International Mathematical Forum, Vol. 7, 2012, no. 6, 297-303 Generalized Semi-Pseudo Ricci Symmetric Manifold Musa A. Jawarneh and Mohammad A. Tashtoush AL-Balqa' Applied University, AL-Huson University College Department of Basic Sciences, P. O. Box 50, AL-Huson 21510 musa_jawarneh, mhmdmath@hotmail.com Abstract. The object of this paper is to introduce an example of Generalized Semi- Pseudo Ricci Symmetric Manifolds defined, and to investigate some applications of the manifold when admitting a semi-symmetric metric connection. Mathematics Subject Classification: 53C55 (Primary); 58C25 (Secondary) Keywords: Generalized Semi-Pseudo Ricci Symmetric Manifold, Semi Symmetric Metric Connection, Pseudo Ricci Symmetric Manifold, Symmetric Manifolds, Lorentzian Manifold. Introduction The notion of Semi-Pseudo Ricci Symmetric Manifolds introduced by M. Tarafdar and Musa Jawarneh in 1993 [1], a non flat n-dimensional Riemannian manifold whose Ricci tensor S satisfies: 1), where is a non zero 1-form, 2), for every vector field X and denotes the operator of covariant differentiation with respect to the metric tensor g. Such n-dimensional manifold was denoted by (SPRS) n. The object of this paper is to study a type of non flat n-dimensional Riemannian manifold whose Ricci tensor S satisfies: 3),,, X S Y Z A Y S X Y B Z S X Y, where A and B are two non zero 1-forms,
298 M. A. Jawarneh and M. A. Tashtoush 4) A(X) = g(x,λ), B(Y) = g(y,µ), Where as stated above. Such a manifold shall be called a generalized semi-pseudo Ricci symmetric manifold, the 1-forms A and B will be its associated 1-forms and an n-dimensional manifold of this kind shall be denoted by G(SPRS) n. In particular A=B, then equation (3) will reduce to semi-pseudo Ricci symmetric manifold [1]. This will justify the name generalized semi-pseudo Ricci symmetric manifold defined by equation (3) and the symbol G(SPRS) n for it. In the first section of this paper we introduce an example of Generalized Semi-Pseudo Ricci Symmetric Manifolds. Then in the second section we studied semi-symmetric metric connection on Generalized Semi-Pseudo Ricci Symmetric Manifolds and obtained the necessary and sufficient conditions for the Ricci Tensor S % of the semisymmetric metric connection to be symmetric. After that we showed that the Ricci tensor S% r% where r% is the scalar curvature of the semi symmetric metric connection, and is a 1-form. Finally there after we conclude that a viscous fluid space-time obeying Einstein's equation with a cosmological constant is a connected non-compact Lorentzian [G(SPRS) n, ] with closed 1-form. 1. G(SPRS) 4 Let us consider M 4 = be an open subsets of R 4 endowed with the metric, 1.1) Where i, j = 1, 2, 3, 4, and f is continuously differentiable function of and such that and, where (. ) denote the partial differentiation with respect to coordinates. Therefore the only non vanishing Ricci tensors and their covariant derivatives are, 1.2) ;, where (, ) denotes the covariant differentiation with respect to the metric tensor g. Hence under consideration is a Riemannian manifold which is neither Ricci symmetric non Ricci recurrent and it is of vanishing scalar curvature tensor. If we now define the 1-form,
Generalized semi-pseudo Ricci symmetric manifold 299 1.3) A i x f.331 f, i 1.33 0, otherwise 1.4) B i x for any point x 2f.331 f.33, i 1, 0, otherwise M. Then (3) reduces to the form, 1.5), and all other cases vanishes identically. Therefore equation (1.1) holds because we have,. Thus we can state, Theorem 1.1: A Riemannian manifold M 4, g endowed with the metric (1.1) is a generalized semi-pseudo Ricci symmetric manifold with vanishing scalar curvature, which is neither Ricci symmetric nor Ricci recurrent. 2. Semi-symmetric metric connection A linear connection on n-dimensional Riemannian manifold is called semisymmetric connection [8] if its torsion tensor T of the connection satisfies, 2.1) T(X,Y) =, for every vector fields X,Y in M and α is a 1-form associated with the torsion tensor T of the connection given by, 2.2) X g X,, where is a vector field associated with the 1-form α. The 1- form α is called the associated 1-form of the semi-symmetric connection and the vector field is called the associated vector field of the semi-symmetric connection. A semi-symmetric connection is called a semi-symmetric metric connection if it satisfies also [10], 2.3) g = 0. The relation between the Riemannian connection and the semi-symmetric metric connection is given by [9],
300 M. A. Jawarneh and M. A. Tashtoush 2.4) % Y Y Y X g X X X, Y. The covariant differentiation of a 1-form ω with respect to is given by [9] 2.5) % X Y X Y X Y g X, Y. Now if and R denote respectively the curvature tensors of and then [9], 2.6), where H is a tensor field of type (0,2) defined as; 2.7) Contracting (2.6) we get 2.8), where a denote the trace of H. Further contraction yields 2.9), n Now let us define A Riemannian manifold M, g, n 3 as a generalized semipseudo Ricci symmetric manifold admitting a semi-symmetric metric connection such that its non-zero Ricci tensor S % of type (0,2) satisfies the relation, 2.10), where and are the associated 1-forms of the manifold. Such a manifold will be denoted by [G(SPRS) n, ]. If we now consider the 1-form α associated with the torsion tensor T to be closed, then we can have X Y Y X which will imply that the tensor H is symmetric, and consequently the Ricci tensor S % will be symmetric. Conversely if the tensor H is symmetric, then (2.7) will imply that the 1-form is closed. Thus we can state, Theorem 2.1: On [G(SPRS) n, ] the Ricci tensor S % is symmetric if and only if the 1- form associated with the torsion tensor T of the connection is closed. Let us now assume that is closed and Interchange Y and Z in (2.10) then subtract the result from (2.10) we can have, 2.11).
