On a Pseudo Projective - Recurrent Sasakian Manifolds
Transcription
On a Pseudo Projective - Recurrent Sasakian Manifolds
Journal of mathematics and computer science On a Pseudo Projective 14 (2015), 309-314 Recurrent Sasakian Manifolds 1 2 3 A. Singh , R. Kumar Pandey , A. Prakash , S. Khare 4 1 Department of Mathematics, B.B.D. University, Lucknow-226004, Uttar Pradesh, India Department of Mathematics, B.B.D. University, Lucknow-226004, Uttar Pradesh, India 3 Department of Mathematics, N.I.T., Kurukshetra -136119, Haryana, India. 4 Department of Mathematics, B.B.D. University, Lucknow-226004, Uttar Pradesh, India [email protected] 2 [email protected] [email protected] [email protected] Article history: Received June 2014 Accepted November 2014 Available online January 2015 Abstract The object of the present paper is to study the pseudo projective Keywords: Pseudo projective Einstein manifold. recurrent Sasakian manifolds. symmetric manifold, pseudo projective recurrent manifold, 1. Introduction The notion of local symmetry of a Riemannian manifold has been studied by many authors in several ways to a different extent. As a weaker version of local symmetry, in 1977, Takahashi [9] introduced the notion of locally symmetric Sasakian manifold and obtained their several interesting results. The properties of pseudo projective curvature tensor is studied by many geometers [17], [18], [19], [22] and obtained their some interesting results. In this paper we shown that pseudo projective recurrent Sasakian manifold is an Einstein manifold and in a pseudo projective recurrent Sasakian manifold, the characteristic vector field and the vector field associated to the form are co-directional. Finally, we proved that a three dimensional locally pseudo – projective recurrent Sasakian manifold is of constant curvature. 2. Preliminaries Let be an almost contact Riemannian manifold, where is a tensor field, is the structure vector field, is a form and is the Riemannian metric.It is well known that the structure satisfy 309 A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314 (1) (2) (3) (4) (5) (6) for all vector fields then where denotes the operator of covariant differentiation with respect to is called a Sasakian manifold [1]. Sasakian manifolds have been studied by many authors such as De, Shaikh and Biswas [3], Takahashi [9], Tanno [15] and many others. In a Sasakian manifold the following relations hold: [1] (7) (8) (9) (10) for all vector fields where curvature tensor of the manifold. is the Ricci tensor of type and is the Riemannian A Sasakian manifold is said to be an Einstein manifold if the Ricci tensor is of the form where λ is a constant. Definition 2.1. A Sasakian manifold is said to be a locally symmetric manifold if [9] ( for all vector fields ) (11) orthogonal to Definition 2.2. A Sasakian manifold is said to be a locally pseudo projective ̃) if (( ) for all vector fields ̃ ̃ ̃ (13) [ are constants such that ,b ] ][ [ ̃ recurrent Sasakian manifold if where ̃ is a pseudo projective curvature tensor given by [17] for arbitrary vector fields and (12) orthogonal to Definition 2.3. A Sasakian manifold is said to be pseudo projective there exists a non-zero form such that where symmetric manifold If [ ] and (14) Then (14) takes of the form ] . where is the projective curvature tensor [21]. Hence the Projective curvature is a particular case of the tensor ̃ . For the reason ̃ is called Pseudo projective curvature tensor, where is the Riemann curvature tensor is Ricci tensor and is the scalar curvature. 310 A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314 If the 1-form manifold. vanishes, then the manifold reduces to a locally pseudo projective 3. Pseudo projective symmetric recurrent Sasakian manifold In this section we consider a Sasakian manifold which is pseudo projective manifold. Then by virtue of (1) and (3), we get ( ̃) (( ̃) recurrent Sasakian ̃ ) (15) from which it follows that (( ̃) Let manifold. Then putting ) (( ̃) (̃ ) ) (16) be an orthonormal basis of the tangent space at any point of the in (16) and taking summation over we get ( [ ) ] (17) Replacing Z by ξ in (17) and using (2) and (9), we get ( [ )] (18) Now we have using (5), (6) and (9) in the above relation, it follows that (19) In view of (18) and (19), we get ( [ Replacing by )] (20) and using (3), (4) and (10) in (20), we get (21) for all Hence, we can state the following theorem: Theorem 3.1 A Pseudo projective recurrent Sasakian manifold is an Einstein manifold. Now from (15), we have ̃) ( Using (14) in (22), we get (( ̃) ( [ [ ] ] )[ ] [ ] ] From (23) and the Bianchi identity, we get [ ( ( ] )[ [ ] )[ ] (22) ) [ ( ̃ ) [ ] ] [ ] 311 (23) A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314 [ ] [ ( )[ ] By virtue of (8), we obtain from (24) that ] [ [ [ (24) ] ] ] [ ( ] )[ [ ] ] [ ( ] )[ [ ] ] [ ( Putting ] )[ [ ] ] in (25) and taking summation over (25) we get (26) for all vector fields Replacing for any vector field i.e., by in (26), we get where From (27), being the vector field associated to the (27) form we can state the following theorem: Theorem 3.2 In a Pseudo projective Sasakian manifold the characteristic vector field and the vector field associated to the form are co-directional and the form is given by (27). 