CUBO A Mathematical Journal Vol.19, No¯ 02, (01–09). June 2017 On Some Recurrent Properties of Three Dimensional K-Contact Manifolds Venkatesha and R.T. Naveen Kumar Department of Mathematics, Kuvempu University, Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA. vensmath@gmail.com, rtnaveenkumar@gmail.com ABSTRACT In this paper we characterize some recurrent properties of three dimensional K-contact manifolds. Here we study Ricci η-recurrent, semi-generalized recurrent and locally gen- eralized concircularly φ-recurrent conditions on three dimensional K-contact manifolds. RESUMEN En este paper caracterizamos algunas propiedades recurrentes de variedades K-contacto tridimensionales. Estudiamos las condiciones de Ricci η-recurrencia, recurrencia semi- generalizada y φ-recurrencia concircular localmente generalizada en variedades K-contacto tridimensionales. Keywords and Phrases: K-contact manifold, Ricci η-recurrent, semi-generalized recurrent, lo- cally generalized concircularly φ-recurrent, scalar curvature, Einstein manifold. 2010 AMS Mathematics Subject Classification: 53C25, 53D15. 2 Venkatesha & R.T. Naveen Kumar CUBO 19, 2 (2017) 1 Introduction In 1950, Walker [17] introduced the notion of recurrent manifolds. In the last five decades, recurrent structures have played an important role in the geometry and the topology of manifolds. In [3], the authors De and Guha introduced the idea of generalized recurrent manifold with the non- zero 1-form A and another non-zero associated 1-form B. If the associated 1-form B becomes zero, then the manifold reduces to a recurrent manifold given by Ruse [11]. As a generalization of recurrency, Khan [6] introduced the notion of generalized recurrent Sasakian manifold. Semi- generalized recurrent manifolds were first introduced and studied by Prasad [10]. The notion of recurrency in a Riemannian manifold has been weakened by many authors in several ways to different extent viz., [1, 8, 12] etc., A K-contact manifold is a differentiable manifold with a contact metric structure such that ξ is a Killing vector field [2, 13]. These are studied by several authors like [4, 9, 14, 15] and many others. It is well known that every Sasakian manifold is K-contact, but the converse ia not true, in general. However a three-dimensional K-contact manifold is Sasakian [5]. Motivated by the above studies, in this study we consider some recurrent properties of three dimensional K-contact manifolds. The paper is organized in the following way: In Section 2, we give the definitions and some results concerning the K-contact manifolds that will be needed hereafter. In Section 3, we discuss the Ricci η-recurrent property of three dimensional K-contact manifold. In particular, we obtain the 1-form A is η parallel and give the expression for Ricci tensor. The Section 4 is devoted to three dimensional semi-generalized recurrent K-contact manifolds. Here we prove some interesting results, such as the facts that a specific linear combination of the 1-forms A and B is always zero and that the manifold is Einstein. In Section 5, we consider three dimensional locally generalized concircularly φ-recurrent K-contact manifolds. In this case the manifold is a space of constant curvature. 