Generic sub manifold of quaternion Kaehler manifold

Introduction

It is well known that any quaternion-Kaehler manifold is a connection-oriented Riemannian 4n-manifold, 2 n  with holonomy group in

 , A group that appears on the database, classification theorem given by Berger's [4] . For possible holonomy of nonsymmetric, non-reducible multiple groups. The geometry of Quaternion-Kaehler comes naturally from the concept of potential holonomies. Einstein also is the quaternion-Kaehler manifold a quaternion Kaehler manifold has a positive scalar curvature, it is considered positive. The sign of the scalar curvature gives this class of manifolds a coarse classification; when 0 s  . The scalar curvature case is stringently distinct from zero, and thoroughly characterizes the geometry of quaternion-Kaehler. A submanifold of M called quaternion CR-submanifold [8] if there exits two orthogonal complementary distribution D and

and D ^ is real. The quaternion CR-submanifold includes quaternion and a completely real submanifold as subcases. A submanifold M of a quaternion Kaehler manifold is called Qr-submanifolds [2] if there exists a vector subbundle of the normal bundle

TM ^of M such that we have ()

, where p v ^is the complementary orthogonal subbundle to v in TM ^.it is known that every real hypersurface of a quaternion Kaehler manifold is a QR-submanifold [9] . In comparison, A. Bejancu [1] defines a CR-submanifold in Kaehler manifold, and studies it. Several articles have appeared on this subject since then [3,5,6,7] . In addition, the quaternion CR-submanifold of a quaternion Kaehler manifold was described by Barros, Chen and Urbano as an analogy with the CR-submanifold of a Kaehler manifold [1] . The geometry of a submanifold of a quaternion Kaehler manifold is mainly based on the action of the local basis 1 2 3 { , , } J J J on each tangent space to M . We know that if the manifold is non-Kaehler than it is non-integrable. However, the concept of a generic submanifold of quaternion Kaehler manifolds has not been studied as yet, as so far we know. Therefore in this paper, we study the Generic submanifold of quaternion Kaehler manifold.

Preliminaries

Let

M be a 4m-dimensional Riemannian manifold with a 3-dimensional vector bundle V consisting of the tensor of type (1,1) with a local basis of almost Hermitian structure  

Where  denotes the Riemannian connection on M . And X being any arbitrary vector field on M . We see that condition (b) is the equivalent of the following:

For all vector fields C tangent to M , Where Ñ is the Levi-Civita connection and

Let M be a sub-manifold of a quaternion space generated by ()

For any X, Y tangent to M and any vector field  normal to M.

The second fundamental form h and the shape operator A  are connected by (2.9) ( ( , ), ) ( , )

International Journal of Statistics and Applied Mathematics http://www.mathsjournal.com

For the second fundamental form h, we define the covariant differentiation  concerning the connection in

For any vector field X, Y, and Z tangent to M. The equations of Gauss and Codazzi and Ricci are then respectively given by (2.11) ( , , , ) ( , , , ) ( ( , ), ( , )) ( ( , ), ( ,

For any vector fields , , ,    and W tangent to M ,  and  are normal to M . let us we put (2.14)

Integrability of Generic Submanifolds in

For vector field ,     and    .

Proof: Since

M is a quaternion Kaehler manifold, using (2.3) and (2.7) we have

for vector field X, Y in  . the holomorphic distribution  is integrable, then RHS of (3.4) lies in TM ; thus we obtain

The proposition is proved.

Theorem 3.2: any generic sub-manifold M of a quaternion Kaehler manifold M . If   is integrable and its leaves are geodesic in M , then

   we have the theorem.

Parallel Canonical Structure with Generic Submanifold of Qurternion Kaehler Manifold

Let us define the endomorphism :

Now compare the tangential parts, we have ( ) ( , ) ( , )

Therefore, for any vector fields ,,

Then the endomorphism will be parallel if

Acknowledgement

AcknowledgementAll authors are grateful for suggestions for the improvement of research work.

References

1. Bejancu A. CR-Submanifolds of a Kaehler Manifold, I-II, Proc. Amer. Math. Soc., 1978;69:135-142. Trans. Amer. Math. Soc 1979;250:333345. International Journal of Statistics and Applied Mathematics http://www.mathsjournal.com
2. Bejancu A. QR-submanifolds of Quaternion Kaehlerian Manifolds, Chinese J Math 1986;14:2.
3. Barros M, Chen BY, Urbano F. Quaternion CR-Submanifolds of Quaternion Kaehler Manifolds, Kodai Math 1981;4:399-418.
4. Berger M. Sur les groupes d'holonomie des vari´et´es `a connexion affine et des vari´et´es riemanniennes, Bull. Soc. Math. France 1955;83:279-330. MR 18:149a
5. Blair DE, Chen BY. On CR-Submanifolds of Hermitian Manifolds, Israel J Math 1979;34:353-363.
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7. Chen BY. Cohomology of CR-Submanifolds, An. Faculte des Sciences Toulouse 1981;3:167-172.
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9. Chen BY. Slant immersions, Bull. Austral. Math. Soc. 1990;41:135-147.