We study submersion of CR-submanifolds
of an l.c.q.K. manifold. We have shown that if an almost Hermitian
manifold B admits a Riemannian submersion π:M→B
of a CR-submanifold M of a locally conformal quaternion Kaehler
manifold M¯, then B is a locally conformal quaternion Kaehler
manifold.

1. Introduction

The concept of locally conformal Kaehler manifolds was introduced by Vaisman in [1]. Since then many papers appeared on these manifolds and their submanifolds (see [2] for details). However, the geometry of locally conformal quaternion Kaehler manifolds has been studied in [2–4] and their QR-submanifolds have been studied in [5].

A locally conformal quaternion Kaehler manifold (shortly, l.c.q.K. manifold) is a quaternion Hermitian manifold whose metric is conformal to a quaternion Kaehler metric in some neighborhood of each point. The main difference between locally conformal Kaehler manifolds and l.c.q.K. manifolds is that the Lee form of a compact l.c.q.K. manifold can be chosen as parallel form without any restrictions [2].

The study of the Riemannian submersion π:M→B of a Riemannian manifold M onto a Riemannian manifold B was initiated by O’Neill [6]. A submersion naturally gives rise to two distributions on M called the horizontal and vertical distributions, respectively, of which the vertical distribution is always integrable giving rise to the fibers of the submersion which are closed submanifold of M. The notion of Cauchy-Riemann (CR) submanifold was introduced by Bejancu [7] as a natural generalization of complex submanifolds and totally real submanifolds. A CR-submanifolds M of a l.c.q.K. manifold M- requires a differentiable holomorphic distribution D, that is, JxDx=Dx for all x∈M, whose orthogonal complement D⊥ is totally real distribution on M, that is, JxDx⊥⊂TM⊥ for all x∈M. A CR-submanifold is called holomorphic submanifold if dim Dx⊥=0, totally real if dim Dx=0 and proper if it is neither holomorphic nor totally real.

A CR-submanifold of a l.c.q.K. manifold M- is called a CR-product if it is Riemannian product of a holomorphic submanifold N⊤ and a totally real submanifold N⊥ of M-. Kobayashi [8] has proved that if an almost Hermitian manifold B admits a Riemannian submersion π:M→B of a CR-submanifold M of a Kaehler Manifold M-, then B is a Kaehler manifold. However, Deshmukh et al. [9] studied similar type of results for CR-submanifolds of manifolds in different classes of almost Hermitian manifolds, namely, Hermitian manifolds, quasi-Kaehler manifolds, and nearly Kaehler manifolds.

In the present paper, we investigate submersion of CR-submanifold of a l.c.q.K. manifold M- and prove that if an almost Hermitian manifold B admits a Riemannian submersion π:M→B of a CR-submanifold M of a l.c.q.K. manifold M-, then B is an l.c.q.K. manifold.

2. Preliminaries

Let (M-,g,H) be a quaternion Hermitian manifold, where H is a subbundle of end (TM-) of rank 3 which is spanned by almost complex structures J1, J2, and J3. The quaternion Hermitian metric g is said to be a quaternion Kaehler metric if its Levi-Civita connection ∇- satisfies ∇-H⊂H.

A quaternion Hermitian manifold with metric g is called a locally conformal quaternion Kaehler (l.c.q.K.) manifold if over neighborhoods {Ui} covering M-, g|Ui=efigi where gi is a quaternion Kaehler metric on Ui. In this case, the Lee form ω is locally defined by ω|Ui=dfi and satisfies [3]
(2.1)dθ=w∧θ,dw=0.
Let M- be l.c.q.K. manifold and ∇- denotes the Levi-Civita connection of M-. Let B be the Lee vector field given by
(2.2)g(X,B)=w(X).
Then for l.c.q.K. manifold, we have [3]
(2.3)(∇~XJa)Y=12{θ(Y)X-w(Y)JaX-g(X,Y)A-Ω(X,Y)B}+Qab(X)JbY+Qac(X)JcY
for any X,Y∈TM-, where Qab is skew-symmetric matrix of local forms θ=w∘Ja and A=-JaB.