2 Preliminaries A Riemannian manifold M is said to admit an almost contact metric structure (φ, ξ, η, g) if it carries a tensor field φ of type (1, 1), a vector field ξ, 1-form η and compatible Riemannian metric g on M, such that φ2X = −X + η(X)ξ, φξ = 0, η(φX) = 0, (2.1) η(ξ) = 1, g(X, ξ) = η(X), (2.2) g(φX, φY) = g(X, Y) − η(X)η(Y), (2.3) g(φX, Y) = −g(X, φY), g(φX, X) = 0. (2.4) CUBO 19, 2 (2017) On Some Recurrent Properties of Three Dimensional . . . 3 If moreover ξ is Killing vector field, then M is called a K-contact manifold [2, 13]. A K-contact manifold is called Sasakian [2], if the relation (∇Xφ)(Y) = g(X, Y)ξ − η(Y)X, (2.5) holds on M, where ∇ denotes the operator of covariant differentiation with respect of metric g. In a K-contact manifold, the following relations hold: ∇Xξ = −φX, (2.6) (∇Xη)(Y) = g(∇Xξ, Y). (2.7) Also in a three dimensional K-contact manifold, the curvature tensor is given by R(X, Y)Z = r − 4 2 [g(Y, Z)X − g(X, Z)Y] − r − 6 2 [g(Y, Z)η(X)ξ (2.8) − g(X, Z)η(Y)ξ + η(Y)η(Z)X − η(X)η(Z)Y], S(X, Y) = 1 2 [(r − 2)g(X, Y) − (r − 6)η(X)η(Y)], (2.9) QX = 1 2 [(r − 2)X − (r − 6)η(X)ξ], (2.10) S(φX, φY) = S(X, Y) − 2η(X)η(Y), (2.11) where r, S and Q are the scalar curvature, Ricci tensor and Ricci operator respectively. Definition 1. A K-contact manifold is said to be Einstein if the Ricci tensor S is of the form S(X, Y) = ag(X, Y), where a is constant. 3 On three dimensional Ricci η-recurrent K-contact mani- fold Definition 2. The Ricci tensor of an three dimensional K-contact manifold is said to be η-recurrent if its Ricci tensor satisfies the following: (∇XS)(φ(Y), φ(Z)) = A(X)S(φ(Y), φ(Z)), (3.1) for all vector fields X, Y, Z ∈ TM, where A(X) = g(X, ρ), ρ is called the associated vector field of 1-form A. In particular, if the 1-form A vanishes then the Ricci tensor is said to be η-parallel and this notion for Sasakian manifold was first introduced by Kon [18]. 4 Venkatesha & R.T. Naveen Kumar CUBO 19, 2 (2017) Now consider three dimensional Ricci η-Recurrent K-contact manifold. From (3.1), it follows that ∇ZS(φ(X), φ(Y)) − S(∇ZφX, φY) − S(φX, ∇ZφY) = A(Z)S(φ(X), φ(Y)). (3.2) By using (2.5), (2.6) and (2.11) in (3.2), yields (∇ZS)(X, Y) = −η(X)[2g(φZ, Y) + S(Z, φY)] − η(Y)[2g(φZ, X) + S(φX, Z)] (3.3) + A(Z)[S(X, Y) − 2η(X)η(Y)]. Hence we can state the following: Theorem 3.1. In a three dimensional K-contact manifold, the Ricci tensor is η-recurrent if and only if (3.3) holds. By virtue of (3.3), let {ei} is an local orthonormal basis of the tangent space at each point of the manifold and taking summation over i, 1 ≤ i ≤ 3, we have dr(Z) = [r − 2]A(Z). (3.4) If the manifold has a constant scalar curvature r (r 6= 2 because the 1-form A is definite), then from (3.4) it follows that A(Z) = 0, ∀ Z. This leads to the following: Theorem 3.2. In a three dimensional Ricci η-recurrent K-contact manifold M if the scalar cur- vature is constant then the 1-form A is η-parallel. Again putting X = Z = ei in (3.3), and taking summation over i, 1 ≤ i ≤ 3, we get 1 2 dr(Y) + µη(Y) = S(Y, ρ) − 2η(ρ)η(Y), (3.5) where µ = Σ3 i=1S(φei, ei). By using (3.4) in (3.5), we obtain 1 2 A(Y)[r − 2] + µη(Y) = S(Y, ρ) − 2η(ρ)η(Y), (3.6) Putting Y = ξ in (3.6), yields µ = ( 1 − r 2 ) η(ρ). (3.7) Considering (3.7) in (3.6), we get S(Y, ρ) = ( r 2 − 1 ) g(Y, ρ) + ( 3 − r 2 ) η(ρ)η(Y). (3.8) Thus we have the following result: CUBO 19, 2 (2017) On Some Recurrent Properties of Three Dimensional . . . 