇′X*ζ=∇′ζX*=-(θX)*,(41)[X*,ξ]=0,[X*,η]=0,[X*,ζ]=0,
where g′ denotes the Riemannian metric of π-1(M) induced from g~ in Sn+p+3 and ∇′ the Levi-Civita connection with respect to g′.  The same equations are valid for the submersion π~ by replacing ϕ, ψ, and θ (resp., ξ, η, and ζ) with F, G, and H (resp., ξ~, η~, and ζ~), respectively. We denote by ∇′⊥ the normal connection of π-1(M) induced from ∇~. Since the diagram is commutative, ∇~X*Nα* implies
(42)∇′X*⊥Nα*-Aα′X*=(∇¯XNα)*+g~((FX)*,Nα*)ξ~+g~((GX)*,Nα*)η~+g~((HX)*,Nα*)ζ~=-(AαX)*+g(Uα,X)*ξ+g(Vα,X)*η+g(Wα,X)*ζ+(∇X⊥Nα)*
because of (10), (26), and (38), from which, comparing the tangential part, we have
(43)Aα′X*=(AαX)*-g(Uα,X)*ξ-g(Vα,X)*η-g(Wα,X)*ζ.
Next, calculating ∇~ξNα* and using (10), (26), and (40), we have
(44)∇′ξ⊥Nα*-Aα′ξ=-(FNα)*=Uα*-(P1Nα)*,
which yields
(45)Aα′ξ=-Uα*
and similarly
(46)Aα′ξ=-Uα*,Aα′η=-Vα*,Aα′ζ=-Wα*.
Hence, (43) and (46) with α=1 imply
(47)A1′X*=(A1X)*-g(U,X)*ξ-g(V,X)*η-g(W,X)*ζ,A1′ξ=-U*,A1′η=-V*,A1′ζ=-W*.


## 4. Co-Gauss Equations for the Submersionπ:π-1(M)→M

In this section, we derive the co-Gauss and co-Codazzi equations of the submersionπ:π-1(M)→M for later use.Differentiating (38) with Y=U covariantly along π-1(M) and using (24), (38), and (39), we have
(48)′∇Y*∇′X*U*0000=(∇Y∇XU)*+{v(X)θY-w(X)ψY}*000000+g(ϕY,∇XU)*ξ0000+{g(ψY,∇XU)+g(∇YX,W)+g(X,∇YW)}*η0000+{g(θY,∇XU)-g(∇YX,V)-g(X,∇YV)}*ζ.
Similarly (38) with Y=V and (38) with Y=W give
(49)′∇Y*∇′X*V*0000=(∇Y∇XV)*+{w(X)ϕY-u(X)θY}*000000+{g(ϕY,∇XV)-g(∇YX,W)-g(X,∇YW)}*ξ000000+g(ψY,∇XV)*η000000+{g(θY,∇XV)+g(∇YX,U)+g(X,∇YU)}*ζ,(50)′∇Y*∇′X*W*0000=(∇Y∇XW)*000000-{v(X)ϕY-u(X)ψY}*000000+{g(ϕY,∇XW)+g(∇YX,V)+g(X,∇YV)}*ξ000000+{g(ψY,∇XW)-g(∇YX,U)-g(X,∇YU)}*η000000+g(θY,∇XW)*ζ,
respectively. On the other hand, it follows from (19), (24), (38), and (39) that
(51)′∇[Y*,X*]U*=(∇[Y,X]U)*+2g(ψY,X)*W*-2g(θY,X)*V*+g([Y,X],W)*η-g([Y,X],V)*ζ,(52)′∇[Y*,X*]V*=(∇[Y,X]V)*-2g(ϕY,X)*W*+2g(θY,X)*U*-g([Y,X],W)*ξ+g([Y,X],U)*ζ,(53)′∇[Y*,X*]W*=(∇[Y,X]W)*+2g(ϕY,X)*V*-2g(ψY,X)*U*+g([Y,X],V)*ξ-g([Y,X],U)*η.