4. On a Manifold On a dimensional Locally Pseudo Projective Recurrent Sasakian dimensional Sasakian Manifold Ricci tensor and curvature tensor has the following form ( ) ( ) )[ ( ( (28) ] )[ ] (29) Taking covariant differentiation of (29), we get [ ( )[ ] Taking orthogonal to and using (5) and (6), we get 312 (30) A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314 [ ] )[ ( ] (31) from (31) it follows that [ orthogonal to and using (1) and (2) in (32), we get Now, taking [ [ Using (28)in (34) and then taking Now, applying ] orthogonal to (34) we get ][ ] (35) to the both side of (35), we get [ [ [ orthogonal to ] ] is a scalar, since [ ] (36) ] ] [ , we get ̃ Putting in (38), where of the manifold and taking summation over ̃ [ ][ [ ̃ where ] ̃) ̃) ( Using (13), (33), (1) in (36), we obtain taking (33) we get ̃) [ ( (32) ] Differentiating covariantly (14) with respect to W ( ] ] ] [ (37) (38) is an orthonormal basis of the tangent space at any point we obtain [ ] is a non-zero form. Then, by Schur’s theorem will be a constant on the manifold Hence, we can state the following theorem: Theorem 4.1 On a dimensional locally Pseudo-projective projective curvature tensor is of the form of constant curvature. recurrent Sasakian manifold, pseudo- Acknowledgements. The authors are thankful to Professor S. Ahmad Ali Dean School of Applied Sciences, B.B.D. University, Lucknow, providing suggestions for the improvement of this paper. References [1] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math, 509 Berlin-Heidelberg-New York, (1976). [2] T. Q. Binh, L. Tamassy, U.C. De, M. Tarafdar, Some remarks on almost Kenmotsu manifolds, Mathematica Pannonica, 13, 3139, (2002). 313 A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314 [3] U.C. De, A.A. Shaikh, S. Biswas, On [4] U.C. De, G. Pathak, On Recurrent Sasakian manifolds, Novi Sad J. Math, 33(2) , 43-48, (2003). dimensional Kenmotsu Manifolds, Indian J. pure appl. Math. 35(2) , 159-165, (2004). [5] U.C. De, A. Yildiz, A.F. Yaliniz, On Recurrent Kenmotsu manifolds,Turk J Math, 33, 17-25, (2009). [6] J.B. Jun , U.C. De, G. Pathak, On Kenmotsu manifolds, J. Korean Math .Soc., 42, 435-445, (2005). [7] T. Adati, K. Matsumoto, On conformally recurrent and conformally symmetric 32, (1977). Sasakian manifolds, TRU Math., 17, 25- [8] C. Ozgur, U.C. De, On the quasi-conformal curvature tensor of a Kenmotsu manifold, Mathematica Pannonica, 17(2), 221228, (2006). [9] T. Takahashi, Sasakian symmetric spaces, Tohoku Math. J., 29, 91-113, (1977). [10] K. Yano, Concircular geometry, Proc. Imp. Acad., Tokyo, 16, 195-200, (1940). [11] U.C. De, On symmetric Kenmotsu manifolds, Int. Electronic J. Geometry, 1(1) , 33-38, (2008). [12] S.S. Shukla, M.K. Shukla, On [13] T. Adati, T. Miyazawa, On Symmetric para-Sasakian manifolds, Int. Journal of Math. Analysis, 4, 761-769, (2010). Sasakian manifolds satisfying certain conditions, Tensor, (N.S.), 33, 173-178, (1979). [14] I. Sato, On a structure similar to the almost contact structure, Tensor, (N.S.), 30, 219-224, (1976). [15] S. Tanno, Isometric Immersions of Sasakian manifold in spheres, Kodai Math. Sem. Rep., 21, 448-458, (1969). [16] D. Tarafdar, U.C. De, On a type of Sasakian manifolds, Extracta Mathematicae, 8(1), 31-36, (1993). [17] B. Prasad, A pseudo projective curvature tensor on a Riemannian manifolds, Bull. Cal. Math. Soc. 94 (3), 163-166, (2002). [18] M. Tarafdar, A. Bhattacharyya, D. Debnath, A type of pseudo projective recurrent Trans - Sasakian manifolds, Analele stiintifice ale Universitatii AL.I. Cuza Iasi Tomul LII, f-2, S.I, Mathematica, 417-422, (2006). [19] Venkatesha, C.S.Bagewadi, On pseudo projective 32(3) , 1-7, (2006). recurrent Kenmotsu manifolds, Soochow Journal of Mathematics, [20] Q. Khan, On an Einstein projective Sasakian manifolds, NOVI SAD J. Math., 36(1) , 97-102, (2006). [21] R.S. Mishra, Structure on a differentiable manifold and their applications, Chandrama Prakashan, 50 A, Balrampur House, Allahabad, India (1984). [22] D. Narain, A. Prakash, B. Prasad, A pseudo projective curvature tensor on a lorentzian Para-Sasakian manifolds, Analele stiintifice ale Universitatii AL.I. Cuza din iasi (S.N.) Mathematica, Tomul LV, fase.2, 275-284, (2009). 314