We also have
(2.4)θ(X)=g(JaX,B),Ω(X,Y)=g(X,JaY).

Let M be a Riemannian manifold isometrically immersed in M-. Let T(M) be the Lie algebra of vector fields in M and TM⊥, the set of all vector fields normal to M. Denote by ∇ the Levi connection of M. Then the Gauss and Weingarten formulas are given by
(2.5)∇-XY=∇XY+h(X,Y),(2.6)∇-XN=-A-NX+∇X⊥N
for any X,Y∈T(M), and N∈TM⊥, where ∇⊥ is the connection in the normal bundle TM⊥, h is the second fundamental form, and A-N is the Weingarten endomorphism associated with N. The second fundamental form and shape operator are related by
(2.7)g(A-NX,Y)=g(h(X,Y),N).
The curvature tensor R of the submanifold M is related to the curvature tensor R- of M- by the following Gauss formula:
(2.8)R-(X,Y;Z,W)=R(X,Y;Z,W)+g(h(X,Z),h(Y,W))-g(h(X,W),h(Y,Z)),
for any X,Y,Z,W∈T(M).

For submersion of a l.c.q.K. manifold onto an almost Hermitian manifold, we have the following.

Definition 2.1.

Let M be a CR-submanifold of a locally conformal quaternion Kaehler manifold M-. By a submersion π:M→B of M onto an almost Hermitian manifold B, we mean a Riemannian submersion π:M→B together with the following conditions:

D⊥ is the kernel of π*, that is, π*D⊥={0},

π*:Dp→Dπ(p)* is a complex isometry of the subspace Dp onto Dπ(p)* for every p∈M, where Dπ(p)* denotes the tangent space of B at π(p),

J interchanges D⊥ and ν, that is, JD⊥=TM⊥.

For a vector field X on M, we set [8]
(2.9)X=HX+VX,
where H and V denoted the horizontal and vertical part of X.

We recall that a vector field X on M for submersion π:M→B is said to be a basic vector field if X∈D and X is π related to a vector field on B, that is, there is a vector field X* on B such that
(2.10)(π*X)p=(X*)π(p)foreachp∈M.

If J and J′ are the almost complex structures on M- and B, respectively, then from Definition 2.1(ii) we have π*∘J=J′∘π* on D.

We have the following lemma for basic vector fields [6].

Lemma 2.2.

Let X and Y be basic vector fields on M. Then

g(X,Y)=g*(X*,Y*)∘π, g is the metric on M, and g* is the Riemannian metric on B;

the horizontal part H[X,Y] of [X,Y] is a basic vector field and corresponds to [X*,Y*], that is,
(2.11)π*H[X,Y]=[X*,Y*]∘π;

H(∇XY) is a basic vector field corresponding to ∇X**Y*, where ∇* is a Riemannian connection on B;

[X,W]∈D⊥ for W∈D⊥.

For a covariant differentiation operator ∇*, we define a corresponding operator ∇~* for basic vector fields of M by
(2.12)∇~X*Y=H(∇XY),X,Y∈D,
then ∇~X*Y is a basic vector field, and from the above lemma we have
(2.13)π*(∇~X*Y)=∇X**Y*∘π.
Now, we define a tensor field C on M by setting
(2.14)∇XY=H(∇XY)+C(X,Y),X,Y∈D,
that is, C(X,Y) is the vertical component of ∇XY.

In particular, if X and Y are basic vector fields, then we have
(2.15)∇XY=∇~X*Y+C(X,Y).
The tensor field C is skew-symmetric and it satisfies
(2.16)C(X,Y)=12V[X,Y],X,Y∈D.
For X∈D and V∈D⊥ define an operator A on M by setting ∇XV=ν(∇XV)+AXV, that is, AXV is the horizontal component of ∇XV. Using (iv) of Lemma 2.2 we have
(2.17)H(∇XV)=H(∇VX)=AXV.
The operator C and A are related by
(2.18)g(AXV,Y)=-g(V,C(X,Y)),X,Y∈D,V∈D⊥.