By means of (48) and (51), we have
(54)R′(Y*,X*)U*={R(Y,X)U}*+{w(Y)ψX-w(X)ψY-v(Y)θX000+v(X)θY+2g(θY,X)V000-2g(ψY,X)W}*+{g(ϕY,∇XU)-g(ϕX,∇YU)}*ξ+{g(ψY,∇XU)-g(ψX,∇YU)000+g(X,∇YW)-g(Y,∇XW)}*η+{g(θY,∇XU)-g(θX,∇YU)000-g(X,∇YV)+g(Y,∇XV)}*ζ,
where R′ denotes the curvature tensor of π-1(M) with respect to the connection ∇′. Using (30), (31), (33), and (35), we can easily see that
(55)R′(Y*,X*)U*={u(A1X)A1Y-u(A1Y)A1Xu(X)Y-u(Y)X0+u(A1X)A1Y-u(A1Y)A1X}*+{u(A1X)A1Y-u(A1Y)A1Xr(Y)w(X)-r(X)w(Y)000+q(Y)v(X)-q(X)v(Y)000+u(X)u(A1Y)-u(Y)u(A1X)}*ξ+{u(A1X)A1Y-u(A1Y)A1Xp(X)v(Y)-p(Y)v(X)000+v(X)u(A1Y)-v(Y)u(A1X)}*η+{u(A1X)A1Y-u(A1Y)A1Xp(X)w(Y)-p(Y)w(X)000+w(X)u(A1Y)-w(Y)u(A1X)}*ζ.
By the same method, we can easily verify that (49), (50), (52), and (53) yield
(56)R′(Y*,X*)V*={v(X)Y-v(Y)X+v(A1X)A1Y0-v(A1Y)A1X}*+{q(X)u(Y)-q(Y)u(X)00-u(Y)v(A1X)+u(X)v(A1Y)}*ξ+{+v(X)v(A1Y)-v(Y)v(A1X)r(Y)w(X)-r(X)w(Y)00+p(Y)u(X)-p(X)u(Y)00+v(X)v(A1Y)-v(Y)v(A1X)}*η+{+v(X)v(A1Y)-v(Y)v(A1X)q(X)w(Y)-q(Y)w(X)00-w(Y)v(A1X)+w(X)v(A1Y)}*ζ,R′(Y*,X*)W*={w(X)Y-w(Y)X+w(A1X)A1Y0-w(A1Y)A1X}*+{+v(X)v(A1Y)-v(Y)v(A1X)r(X)u(Y)-r(Y)u(X)00-u(Y)w(A1X)+u(X)w(A1Y)}*ξ+{+v(X)v(A1Y)-v(Y)v(A1X)r(X)v(Y)-r(Y)v(X)00-v(Y)w(A1X)+v(X)w(A1Y)}*η+{q(Y)v(X)-q(X)v(Y)00+p(Y)u(X)-p(X)u(Y)00+w(X)w(A1Y)-w(Y)w(A1X)}*ζ.Differentiating (38) with X=U covariantly along π-1(M) and using (24), we have
(57)∇′Y*∇′U*X*=(∇Y∇UX)*+{w(X)ψY-v(X)θY}*+g(ϕY,∇UX)*ξ+{g(ψY,∇UX)-g(∇YW,X)-g(W,∇YX)}*η+{g(θY,∇UX)+g(∇YV,X)+g(V,∇YX)}*ζ.
Similarly, (38) with X=V and (38) with X=W, respectively, give
(58)∇′Y*∇′V*X*=(∇Y∇VX)*-{w(X)ϕY-u(X)θY}*+g(ψY,∇VX)*η+{g(ϕY,∇VX)+g(∇YW,X)+g(W,∇YX)}*ξ+{g(θY,∇VX)-g(∇YU,X)-g(U,∇YX)}*ζ,(59)∇′Y*∇′W*X*=(∇Y∇WX)*+{v(X)ϕY-u(X)ψY}*00000+g(θY,∇WX)*ζ00000+{g(ϕY,∇WX)-g(∇YV,X)-g(V,∇YX)}*ξ00000+{g(ψY,∇WX)+g(∇YU,X)+g(U,∇YX)}*η.
Differentiating (38) also covariantly in the direction of  U* and using (24), we have
(60)∇′U*∇′Y*X*=(∇U∇YX)*+g(ψY,X)*W*-g(θY,X)*V*+{g(∇U(ϕY),X)+g(ϕY,∇UX)}*ξ+{g(∇U(ψY),X)+g(ψY,∇UX)-g(W,∇YX)}*η+{g(∇U(θY),X)+g(θY,∇UX)+g(V,∇YX)}*ζ.
