ON LORENTZIAN TRANS-SASAKIAN MANIFOLDS

. The object of the present paper is to study the Trans-Sasakian structure on a manifold with Lorentzian metric. Several interesting results are obtained on the manifold. Also conformally (cid:135)at Lorentzian Trans-Sasakian manifolds have been studied. Next, in three- dimensional Lorentzian Trans-Sasakian manifolds, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained. Also it is proved that a three-dimensional Lorentzian Trans-Sasakian manifold of type ( (cid:11);(cid:12) ) is locally (cid:30) - symmetric if and only if the scalar curvature r is constant provided (cid:11) and (cid:12) are constants. Finally, we give some examples of three-dimensional Lorentzian Trans-Sasakian manifold.


Introduction
Let M be an odd dimensional manifold with Riemannian metric g. It is well known that an almost contact metric structure ( ; ; ) (with respect to g) can be de…ned on M by a tensor …eld of type (1; 1), a vector …eld and a 1-form . If M has a Sasakian structure (Kenmotsu structure), then M is called a Sasakian manifold (Kenmotsu manifold). Sasakian manifolds and Kenmotsu manifolds have been studied by several authors. In the classi…cation of Gray and Hervella [8] of almost Hermitian manifolds there appears a class, W 4 , of Hermitian manifolds which are closely related to locally conformally Kaehler manifolds. An almost contact metric structure ( ; ; ; g) on M is Trans-Sasakian [17] if (M R,J; G) belongs to the class W 4 , where J is the almost complex structure on M R de…ned by for all vector …elds X on M , f is a smooth function on M R and G is the product metric on M R. This may be expressed by the condition [2] (r X )Y = (g(X; Y ) (Y )X) + (g( X; Y ) (Y ) X) (1.1) for smooth functions and on M . Hence we say that the Trans-Sasakian structure is of type ( , ). In particular, it is normal and it generalizes both -Sasakian and -Kenmotsu structures. From the formula (1:1) one easily obtains r X = ( X) + (X (X) ): In 1981, Janssens and Vanhecke introduced the notion of -Sasakian and -Kenmotsu manifolds where and are non zero real numbers. It is known that [6] Trans-Sasakian structures of type (0,0) , (0, ) and ( ,0) are cosymplectic ([1], [2]), -Kenmotsu ( [6]) and -Sasakian ( [6]) respectively. The local structure of Trans-Sasakian manifolds of dimension n 5 has been completely characterized by Marrero [10]. He proved that a Trans-Sasakian manifold of dimension n 5 is either cosymplectic or -Sasakian or -Kenmotsu manifold. Trans-Sasakian manifolds have been studied by several authors ( [3], [4], [5], [11], [18]).
Let M be a di¤erentiable manifold. When M has a Lorentzian metric g, that is, a symmetric non degenerate (0; 2) tensor …eld of index 1, then M is called a Lorentzian manifold. Since the Lorentzian metric is of index 1, Lorentzian manifold M has not only spacelike vector …elds but also timelike and lightlike vector …elds. This di¤erence with the Riemannian case give interesting properties on the Lorentzian manifold. A di¤erentiable manifold M has a Lorentzian metric if and only if M has a 1-dimensional distribution. Hence odd dimensional manifold is able to have a Lorentzian metric. Therefore, it is very natural and interesting idea to de…ne both a Trans-Sasakian structure and a Lorentzian metric on an odd dimensional manifold.
The paper is organized as follows. In Section 1, we give a brief account of Lorentzian Trans-Sasakian manifolds. After preliminaries, some basic results are given. In Section 4, we study conformally ‡at Lorentzian Trans-Sasakian manifolds.
In the next section, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained for three-dimensional Trans-Sasakian manifolds. Also it is proved that a three-dimensional Lorentzian Trans-Sasakian manifold of type ( ; ) is locally -symmetric if and only if the scalar curvature r is constant provided and are constants. Finally we construct some examples of three-dimensional Lorentzian Trans-Sasakian manifolds.

Lorentzian Trans-Sasakian manifolds
A di¤erentiable manifold M of dimension (2n + 1) is called a Lorentzian Trans-Sasakian manifold if it admits a (1; 1) tensor …eld , a contravariant vector …eld , a covariant vector …eld and the Lorentzian metric g which satisfy ( ) = 1; (2.1) Also a Lorentzian Trans-Sasakian manifold M satis…es where r denotes the operator of covariant di¤erentation with respect to the Lorentzian metric g . If = 0 and "R, the set of real numbers, then the manifold reduces to a Lorentzian -Kenmotsu manifold studied by Funda Yaliniz, Yildiz, and Turan [20]. If = 0 and "R, then the manifold reduces to a Lorentzian -Sasakian manifold studied by Yildiz, Turan and Murathan [21]. If = 0 and = 1, then the manifold reduces to a Lorentzian Kenmotsu manifold introduced by Mihai, Oiaga and Rosca [15]. Furthermore, if = 0 and = 1, then the manifold reduces to a Lorentzian Sasakian manifold studied by Ikawa and Erdogan [15]. Also Lorentzian para contact manifolds were introduced by Matsumoto [12] and further studied by the authors ( [13], [14], [16]). Trans Lorentzian para Sasakian manifolds have been used by Gill and Dube [7].

SOME BASIC RESULTS
In this section, we prove some Lemmas which are needed in the rest of the sections.
Lemma 3.1. In a Lorentzian Trans-Sasakian manifold, we have where R is the curvature tensor.
Proof. We have where (2.2) and (2.6) have been used. Hence, in view of the above equation and (2.6), we get which in view of (2.5) and (2.7) gives (3.1).
Lemma 3.2. For a Lorentzian Trans-Sasakian manifold, we have Proof. We have from (3.1), Now interchanging and Z in the above equation, we get After simpli…cation, we …nd, For a Lorentzian Trans-Sasakian manifold, we have Proof. Replacing X by in (3.1), we get (3.3).
Lemma 3.4. In a (2n + 1)-dimensional Lorentzian Trans-Sasakian manifold, we have where S is the Ricci curvature and Q is the Ricci operator given by Proof. Let M be an (2n + 1) dimensional Lorentzian Trans-Sasakian manifold. Then the Ricci tensor S of the manifold M is de…ned by where i = g(e i ; e i ); i = 1. From (3.1), we have and hence from (3.4), we get (3.5). and respectively.

Three-dimensional Lorentzian Trans-Sasakian manifolds
Since the conformal curvature tensor vanishes in a three-dimensional Riemannian manifold, therefore we get where Q is the Ricci operator, that is, g(QX; Y ) = S(X; Y ) and r is the scalar curvature of the manifold. From Lemma 2:4, in a three-dimensional Lorentzian Trans-Sasakian manifold we have S(X; ) = (2( 2 + 2 ) ) (X) + (X ) Now, in the following theorem, we obtain an expression for Ricci operator in a three-dimensional Lorentzian Trans-Sasakian manifold.

Locally -symmetric three-dimensional Lorentzian
Trans-Sasakian manifolds with trace = = 0 The notion of locally -symmetry was …rst introduced by T.Takahashi [19] on a Sasakian manifold. In this paper we study locally -symmetric three-dimensional Lorentzian Trans-Sasakian manifolds.