Generalized semi-pseudo Ricci symmetric manifold 301 Let E% Z g X, A% X B% X, for every vector field X and is a vector field associated with the 1-form. Then (2.11) will reduce to the form, 2.12). Contraction of the above equation with respect to X and Z yields, 2.13), where. Also from (2.12) we can have, 2.14) E% S% X, Y E% Z S% X, E% Y g QX %, E% Y E% QX % Which is by virtue of (2.13) yields, r S% % X, Y E% X E% Y r% X Y, E% 1 where, X g X, E% X, and is a unit vector field associated with the 1-form E% called the generator of [G(SPRS)n, % ]. Thus we can state, Theorem 3.1: If in a [G(SPRS) n, ] the 1-form α associated with the torsion tensor T is closed then its Ricci tensor is of the form. Now from (2.8) we can have by virtue of (2.9) and (2.14) that, 2.15). It is known [11] that a non zero vector field W on a manifold is called time like (res., non-space like, null, space like) if it satisfies g(w,w) 0 (res.,. If a [G(SPRS) n, ] is a connected and non compact Riemannian manifold with metric g, then since is a unit vector field it can be shown [11] that g 2 is a Lorentz metric on [G(SPRS) n, ]. Hence becomes a time like and therefore the resulting Lorentz manifold is a time oriantable. Finally let us consider a connected Lorentz manifold (M 4,g) with signature (-,+,+,+) which is a viscous fluid space time. Einstein equation is given by the relation, 2.16), where λ is the cosmological constant, k is the gravitational constant and T is the energy momentum tensor of type (0,2). The energy momentum tensor T of the viscous fluid space time has the following form,
302 M. A. Jawarneh and M. A. Tashtoush 2.17), Where σ denotes the energy density, p denotes the isotropic pressure, and H denotes the anisotropic pressure tensor of the fluid. From (2.16) and (2.17) we can have, 2.18). Which coincide with (2.15), and therefore the space time under consideration is a [G(SPRS) 4, ]. Hence we can state, Theorem 4.1: A viscous fluid space time obeying Einstein equation cosmological constant is a connected non-compact Lorantzian [G(SPRS) 4, ] with closed 1-form α. References 1. M. Tarafdar and Musa A. A. Jawarneh: Semi-Pseudo Ricci Symmetric manifold, J. Indian. Inst. of Science. 1993, 73, 591 596. 2. L. Tamassy and T. Q. Binh: On weekly symmetric and weekly projective symmetric Riemannian manifolds, Colloquia Mathematical Societitia ja`nos Bolyai, 56 (1989), 663 670. 3. K. Yano and M. Kon: Structures on Manifolds, World scientific 1989. 4. L. P. Eisenhart : Riemannian Geometry, 1926. 5. B. Smaranda: Pseudo Riemannian recurrent manifolds with almost constant curvature. The xvlll Inst. Conf.on Geometry and Topology (ordadea, 1989) 175-180. 6. J. A. Schouten: Ricci Calculus, Springer-verlag, Barlin, 1954. 7. B. Barua and A.K. Ray: Some properties of semi-symmetric metric connection in a Riemannian manifold. Indian J. Pure Appl. Math.,1985, 16(7), 736-740. 8. A. Friedmann and J. A. Schouten: J. A.Uber die geometrie der halbbsymmetrischen Ubertragung. Math. Zeitschr., 1924, 21, 211-223. 9. K. Yano: On semi-symmetric metric connection. Rav. Roum. Math. Pures Appl. (Bucharest), 1970, 15(9), 1579-1586.
Generalized semi-pseudo Ricci symmetric manifold 303 10. H. A. Hayden: Subspaces of a space with torsion. Proc. London Math. Sos., 1932, 34, 821-842. 11. B. O'Neill: semi-riemannian geometry with applications to relativity. Academic Press, New York, 1983. 12. K. Yano: Differential geometry on complex and almost complex spaces, Pergamon Press, 1965. 13. J. A. Schouten : Ricci Calculus, Springer-verlag, Barlin, 1954. Received: August, 2011