Similarly, differentiating (38) covariantly in the direction of V* and W*, respectively, we have
(61)∇′V*∇′Y*X*=(∇V∇YX)*-g(ϕY,X)*W*-g(θY,X)*U*+{g(∇V(ϕY),X)+g(ϕY,∇VX)+g(W,∇YX)}*ξ+{g(∇V(ψY),X)+g(ψY,∇VX)}*η+{g(∇V(θY),X)+g(θY,∇VX)-g(U,∇YX)}*ζ,(62)∇′W*∇′Y*X*=(∇W∇YX)*-g(ψY,X)*U*+g(ϕY,X)*V*+{g(∇W(ϕY),X)+g(ϕY,∇WX)-g(V,∇YX)}*ξ+{g(∇W(ψY),X)+g(ψY,∇WX)+g(U,∇YX)}*η+{g(∇W(θY),X)+g(θY,∇WX)}*ζ.On the other hand, (38) and (39) with X=U imply
(63)∇′[Y*,U*]X*=(∇[Y,U]X)*-2{w(Y)ψX-v(Y)θX}*+g(ϕ[Y,U],X)*ξ+g(ψ[Y,U],X)*η+g(θ[Y,U],X)*ζ.
Similarly,  from (39) with X=V and (39) with X=W, respectively, we find that
(64)∇′[Y*,V*]X*=(∇[Y,V]X)*+2{w(Y)ϕX-u(Y)θX}*+g(ϕ[Y,V],X)*ξ+g(ψ[Y,V],X)*η+g(θ[Y,V],X)*ζ,(65)∇′[Y*,W*]X*=(∇[Y,W]X)*-2{v(Y)ϕX-u(Y)ψX}*+g(ϕ[Y,W],X)*ξ+g(ψ[Y,W],X)*η+g(θ[Y,W],X)*ζ.Using (28)–(31), it follows from (57), (60), and (63) that
(66)R′(Y*,U*)X*={R(Y,U)X}*+{w(X)ψY-v(X)θY000-g(ψY,X)W+g(θY,X)V000+2w(Y)ψX-2v(Y)θX}*-{r(U)g(ψY,X)-q(U)g(θY,X)000+u(Y)u(A1X)+r(Y)w(X)000+q(Y)v(X)-g(A1Y,X)}*ξ-{-p(Y)v(X)-r(U)g(ϕY,X)000+p(U)g(θY,X)+v(Y)u(A1X)}*η-{-p(Y)w(X)+q(U)g(ϕY,X)000-p(U)g(ψY,X)+w(Y)u(A1X)}*ζ,
from which, taking account of (35) and using (24) and (33), we obtain
(67)R′(Y*,U*)X*={u(X)Y-g(Y,X)U0+u(A1X)A1Y-g(A1Y,X)A1U}*-{r(U)g(ψY,X)-q(U)g(θY,X)00+u(Y)u(A1X)+r(Y)w(X)00+q(Y)v(X)-g(A1Y,X)}*ξ-{-p(Y)v(X)-r(U)g(ϕY,X)00+p(U)g(θY,X)+v(Y)u(A1X)}*η-{-p(Y)w(X)+q(U)g(ϕY,X)00-p(U)g(ψY,X)+w(Y)u(A1X)}*ζ.
Similarly, by using (58), (59), (61), (62), (64), and (65), we can easily obtain
(68)R′(Y*,V*)X*={v(X)Y-g(Y,X)V00+v(A1X)A1Y-g(A1Y,X)A1V}*-{-r(V)g(ϕY,X)+p(V)g(θY,X)000+v(Y)v(A1X)+r(Y)w(X)000+p(Y)u(X)-g(A1Y,X)}*η+{q(Y)u(X)-r(V)g(ψY,X)000+q(V)g(θY,X)-u(Y)v(A1X)}*ξ-{-q(Y)w(X)+q(V)g(ϕY,X)000-p(V)g(ψY,X)+w(Y)v(A1X)}*ζ,R′(Y*,W*)X*={w(X)Y-g(Y,X)W0+w(A1X)A1Y-g(A1Y,X)A1W}*-{q(W)g(ϕY,X)-p(W)g(ψY,X)00+w(Y)w(A1X)+q(Y)v(X)00+p(Y)u(X)-g(A1Y,X)}*ζ-{-r(Y)u(X)+r(W)g(ψY,X)000-q(W)g(θY,X)+u(Y)w(A1X)}*ξ+{r(Y)v(X)+r(W)g(ϕY,X)00-p(W)g(θY,X)-v(Y)w(A1X)}*η.