For a CR-submanifold M in a locally conformal quaternion Kaehler manifold M-, we denote by ν the orthogonal complement of JD⊥ in TM⊥. Hence, we have the following orthogonal decomposition of the normal bundle:
(2.19)TM⊥=JD⊥⊕ν,JD⊥⊥ν.
Set
(2.20)PX=tan(JX),FX=nor(JX),forX∈TM,tN=tan(JN),fN=nor(JN),forN∈TM⊥.
Here, tanx and norx are the natural projections associated with the orthogonal direct sum decomposition
(2.21)TxM-=TxM⊕TMx⊥foranyx∈M.
Then the following identities hold:
(2.22)P2=-I-tF,FP+tF=0,Pt+tf=0,f2=-I-Ft,
where I is the identity transformation.

We have following results.

Lemma 2.3.

Let M be a CR-submanifold in a l.c.q.K. manifold M-. Then

holomorphic distribution D is integrable iff
(2.23)h(X,JaY)-h(JaX,Y)+Ω(X,Y)
nor
(B)=0,∀X,Y∈D
or equivalently,
(2.24)g~(h(X,JaY)-h(JaX,Y)+Ω(X,Y)B,JaZ)=0,∀X,Y∈D,Z∈D⊥;

anti-invariant distribution D⊥ of M is integrable iff
(2.25)AJaWT=AJaTW,∀W,T∈D⊥.

Proof.

(i) Using (2.3), we have
(2.26)∇XJaY=Pa∇XY+tah(X,Y)+12{θ(Y)X-w(Y)JaX-g(X,Y)tan(A)-Ω(X,Y)tan(B)}+Qab(X)JbY+Qac(X)JcY,h(X,JaY)=Fa∇XY+fah(X,Y)-12{g(X,Y)nor(A)+Ω(X,Y)nor(B)}.
From the second of these equations, we have
(2.27)h(X,JaY)-h(Y,JaX)+Ω(X,Y)nor(B)=Fa[X,Y],∀X,Y∈D.
If we need D to be integrable, we have
(2.28)h(X,JaY)-h(Y,JaX)+Ω(X,Y)nor(B)=0
or
(2.29)g(h(X,JaY)-h(Y,JaX)+Ω(X,Y)(B),JaZ)=0,∀Z∈D⊥.ν-part of h(X,JaY)-h(JaX,Y)+Ω(X,Y)B vanishes for all X,Y∈D.

(ii) We have
(2.30)∇-XJaY=Ja∇-XY+12{θ(Y)X-ω(Y)JaX-Ω(X,Y)B+g(X,Y)JaB}+Qab(X)JbY+Qac(X)JcY.
Then for any T, W∈D⊥, and X∈D, we have
(2.31)g(∇-TJaW,X)=g(Ja∇-TW,X)+12θ(W)g(T,X)-12ω(W)g(JaT,X)-12Ω(T,W)g(B,X)+12g(T,W)g(JaB,X)+Qab(T)g(JbW,X)+Qac(T)g(JcW,X)(2.32)⇒〈-AJaWT,X〉=-〈∇-T,JaX〉-12g(T,W)g(B,JaX).
So, we have
(2.33)〈AJaWT,X〉=〈∇-TW,JaX〉+12g(T,W)g(B,JaX),〈AJaTW,X〉=〈∇-WT,JaX〉+12g(W,T)g(B,JaX).
From these two equations, we have
(2.34)〈AJaWT-AJaTW,X〉=〈∇-TW-∇-WT,JaX〉⇒〈AJaWT-AJaTW,X〉=〈[W,T],JaX〉.
So, we conclude that if AJaWT=AJaTW then D⊥ is integrable. Converse is obvious.

Lemma 2.4.

Let M be a CR-submanifold of l.c.q.K. manifold. Then
(2.35)∇XJaY=∇YJaX
iff Lee vector field B is orthogonal to anti-invariant distribution D⊥.

Proof.

