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Volume 2012 (2012), Article ID 309145, 13 pages
On Submersion of CR-Submanifolds of l.c.q.K. Manifold
Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India
Received 27 September 2012; Accepted 15 October 2012
Academic Editors: A. Ferrandez and T. Friedrich
Copyright © 2012 Majid Ali Choudhary et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The concept of locally conformal Kaehler manifolds was introduced by Vaisman in . Since then many papers appeared on these manifolds and their submanifolds (see  for details). However, the geometry of locally conformal quaternion Kaehler manifolds has been studied in [2####^~^~^~^~^~^####x2013;4] and their QR-submanifolds have been studied in .
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 .
The study of the Riemannian submersion of a Riemannian manifold onto a Riemannian manifold was initiated by O####^~^~^~^~^~^####x2019;Neill . A submersion naturally gives rise to two distributions on 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 . The notion of Cauchy-Riemann (CR) submanifold was introduced by Bejancu  as a natural generalization of complex submanifolds and totally real submanifolds. A CR-submanifolds of a l.c.q.K. manifold requires a differentiable holomorphic distribution , that is, for all , whose orthogonal complement is totally real distribution on , that is, for all . A CR-submanifold is called holomorphic submanifold if dim , totally real if dim and proper if it is neither holomorphic nor totally real.
A CR-submanifold of a l.c.q.K. manifold is called a CR-product if it is Riemannian product of a holomorphic submanifold and a totally real submanifold of . Kobayashi  has proved that if an almost Hermitian manifold admits a Riemannian submersion of a CR-submanifold of a Kaehler Manifold , then is a Kaehler manifold. However, Deshmukh et al.  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 and prove that if an almost Hermitian manifold admits a Riemannian submersion of a CR-submanifold of a l.c.q.K. manifold , then is an l.c.q.K. manifold.
Let be a quaternion Hermitian manifold, where is a subbundle of end of rank 3 which is spanned by almost complex structures , , and . The quaternion Hermitian metric is said to be a quaternion Kaehler metric if its Levi-Civita connection satisfies .
A quaternion Hermitian manifold with metric is called a locally conformal quaternion Kaehler (l.c.q.K.) manifold if over neighborhoods covering , where is a quaternion Kaehler metric on . In this case, the Lee form is locally defined by and satisfies  Let be l.c.q.K. manifold and denotes the Levi-Civita connection of . Let be the Lee vector field given by Then for l.c.q.K. manifold, we have  for any , where is skew-symmetric matrix of local forms and .
We also have
Let be a Riemannian manifold isometrically immersed in . Let be the Lie algebra of vector fields in and , the set of all vector fields normal to . Denote by the Levi connection of . Then the Gauss and Weingarten formulas are given by for any , and , where is the connection in the normal bundle , is the second fundamental form, and is the Weingarten endomorphism associated with . The second fundamental form and shape operator are related by The curvature tensor of the submanifold is related to the curvature tensor of by the following Gauss formula: for any .
For submersion of a l.c.q.K. manifold onto an almost Hermitian manifold, we have the following.
Definition 2.1. Let be a CR-submanifold of a locally conformal quaternion Kaehler manifold . By a submersion of onto an almost Hermitian manifold , we mean a Riemannian submersion together with the following conditions:(i) is the kernel of , that is, ,(ii) is a complex isometry of the subspace onto for every , where denotes the tangent space of at ,(iii) interchanges and , that is, .
For a vector field on , we set  where and denoted the horizontal and vertical part of .
We recall that a vector field on for submersion is said to be a basic vector field if and is related to a vector field on , that is, there is a vector field on such that
If and are the almost complex structures on and , respectively, then from Definition 2.1(ii) we have on .
We have the following lemma for basic vector fields .
Lemma 2.2. Let and be basic vector fields on . Then(i), is the metric on , and is the Riemannian metric on ;(ii)the horizontal part of is a basic vector field and corresponds to , that is, (iii) is a basic vector field corresponding to , where is a Riemannian connection on ;(iv) for .
For a covariant differentiation operator , we define a corresponding operator for basic vector fields of by then is a basic vector field, and from the above lemma we have Now, we define a tensor field on by setting that is, is the vertical component of .
In particular, if and are basic vector fields, then we have The tensor field is skew-symmetric and it satisfies For and define an operator on by setting , that is, is the horizontal component of . Using (iv) of Lemma 2.2 we have The operator and are related by
For a CR-submanifold in a locally conformal quaternion Kaehler manifold , we denote by the orthogonal complement of in . Hence, we have the following orthogonal decomposition of the normal bundle: Set Here, and are the natural projections associated with the orthogonal direct sum decomposition Then the following identities hold: where is the identity transformation.
We have following results.
Lemma 2.3. Let be a CR-submanifold in a l.c.q.K. manifold . Then (i)holomorphic distribution is integrable iff or equivalently, (ii)anti-invariant distribution of is integrable iff
Proof. (i) Using (2.3), we have
From the second of these equations, we have
If we need to be integrable, we have
-part of vanishes for all .
(ii) We have Then for any , , and , we have So, we have From these two equations, we have So, we conclude that if then is integrable. Converse is obvious.
Lemma 2.4. Let be a CR-submanifold of l.c.q.K. manifold. Then iff Lee vector field is orthogonal to anti-invariant distribution .
Proof. Since is metric connection, for , and , using (2.3), we have
The above two equations give or This gives iff .
3. Submersions of CR-Submanifolds
On a Riemannian manifold , a distribution is said to be parallel if , , where is a Riemannian connection on . It is proved earlier that horizontal distribution is integrable. If, in addition, is parallel, then we prove the following.
Proposition 3.1. Let be a submersion of a CR-submanifold of a locally conformal quaternion Kaehler manifold onto an almost Hermitian manifold . If (horizontal distribution) is integrable and (vertical distribution) is parallel, then is a CR-product (Rienannian product , where is an invariant submanifold and is a totally real submanifold of ).
Proof. Since the horizontal distribution is integrable for , we have . Therefore, . Then from (2.16), we have Thus, from the definition of , we have Since and are both parallel, using de Rham####^~^~^~^~^~^####x2019;s theorem, it follows that is the product , where is invariant submanifold of and is totally real submanifold of . Hence, is a CR-product.
In , Simons defined a connection and an invariant inner product on , where is vector bundle over and be tangent bundle of . In fact, if , we set Define , which implies .
Let be dimensional distribution whose basis is given by where , , ####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;and , , .
Now, component of is defined as follows: So, we have or Applying and using Lemma 2.2, we get or
Now, we prove the main result of this paper.
Theorem 3.2. Let be an l.c.q.K. manifold and be a CR-submanifold of . Let be an almost Hermitian manifold and be a submersion. Then is an l.c.q.K. manifold.
Proof. Let be basic vector fields. Then from (2.5) and (2.15), we have Replacing by in (3.9), we have Using (2.3), we get or Thus, we have Comparing horizontal, vertical, and normal components in the above equation to get from (3.14), we have Then for any , and being almost complex structure on , we have after operating on the above equation This gives This shows that is l.c.q.K. manifold.
Now, using the above equation together with (2.8) and using the fact that is skew-symmetric, we obtain where and are the holomorphic sectional curvature tensors of and , respectively.
Thus, we have the following result.
Theorem 3.3. Let be a CR-submanifold of a l.c.q.K. manifold with integrable . Let be an almost Hermitian manifold and be a submersion. Then holomorphic sectional curvatures and of and , respectively, satisfy
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.
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