## 5.   Main Results

It is well known [3] that if π-1(M) is locally symmetric then ∇′A1=0 which implies identities (2) in Theorem K-P. In this point of view, we consider the following assumptions in (69) which are weaker conditions than the locally symmetry of π-1(M).In order to obtain our main results, letM be n-dimensional QR-submanifolds of (p-1)QR-dimension in QP(n+p)/4 with the assumptions
(69)(∇′ξR′)(Y*,X*)U*=0,(∇′ηR′)(Y*,X*)V*=0,(∇′ζR′)(Y*,X*)W*=0,(∇′ξR′)(Y*,U*)X*=0,(∇′ηR′)(Y*,V*)X*=0,(∇′ζR′)(Y*,W*)X*=0.We notice here that the above curvature conditions in (69) are different from those in [18] due to Pak and Sohn.We first consider the assumption(70)(∇′ξR′)(Y*,X*)U*=0.Differentiating (55) covariantly in the direction of ξ and using (19), (36), and (40), and the assumption (∇ξ′R′)(Y*,X*)U*=0, we have
(71)-R′((ϕY)*,X*)U*-R′(Y*,(ϕX)*)U*0000={-u(X)ϕY+u(Y)ϕX00000000-u(A1X)ϕA1Y+u(A1Y)ϕA1X}*000000+{p(X)w(Y)-p(Y)w(X)000000000+w(X)u(A1Y)-w(Y)u(A1X)}*η000000-{p(X)v(Y)-p(Y)v(X)000000000+v(X)u(A1Y)-v(Y)u(A1X)}*ζ,
from which, taking the vertical component of ξ,η, and ζ, respectively, and using (22)–(24) and (55) itself, we can get
(72)r(X)v(Y)-r(Y)v(X)-q(X)w(Y)0000+q(Y)w(X)-r(ϕY)w(X)+r(ϕX)w(Y)0000-q(ϕY)v(X)+q(ϕX)v(Y)0000-u(X)u(A1ϕY)+u(Y)u(A1ϕX)=0,(73)-u(A1ϕY)v(X)+u(A1ϕX)v(Y)+p(ϕY)v(X)0000-p(ϕX)v(Y)=0,(74)-u(A1ϕY)w(X)+u(A1ϕX)w(Y)+p(ϕY)w(X)0000-p(ϕX)w(Y)=0.PuttingY=U in (72) and using (19) and (24), we have
(75)ϕA1U+r(U)V-q(U)W=0,
and consequently,
(76)r(U)=w(A1U)=u(A1W),q(U)=v(A1U)=u(A1V).
Putting Y=W and X=V in (73) and using (15) and (24) yield
(77)p(V)=v(A1U)=u(A1V).
Also, putting Y=V and X=W in (74) and using (15) and (24), we have
(78)p(W)=w(A1U)=u(A1W).Summing up, we have(79)A1U=u(A1U)U+p(V)V+p(W)W,p(V)=v(A1U)=u(A1V)=q(U),p(W)=w(A1U)=u(A1W)=r(U).Thus we get the following lemma.Lemma 1.
LetM be an n-dimensional QR-submanifold of  (p-1)QR-dimension in a quaternionic projective space QP(n+p)/4, and let the normal vector field N1 be parallel with respect to the normal connection. If the equalities in (69) are established, then
(80)A1U=u(A1U)U+p(V)V+p(W)W,A1V=q(U)U+v(A1V)V+q(W)W,A1W=r(U)U+r(V)V+w(A1W)W,p(V)=v(A1U)=u(A1V)=q(U),p(W)=w(A1U)=u(A1W)=r(U),q(W)=w(A1V)=v(A1W)=r(V).Next, we assume the additional condition(81)(∇ξ′R′)(Y*,U*)X*=0.Differentiating (67) covariantly in the direction of ξ and using (36), (40) and the assumption (∇ξ′R′)(Y*,U*)X*=0, we have
(82)-R′((ϕY)*,U*)X*-R′(Y*,U*)(ϕX)*0000={-u(X)ϕY-u(A1X)ϕA1Y00000000+g(A1Y,X)ϕA1U}*000000-{-p(Y)w(X)+q(U)g(ϕY,X)000000000-p(U)g(ψY,X)+w(Y)u(A1X)}*η000000+{-p(Y)v(X)-r(U)g(ϕY,X)000000000+p(U)g(θY,X)+v(Y)u(A1X)}*ζ,
from which, taking the vertical component of ξ,η, and ζ, respectively, and using (22)–(24) and (67) itself, we can find
(83)r(U){-2g(θY,X)+u(Y)v(X)-v(Y)u(X)}0000-q(U){2g(ψY,X)+u(Y)w(X)-w(Y)u(X)}0000+u(Y)u(A1ϕX)+r(ϕY)w(X)+r(Y)v(X)0000+q(ϕY)v(X)-q(Y)w(X)0000+g(ϕA1Y-A1ϕY,X)=0,(84)-q(U)g(ϕY,X)+p(ϕY)v(X)-v(Y)u(A1ϕX)0000-p(U){g(ψY,X)+u(Y)w(X)00000000000-w(Y)u(X)}=0,(85)-r(U)g(ϕY,X)+p(ϕY)w(X)-w(Y)u(A1ϕX)0000-p(U){g(θY,X)+v(Y)u(X)-u(Y)v(X)}=0.Taking the skew-symmetric part of (83) with respect to X and Y, we have
(86)2r(U){-2g(θY,X)+u(Y)v(X)-v(Y)u(X)}0000-2q(U){2g(ψY,X)+u(Y)w(X)-w(Y)u(X)}0000+u(Y)u(A1ϕX)-u(X)u(A1ϕY)+r(ϕY)w(X)0000-r(ϕX)w(Y)+r(Y)v(X)-r(X)v(Y)0000+q(ϕY)v(X)-q(ϕX)v(Y)0000-q(Y)w(X)+q(X)w(Y)=0.