Since ∇- is metric connection, for X,Y∈D, and Z∈D⊥, using (2.3), we have
(2.36)〈∇-XJaY,Z〉=〈Ja∇-XY,Z〉+12θ(Y)〈X,Z〉-12Ω(X,Y)g(B,Z)-12ω(Y)g(JaX,Z)+12g(X,Y)g(JaB,Z)+Qab(X)g(JbY,Z)+Qac(X)g(JcY,Z)=-〈∇-XY,JaZ〉-12Ω(X,Y)ω(Z)-12g(X,Y)g(B,JaZ)=〈Y,∇-XJaZ〉-12Ω(X,Y)ω(Z)-12g(X,Y)g(B,JaZ)
or
(2.37)〈∇XJaY+h(X,JaY)Z〉=〈Y,-AJaZX+∇-X⊥JaZ〉-12Ω(X,Y)ω(Z)-12g(X,Y)g(B,JaZ).
This gives
(2.38)〈∇XJaY,Z〉=-〈Y,AJaZX〉-12Ω(X,Y)ω(Z)-12g(X,Y)g(B,JaZ),〈∇YJaX,Z〉=-〈X,AJaZY〉-12Ω(Y,X)ω(Z)-12g(Y,X)g(B,JaZ).

The above two equations give
(2.39)〈∇XJaY-∇YJaX,Z〉=-〈AJaZX,Y〉+〈X,AJaZY〉-12Ω(X,Y)ω(Z)+12Ω(Y,X)ω(Z)=Ω(Y,X)ω(Z)
or
(2.40)〈∇XJaY-∇YJaX,Z〉=Ω(Y,X)g(B,Z).
This gives ∇XJaY=∇YJaX iff ω(Z)=0.

3. Submersions of CR-Submanifolds

On a Riemannian manifold M, a distribution S is said to be parallel if ∇XY∈S, X,Y∈S, where ∇ is a Riemannian connection on M. It is proved earlier that horizontal distribution D is integrable. If, in addition, D⊥ is parallel, then we prove the following.

Proposition 3.1.

Let π:M→B be a submersion of a CR-submanifold M of a locally conformal quaternion Kaehler manifold M- onto an almost Hermitian manifold B. If (horizontal distribution) D is integrable and (vertical distribution) D⊥ is parallel, then M is a CR-product (Rienannian product M1×M2, where M1 is an invariant submanifold and M2 is a totally real submanifold of M-).

Proof.

Since the horizontal distribution D is integrable for X,Y∈D, we have [X,Y]∈D. Therefore, V[X,Y]=0. Then from (2.16), we have
(3.1)C(X,Y)=0,∀X,Y∈D.
Thus, from the definition of C, we have
(3.2)∇XY=∇~X*Y∈D,thatis,Disparallel.
Since D and D⊥ are both parallel, using de Rham’s theorem, it follows that M is the product M1×M2, where M1 is invariant submanifold of M- and M2 is totally real submanifold of M-. Hence, M is a CR-product.

In [10], Simons defined a connection and an invariant inner product on H(T,V)=Hom(T(M),V(M)), where V(M) is vector bundle over M and T(M) be tangent bundle of M. In fact, if r,s∈H(T,V)m, we set
(3.3)〈r,s〉=∑i=1p〈r(ei),s(ei)〉,where{ei}isaframeinT(M)m.
Define Qab(X)=〈D-Ja,Jb〉, which implies Qab(X)JaY=〈D-Ja,Jb〉JaY.

Let D be 4n dimensional distribution whose basis is given by {e1,…,en,ea1,…,ean,eb1,…,ebn,ec1,…,ecn} where eai=Ja(ei), ebi=Jb(ei), eci=Jc(ei) and Ja∘Jb=Jc, Jb∘Jc=Ja, Jc∘Ja=Jb.

