Science and Technology Journal, Vol. 3 Issue: II ISSN: 2321-3388 Some Results on α–Kenmotsu Manifolds Rajesh Kumar Department of Mathematics Pachhunga University College, Aizawl–796001, Mizoram, India E-mail: rajesh_mzu@yahoo.com Abstract—The object of the paper is to study -Kenmotsu manifolds which can be derived from almost contact Riemannian manifolds satisfying certain conditions. We first examine the projective flat -Kenmotsu manifold and give some interesting results. Next, it is shown that an -Kenmotsu manifold is a manifold of constant curvature in both the case of Einstein -Kenmotsu manifold satisfying as well as -Kenmotsu manifold satisfying , where is the projective curvature tensor. It is also proved that if -Kenmotsu manifold satisfying then the manifold is projectively flat. Keywords: Almost contact metric manifold, -Kenmotsu manifold, Einstein manifold, scalar curvature and Projective curvature. INTRODUCTION Let be an -dimensional -manifold admitting a vector ield , a tensor ield , and a 1-form such that , and . , there exists a metric tensor where denotes the operator of covariant differentiation with respect to the Riemannian connection , then (2) called an satisfying -Kenmotsu manifold [8, 9]. If and -Kenmotsu manifold is called a Kenmotsu manifold and if is constant then it is called a homothetic Kenmotsu (4) by many authors [ 10, 11, 12]. Further, on an -Kenmotsu manifold , the following relations hold for any vector ield and , then is called an almost contact metric manifold with structure [1]. In 1972 Kenmotsu [2] introduced a new class of almost contact Riemannian manifolds which are now a days called Kenmotsu manifolds. It is well known that odd dimensional spheres admit Sasakian structures where as odd dimensional hyperbolic spaces cannot admit Sasakian structure, but have so-called Kenmotsu structure. Kenmotsu manifolds are normal (noncontact) almost contact Riemannian manifolds. Kenmotsu manifolds have been studied by many authors [3, 4, 5, 6, 7]. Also, if the manifold , is , then a manifold. Recently -Kenmotsu manifold have been studied (3) , (7) And (1) then such a manifold is called an almost contact manifold. If in (6) , satis ies the following relations (5) , (9) , (10) , . (8) , (11) , (12) (13) The projective curvature tensor on a Riemannian manifold is given by [13] (14) Kumar PROJECTIVELY FLAT –KENMOTSU MANIFOLD This shows that either or Let us suppose that in an -Kenmotsu manifold , . Now, if (15) , then from (3), we get then it follows from (14) that , (16) which is not possible. Or Therefore which gives from (21) (17) Taking and (13), we get . in (17) and then using (1), (4), (12) . Theorem 3. A projectively lat Einstein -Kenmotsu manifold is the manifold of constant curvature, provided . (18) Thus, we have the theorem. Theorem 1. A projective lat -Kenmotsu manifold is an Einstein manifold. From (18), we have , where Thus, we have the following theorem. AN EINSTEIN -KENMOTSU MANIFOLD SATISFYING In this section we assume that (19) . . From (14) and (20), we have Contracting (19) with respect to , we have . It can be written as Hence, we have the following result. Theorem 2. The scalar curvature of a projectively lat -Kenmotsu manifold is constant. (24) Let an -Kenmotsu manifold is an Einstein manifold, then its Ricci tensor is of the form , where Putting in (24) and using (8), we get (20) . is a constant. Then (16) reduces to Putting (21) (25) in (25), we get (26) or Again putting . Taking (12), we get (23) in (25), we get . (22) Now, in (22) and then using (1), (4) and 180 (27) Some Results on α–Kenmotsu Manifolds In view of (23), we get AN -KENMOTSU MANIFOLD SATISFYING From (8) and (14), we have . Therefore, (32) Putting . in (32), we get . (33) From this it follows that Again putting in (32), we have . (34) Now, we have , (28) where On putting . in (26), we get Let , then we have . , Therefore, (29) be an orthonormal basis of the Let of tangent space at any point. Then the sum for yields the relation (29) for . . From this it follows that (30) Using (25) and (30), it follows from (28) that , . (35) where (31) . Let be an orthonormal basis of the tangent space at any point. Then the sum for of the relation (35) for yields From (24) and (31), we get , . Therefore, we can state. (36) In virtue of (34) and (36), we have Theorem 4. If in an Einstein -Kenmotsu manifold the hold, then the manifold is of relation . constant curvature, provided . Putting 181 in (37), we get (37) Kumar REFERENCES (38) Hence (37) becomes 1. . (39) Using (32), (34) and (39) it follows from (35) that 2. 3. . 4. Hence, we have the following theorems 5. Theorem 5. If in an -Kenmotsu manifold the relation hold, then the manifold is of scalar curvature . 6. 7. Theorem 6. 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