Replacing Y with θY in (86) and using (19) and (22)–(24), we have
(87)r(U){4g(Y,X)-3w(Y)w(X)00000-2u(Y)u(X)-2v(Y)v(X)}0000-q(U){4g(ϕY,X)+3w(Y)v(X)-2w(X)v(Y)}0000-q(θY)w(X)-q(ψY)v(X)-q(ϕX)u(Y)0000+r(θY)v(X)-r(X)u(Y)-r(ψY)w(X)0000-v(Y)u(A1ϕX)+u(X)u(A1ψY)0000-u(A1U)u(X)w(Y)=0.
Now we consider the following orthonormal basis:
(88){U,V,W,e1,…,em,ϕ(e1),…,ϕ(em),00ψ(e1),…,ψ(em),θ(e1),…,θ(em)},
which will be called Q-basis, where 4m+3=dimM. Taking the trace of the above equation with respect to the Q-basis and using (76), we can easily see 4mr(U)=0; that is,
(89)r(U)=0.Replacing alsoY with ψY in (86) and using (19) and (22)–(24), we have
(90)q(U)=0.
Substituting (89) and (90) into (75), we have ϕA1U=0 and hence
(91)A1U=u(A1U)U.On the other hand, replacingY with ψY in (84) and using (19), (22), (23), and (90), we obtain
(92)p(U){g(Y,X)-u(Y)u(X)-w(Y)w(X)}0000+p(θY)v(X)=0,
from which, taking the trace with respect to the Q-basis and using (15) and (24), we find 4mp(U)=0; that is,
(93)p(U)=0
which together with (84) and (90) implies
(94)p(ϕY)=0.
Replacing Y with ϕY in the above equation and using (17) and (93), we can easily see that
(95)p(Y)=0
for any vector Y tangent to M.Summing up, we have the following lemma.Lemma 2.
LetM be as in Lemma 1, and let the normal vector field N1 be parallel with respect to the normal connection. If the equalities in (69) and (5.2) are established, then
(96)A1U=u(A1U)U,A1V=v(A1V)V,A1W=w(A1W)W,p=q=r=0.Finally, we will prove our main theorem.Theorem 3.
LetM be an n-dimensional QR-submanifold of (p-1)QR-dimension in a quaternionic projective space QP(n+p)/4, and let the normal vector field N1 be parallel with respect to the normal connection. If the equalities in (69) and (5.2) are established, then π-1(M) is locally a product of M1×M2 where M1 and M2 belong to some (4n1+3)- and (4n2+3)-dimensional spheres (π is the Hopf fibration Sn+p+3(1)→QP(n+p)/4).Proof.
By means of (96), it follows easily from (83) that
(97)ϕA1=A1ϕ.
By the quite same method, we can obtain
(98)ψA1=A1ψ,θA1=A1θ.
Combining with those equalities and Theorem K-P, we complete the proof.Corollary 4.
LetM be an n-dimensional QR-submanifold of (p-1)QR-dimension in a quaternionic projective space QP(n+p)/4, and let the normal vector field N1 be parallel with respect to the normal connection. If the following equalities:
(99)∇′ξR′=0,∇′ηR′=0,∇′ζR′=0
are established, then π-1(M) is locally a product of M1×M2 where M1 and M2 belong to some (4n1+3)- and (4n2+3)-dimensional spheres.



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*Source: 193697-2013-06-04.xml*