Now, component of Qab(X) is defined as follows:
(3.4)Qab(X)=〈D-XJa,Jb〉=∑i=1n〈D-XJa(ei),Jb(ei)〉+∑i=1n〈D-XJa(eai),Jb(eai)〉+∑i=1n〈D-XJa(ebi),Jb(ebi)〉+∑i=1n〈D-XJa(eci),Jb(eci)〉+∑j=1q〈D-XJa(ej),Jb(ej)〉.
So, we have
(3.5)Qab(X)=∑i=1n〈D-XJa(ei),Jb(ei)〉+∑i=1n〈(D-XJa)(Jaei),Jb(Jaei)〉+∑i=1n〈D-XJa(Jbei),Jb(Jbei)〉+∑i=1n〈D-XJa(Jcei),Jb(Jcei)〉+∑j=1q〈D-XJa(ej),Jb(ej)〉=∑i=1n〈D-XJa(ei),Jb(ei)〉+∑i=1n〈D-Xei,Jc(ei)〉+∑i=1n〈D-XJa(ei),Jb(ei)〉-∑i=1n〈D-XJc(ei),ei〉-∑i=1n〈D-XJa(ei),Jb(ei)〉-∑i=1n〈D-XJb(ei),Ja(ei)〉+∑i=1n〈D-XJa(ei),Jb(ei)〉+∑i=1q〈D-XJa(ej),Jb(ej)〉
or
(3.6)Qab(X)JbY=∑i=1n〈D-XJa(ei),Jb(ei)〉JbY+∑i=1n〈D-Xei,Jc(ei)〉JbY+∑i=1n〈D-XJaei,Jbei〉JbY-∑i=1n〈D-XJc(ei),ei〉JbY-∑i=1n〈D-XJbei,Jaei〉JbY+∑i=1q〈D-XJa(ej),Jb(ej)〉JbY.
Applying π* and using Lemma 2.2, we get
(3.7)π*Qab(X)JbY=∑i=1n〈D-X**Ja′ei*,Jb′ei*〉Jb′Y*+∑i=1n〈D-X**ei*,Jc′ei*〉Jb′Y*+∑i=1n〈D-X**Ja′ei*,Jb′ei*〉Jb′Y*-∑i=1n〈D-X**Jc′ei*,ei*〉Jb′Y*-∑i=1n〈D-X**Jb′ei*,Ja′ei*〉Jb′Y*+∑i=1q〈D-X**Ja′ej*,Jb′ej*〉Jb′Y*=〈D-X**Ja′,Jb′〉Jb′Y*
or
(3.8)π*Qab(X)JbY=Qab*(X*)Jb*Y*.

Now, we prove the main result of this paper.

Theorem 3.2.

Let M- be an l.c.q.K. manifold and M be a CR-submanifold of M-. Let B be an almost Hermitian manifold and π:M→B be a submersion. Then B is an l.c.q.K. manifold.

Proof.

Let X,Y∈D be basic vector fields. Then from (2.5) and (2.15), we have
(3.9)∇-XY=∇~X*Y+C(X,Y)+h(X,Y).
Replacing Y by JaY in (3.9), we have
(3.10)∇-XJaY=∇~X*JaY+C(X,JaY)+h(X,JaY).
Using (2.3), we get
(3.11)∇~X*JaY+C(X,JaY)+h(X,JaY)=Ja∇~X*Y+12{θ(Y)X-ω(Y)JaX-Ω(X,Y)B+g(X,Y)JaB}+Qab(X)JbY+Qac(X)JcY
or
(3.12)∇~X*JaY+C(X,JaY)+h(X,JaY)=Ja∇~X*Y+JaC(X,Y)+Jah(X,Y)+12{θ(Y)X-ω(Y)JaX-Ω(X,Y)B+g(X,Y)JaB}+Qab(X)JbY+Qac(X)JcY.
Thus, we have
(3.13)(∇~X*Ja)Y+C(X,JaY)+h(X,JaY)-JaC(X,Y)-Jah(X,Y)-12{θ(Y)X-ω(Y)JaX-Ω(X,Y)B+g(X,Y)JaB}-Qab(X)JbY-Qac(X)JcY=0.
Comparing horizontal, vertical, and normal components in the above equation to get
(3.14)(∇~X*Ja)Y-12{θ(Y)X-ω(Y)JaX-Ω(X,Y)B+g(X,Y)JaB}-Qab(X)JaY-Qac(X)JcY=0,(3.15)C(X,JaY)=Jah(X,Y),(3.16)h(X,JaY)=JaC(X,Y)
from (3.14), we have
(3.17)∇~X*JaY-Ja∇~X*Y-12{g(JaY,B)X-g(B,Y)JaX-g(X,JaY)B+g(X,Y)JaB}-Qab(X)JaY-Qac(X)JcY=0.
Then for any X*,Y*∈χ(B), and J' being almost complex structure on B, we have after operating π* on the above equation
(3.18)∇X**Ja′Y*-Ja′∇X**Y*-12{g*(Ja′Y*,B*)X*-g*(B*,Y*)Ja′X*-g*(X*,Ja′Y*)B*+g*(X*,Y*)Ja′B*}-Qab*(X*)Ja*(Y*)-Qac*(X*)Jc*(Y*)=0.
This gives
(3.19)(∇X**Ja′)Y*-12{θ′(Y*)X*-ω′(Y*)Ja′X*-Ω′(X*,Y*)B*+g*(X*,Y*)Ja′B*}-Qab′(X*)Ja′Y*-Qac′(X*)Jc′Y*=0.
This shows that B is l.c.q.K. manifold.

