On Submersion of CR-Submanifolds of l.c.q.K. Manifold

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


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 g X, B w X .

2.2
Then for l.c.q.K. manifold, we have 3 for any X, Y ∈ T M, where Q ab is skew-symmetric matrix of local forms θ w • J a and A −J a B.

ISRN Geometry 3
We also have θ X g J a X, B , Ω X, Y g X, J a Y .

2.4
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 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 The curvature tensor R of the submanifold M is related to the curvature tensor R of M by the following Gauss formula: 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.
For a vector field X on M, we set 8 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 π * X p X * π p for each p ∈ M.

2.10
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 is the metric on M, and g * is the Riemannian metric on B; ii the horizontal part H X, Y of X, Y is a basic vector field and corresponds to X * , Y * , that is, For a covariant differentiation operator ∇ * , we define a corresponding operator ∇ * for basic vector fields of M by then ∇ * X Y is a basic vector field, and from the above lemma we have Now, we define a tensor field C on M by setting In particular, if X and Y are basic vector fields, then we have The tensor field C is skew-symmetric and it satisfies The operator C and A are related by 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: Set PX tan JX , FX nor JX , for X ∈ TM, tN tan JN , fN nor JN , for N ∈ TM ⊥ .

2.20
Here, tan x and nor x are the natural projections associated with the orthogonal direct sum decomposition for any x ∈ M.

2.21
Then the following identities hold: 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
or equivalently, Proof. i Using 2.3 , we have

2.26
From the second of these equations, we have

2.27
If we need D to be integrable, we have

2.30
Then for any T , W ∈ D ⊥ , and X ∈ D, we have A J a T W, X ∇ W T, J a X 1 2 g W, T g B, J a X .

2.33
From these two equations, we have

2.34
So, we conclude that if A J a W T A J a T W then D ⊥ is integrable. Converse is obvious.

Lemma 2.4. Let M be a CR-submanifold of l.c.q.K. manifold. Then
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

ISRN Geometry
This gives

2.38
The above two equations give

Submersions of CR-Submanifolds
On a Riemannian manifold M, a distribution S is said to be parallel if ∇ X Y ∈ 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. 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 Thus, from the definition of C, we have Since D and D ⊥ are both parallel, using de Rham's theorem, it follows that M is the product