ON KENMOTSU MANIFOLDS SATISFYING CERTAIN CURVATURE CONDITIONS

The purpose of the present paper is to study certain curvature conditions on Kenmotsu manifolds. It was proved that Kenmotsu manifolds satisfying curvature conditions     0 . , ~ , 0 . ,   B X C B X R   and   0 . ,  B X S  are D-conformally flat. It was also proved that Kenmotsu manifolds satisfying the curvature conditions   , 0 . ,  B X P    0 . ,  B X C  and     0 , ,  W Z Y X B g     are Einstein manifolds with scalar curvature  . 1    n n r Finally, we gave an example of 3-dimensional Kenmotsu manifold.

. A Kenmotsu manifold is normal but not Sasakian.Moreover, it is also not compact since where s is a nonzero constant.He also proved that if Kenmotsu manifold satisfies the then the manifold is of negative curvature -1.Later, Kenmotsu manifolds have been studied by De and Pathak (2004), Jun et al. (2005), De (2008), De et al. (2009).In preliminaries we studied some basic relations of Kenmotsu manifolds and D-conformal curvature tensor.We investigated some results on Kenmotsu manifolds satisfying curvature conditions such as Finally, we studied an example of 3-dimensional Kenmotsu manifold.

PRELIMINARIES
Let M be   -dimensional almost contact manifold equipped with an almost contact metric structure   g , , ,    consisting of a (1, 1) tensor field ,  a contravariant vector field ,  a 1-form  and a compatible Riemannian metric g satisfying (Blair, 1976(Blair, & 2002)) where  denotes the Riemannian connection of g (Kenmotsu, 1972).In an -dimensional Kenmotsu manifold the following relations hold: for any vector fields and Q are the Riemannian curvature, the Ricci tensor and the Ricci operator respectively (Kenmotsu, 1972).The D-conformal curvature tensor in an (Chuman, 1983).From (13), we also have then the manifold is said to be an Einstein manifold.

RESULTS AND DISCUSSION
We proved the following theorems: Theorem 1.Let M be an n -dimensional Kenmotsu manifold satisfying the condition   Then the manifold M is D-conformally flat.Proof.Let us consider an n -dimensional Kenmotsu manifold M which satisfies the condition Then, by definition we have By virtue of ( 14), ( 15) and (18) we have Taking inner product on both sides of ( 19) by  and using ( 1) and ( 15) we get Thus the manifold is D-conformally flat.This completes the proof of the theorem.
then the manifold is Einstein and the scalar curvature is Proof.Let M be an n -dimensional Kenmotsu manifold.The Weyl projective curvature tensor (Yano & Kon, 1984).From ( 22), it follows that Then by definition we have Using ( 23) in ( 24) we obtain Using ( 14) in ( 25) we get Taking inner product on both sides of (26) by  we get Thus the manifold is an Einstein manifold.Now, taking an orthonormal frame field and contracting over X and W in (28) we have where r is the scalar curvature.In view of ( 28) and ( 29), the theorem is proved.
(30) (Yano & Kon, 1984).From (30), we have We suppose that the manifold M satisfies the By virtue of ( 31) and (32), we obtain By the use of ( 14) and ( 15) in ( 33), (33) reduces to Taking inner product on both sides of (34) by  and using (1) and ( 15), we get This implies that either the scalar curvature is Hence the manifold is D-conformally flat.This completes the proof of the theorem.Theorem 4. In an n -dimensional Kenmotsu holds, then the manifold is an Einstein manifold with scalar curvature  .
Proof.Let us consider an n -dimensional Kenmotsu manifold .
M The Weyl conformal curvature tensor (37) (Yano & Kon, 1984).From (37), we have By the use of ( 38) in (39), we obtain Using ( 14) and ( 15) in (40), we get Taking inner product on both sides of (41) by  and using ( 1) and ( 15) we obtain Taking an orthonormal frame field and contracting over X and W in (43), we get  .
In view of ( 43) and ( 44), the theorem is proved.
where the endomorphism In view of ( 45), ( 46) and ( 47), we get By the use of ( 10) and ( 14) in ( 48), we get Taking inner product on both sides of ( 49) by  and using (1), ( 3) and ( 15), we obtain Thus the manifold is D-conformally flat.This completes the proof of the theorem.
Using ( 2), ( 7) and ( 12) in (52), we get ,..., 2 , 1 :  be an orthonormal basis of the tangent space at any point of the manifold.Putting Thus the manifold is an Einstein manifold.Now, taking an orthonormal frame field and contracting over Y and Z in (55), we get  .
By virtue of ( 55) and ( 56), the theorem is proved.

EXAMPLE OF A 3-DIMENSIONAL KENMO-TSU MANIFOLD
are the standard coordinates of .Then using the linearity of  and , g we have for any vector fields  .
and   .
The Levi-Civita connection  of the Riemannian metric g is given by which is known as Koszul's formula.By virtue of (58), ( 59), ( 63) and (64), we get  Therefore, proceeding same way we obtain Hence the manifold satisfies the condition (5).Again, using ( 60) and ( 65  Similarly, we can verify other relations and the manifold also satisfies the condition (4).From above it follows that the conditions (4) and ( 5) are satisfied by the manifold for   3 e and consequently the manifold under the consideration is a 3-dimensional Kenmotsu manifold.

CONCLUSION
In this paper, we have proved that an  The paper will be useful for those who are working and studying in the field of structures on differentiable manifolds.
conformally flat.It was also proved that Kenmotsu manifolds satisfying the curvature conditions 

Theorem 6 .
If a Kenmotsu manifold n M is  -Dconformally flat, then the manifold is an Einstein manifold with scalar curvature Let us consider an n -dimensional Kenmotsu manifold M which is  -D-conformally flat.Then the condition

Mg
Let  be the Levi-Civita connection with respect to the Riemannian metric .Then by the definition of Lie bracket and (57 can easily verify other relations and we have