Now, using (2.17) and (2.18), we obtain a relation between curvature tensor R on M and curvature tensor R* of B as follows:
(3.20)R(X,Y,Z,W)=R*(X*,Y*,Z*,W*)-g(C(Y,Z),C(X,W))+g(C(X,Z),C(Y,W))+2g(C(X,Y),C(Z,W)),
where π*X=X*, π*Y=Y*, π*Z=Z*, and π*W=W*∈B.

Now, using the above equation together with (2.8) and using the fact that C is skew-symmetric, we obtain
(3.21)H-(X)=R-(X,JaX,JaX,X)=H*(X*)+∥h(X,JaX)∥2-g(h(JaX,JaX),h(X,X))-3∥C(X,JaX)∥2,
where H-(X) and H*(X*) are the holomorphic sectional curvature tensors of M- and B, respectively.

If we assume that D is integrable then using Lemma 2.3(i), we have
(3.22)h(JaX,JaX)=-h(X,X).
Also from (3.15), we have C(X,JaX)=0. Then, (3.21) reduces to
(3.23)H-(X)=H*(X*)+∥h(X,JaX)∥2+∥h(X,X)∥2,∀X∈D.
This gives H-(X)≥H*(X*).

Thus, we have the following result.

Theorem 3.3.

Let M be a CR-submanifold of a l.c.q.K. manifold M- with integrable D. Let B be an almost Hermitian manifold and π:M-→B be a submersion. Then holomorphic sectional curvatures H- and H* of M- and B, respectively, satisfy
(3.24)H-(X)≥H*(X*),forallunitvectorsX∈D.

Note. The above result was obtained in [9] by taking M- to be quasi-Kaehler manifold. Later, similar type of relation was derived in [11], considering M- to be l.c.K manifold.

Acknowledgments

The first author is thankful to the Department of Science and Technology, Government of India, for its financial assistance provided through INSPIRE fellowship no. DST/INSPIRE Fellowship/2009/[XXV] to carry out this research work.

VaismanI.On locally conformal almost kähler manifoldsDragomirS.OrneaL.OrneaL.PiccinniP.Locally conformal kähler structures in quaternionic geometryOrneaL.Weyl structure on quaternioric manifolds, a state of the arthttp://arxiv.org/abs/math/0105041SahinB.GünesR.QR-submanifolds of a locally conformal quaternion kaehler manifoldO'NeillB.The fundamental equations of a submersionBejancuA. CR submanifolds of a kaehler manifold. IKobayashiS.Submersions of CR submanifoldsDeshmukhS.GhazalT.HashemH.Submersions of CR-submanifolds on an almost hermitian manifoldSimonsJ.Minimal varieties in riemannian manifoldsAl-GhefariR.ShahidM. H.Al-SolamyF. R.Submersion of CR-submanifolds of locally conformal kaehler manifold