On Extended Generalized φ-Recurrent ( LCS ) 2 n + 1-Manifolds

In 2003, Shaikh [14] introduced the notion of Lorentzian concircular structure manifolds (briefly (LCS)2n+1-manifolds) with an example, which generalizes the notion of LP-Sasakian manifolds introduced by Matsumoto [7]. The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. As a weaker version of local symmetry, Takahashi [19] introduced the notion of local φ-symmetry on a Sasakian manifold. Generalizing the notion of local φ-symmetry of Takahashi [19], De et al. [2] introduced the notion of φ-recurrent Sasakian manifold. Recently De et al. [3] introduced the notion of φ-recurrent Kenmotsu manifolds. The locally φ-symmetric LP-Sasakian manifolds is also studied by Shaikh and Baishya [15]. Again locally φ-symmetric and locally φ-recurrent (LCS)2n+1-manifolds are respectively studied in [16] and [17]. The notion of generalized recurrent manifolds has been introduced by Dubey [6] and studied by De and Guha [4]. Again, the notion of generalized Ricci-recurrent manifolds has been introduced and studied by De et al. [5].


Introduction
In 2003, Shaikh [14] introduced the notion of Lorentzian concircular structure manifolds (briefly (LCS) 2n+1 -manifolds) with an example, which generalizes the notion of LP-Sasakian manifolds introduced by Matsumoto [7].The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent.As a weaker version of local symmetry, Takahashi [19] introduced the notion of local φ-symmetry on a Sasakian manifold.Generalizing the notion of local φ-symmetry of Takahashi [19], De et al. [2] introduced the notion of φ-recurrent Sasakian manifold.Recently De et al. [3] introduced the notion of φ-recurrent Kenmotsu manifolds.The locally φ-symmetric LP-Sasakian manifolds is also studied by Shaikh and Baishya [15].Again locally φ-symmetric and locally φ-recurrent (LCS) 2n+1 -manifolds are respectively studied in [16] and [17].The notion of generalized recurrent manifolds has been introduced by Dubey [6] and studied by De and Guha [4].Again, the notion of generalized Ricci-recurrent manifolds has been introduced and studied by De et al. [5].
A Riemannian manifold (M n , g), n > 2, is called generalized recurrent ( [4], [6]), if its curvature tensor R satisfies the condition where A and B are non vanishing 1-forms defined by A(•) = g(•, ρ 1 ), B(•) = g(•, ρ 2 ) and the tensor G is defined by for all X, Y, Z ∈ χ(M ); χ(M ) being the Lie algebra of smooth vector fields on M and ∇ denotes the operator of covariant differentiation with respect to metric g.The 1− forms A and B are called the associated 1-forms of the manifold.A Riemannian manifold (M n , g), n > 2, is called generalized Ricci-recurrent [5] if its Ricci tensor S of type (0, 2) satisfies the condition where A and B are non vanishing 1-forms defined in (1.1).
The paper is organized as follows.Section 2 deals with a brief account of (LCS) 2n+1 -manifolds.In section 3, we study generalized φ-recurrent (LCS) 2n+1manifolds and obtain a necessary and sufficient condition for a manifold to be a generalized Ricci-recurrent manifold.Also we study extended generalized concircularly φ-recurrent (LCS) 2n+1 -manifolds and find the nature of its associated 1-forms.Finally; the last section is responsible for the existence of extended generalized φ-recurrent (LCS) 2n+1 -manifolds.

Preliminaries
An (2n + 1)-dimensional Lorentzian manifold M is smooth connected paracontact Hausdroff manifold with Lorentzian metric g, that is, M admits a smooth symmetric tensor field φ of type (0, 2) such that for each point p ∈ M the tensor g p : T p M × T p M → ℜ is a non degenerate inner product of signature (−, +, ...., +), where T p M denotes the tangent space of M at p and ℜ is the real number space.A non-zero vector field v ∈ T p M is said to be time like (resp., non-spacelike, null, and spacelike) if it satisfies g p (v, v) < 0 (resp.,≤0, =, > 0) [9].Definition 2.1.In a Lorentzian manifold (M, g) a vector field ρ defined by g(X, ρ) = A(X) for any X ∈ χ(M ) is said to be a concircular vector field if where α is a non-zero scalar and ω is a closed 1-form.
Let M be a Lorentzian manifold admitting a unit time like concircular vector field ξ, called the generator of the manifold.Then, we have Since, ξ is the unit concircular vector field, there exists a non-zero 1-form η such that g(X, ξ) = η(X), the equation of the following form holds since, ξ is a concircular field, therefore where ω is a 1-form.Also since ξ is a unit vector field from (2.1), which implies g(∇ X ξ, ξ) = 0 and hence, we get from above equation where η(X) = g(X, ξ), Now now, we arrive at the result (2.3) for all vector fields X, Y , where ∇ denotes the operator of covariant differentiation with respect to the Lorentzan metric g and α is a non-zero scalar function satisfies ρ being a certain scalar function given by ρ = −(ξα).
then from (2.3) and (2.5) we have from which it follows that φ is a symmetric (1, 1)-tensor and called the structure tensor of the manifold.Thus the Lorentzian manifold M together with the unit timelike concircular vector field ξ its associated 1-form η and (1, 1)-tensor field φ is said to be a Lorentzian concircular structure manifold (briefly (LCS) 2n+1manifolds) [4].Especially, if we take α = 1, then we can obtain the LP-Sasakian structure of Motsumoto [7].
Lemma 2.2.From [17], Let M 2n+1 (φ, ξ, η, g) be a Lorentzian concircular structure manifold.Then for any vector fields X, Y, W the following relation holds: for all X, Y, Z, W ∈ χ(M ), where ∇ denotes the operator of covariant differentiation with respect to the metric g ,i.e.∇ is the Riemannian connection; A and B are non-vanishing 1-form such that A(X) = g(X, ρ 1 ), B(X) = g(X, ρ 2 ) and G is a tensor of type (1, 3) defined in (1.2).The 1-forms A and B are called the associated 1-forms of the manifold.
Let φ be the (1, 1)-tensor field defined by Then using the linearity of φ and g we have Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g and R be the curvature tensor of g.Then we have Taking E 3 = ξ and using Koszula formula for the Lorentzian metric g, we can easily calculate From the above it can be easily seen that E 3 = ξ is a unit timelike concircular vector field and hence (φ, ξ, η, g) is a (LCS) 3 -structure on M .Consequently M 3 (φ, ξ, η, g) is a (LCS) 3 -manifolds with α = −e 2z = 0 such that (Xα) = ρη(X) where ρ = 2e 4z Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor R as follows: and the components which can be obtained from these by the symmetric properties.