5 Theorem 3.3. If the Ricci tensor in a three dimensional K-contact manifold is η-recurrent, then its Ricci tensor along the associated vector field of the 1-form is given by (3.8). Substituting Y = φY in (3.8) and by virtue of (2.1), we obtain S(Y, L) = Kg(Y, L), (3.9) where L = φρ, K = r 2 − 1. Hence we can state the following: Theorem 3.4. If the Ricci tensor in a three dimensional K-contact manifold is η-recurrent, then K = r 2 − 1 is an eigen value of the Ricci tensor corresponding to the eigen vector φρ. 4 On three dimensional semi-generalized recurrent K-contact manifolds Definition 3. A Riemannian manifold is said to be semi-generalized recurrent manifold if its curvature tensor R satisfies the relation (∇XR)(Y, Z)W = A(X)R(Y, Z)W + B(X)g(Z, W)Y, (4.1) where A and B are two 1-forms, B is non-zero, ρ1 and ρ2 are two vector fields such that g(X, ρ1) = A(X), g(X, ρ2) = B(X), (4.2) for any vector field X and ∇ be the covariant differentiation operator with respect to the metric g. Definition 4. A Riemannian manifold M is said to be three dimensional semi-generalized Ricci recurrent manifold if: (∇XS)(Y, Z) = A(X)S(Y, Z) + 3B(X)g(Y, Z). (4.3) Taking cyclic sum of (4.1) with respect to X, Y, Z, and using second Bianchi’s identity, we get 0 = A(X)R(Y, Z)W + A(Y)R(Z, X)W + A(Z)R(X, Y)W (4.4) + B(X)g(Z, W)Y + B(Y)g(X, W)Z + B(Z)g(Y, W)X. On contracting above equation with respect to Y, yields 0 = A(X)S(Z, W) − g(R(Z, X)ρ1, W) − A(Z)S(X, W) (4.5) + 3B(X)g(Z, W) + g(X, W)g(ρ2, Z) + B(Z)g(X, W). Again putting Z = W = ei in (4.5), and taking summation over i, 1 ≤ i ≤ 3, we obtain rA(X) + 11B(X) − 2S(X, ρ1) = 0. (4.6) 6 Venkatesha & R.T. Naveen Kumar CUBO 19, 2 (2017) Putting X = ξ in (4.6) and by virtue of (4.2) and (2.11), we get r = 1 η(ρ1) [4η(ρ1) − 11η(ρ2)]. (4.7) Since for a contact metric manifold η(ρ1) 6= 0. Hence we can state the following: Theorem 4.1. In a three dimensional semi-generalized recurrent K-contact manifold, the scalar curvature r takes the form (4.7). Again taking Z = ξ in (4.3), we get (∇XS)(Y, ξ) = A(X)S(Y, ξ) + 3B(X)g(Y, ξ). (4.8) Left hand side of the above equation can be written as (∇XS)(Y, ξ) = ∇XS(Y, ξ) − S(∇XY, ξ) − S(Y, ∇Xξ). (4.9) In view of (2.2), (2.9) and (4.9) in (4.8), gives −2g(φX, Y) + S(φX, Y) = 2A(X)η(Y) + 3B(X)η(Y). (4.10) Plugging Y = ξ in (4.10), we obtain 2A(X) + 3B(X) = 0. This leads to the following: Theorem 4.2. In a three dimensional semi-generalized Ricci recurrent K-contact manifold, the linear combination 2A + 3B is always zero. Replace Y by φY in (4.10), we get S(X, Y) = 2g(X, Y). Thus we have the following result: Theorem 4.3. A three dimensional semi-generalized Ricci recurrent K-contact manifold is Ein- stein manifold. 5 On three dimensional locally generalized concircularly φ- recurrent K-contact manifolds Definition 5. A three dimensional K-contact manifold is called the locally generalized concircularly φ-recurrent if its concircular curvature tensor C̃ C̃(X, Y)Z = R(X, Y)Z − r 6 [g(Y, Z)X − g(X, Z)Y], (5.1) CUBO 19, 2 (2017) On Some Recurrent Properties of Three Dimensional . . . 7 satisfies the condition φ2((∇WC̃)(X, Y)Z) = A(W)C̃(X, Y)Z + B(W)[g(Y, Z)X − g(X, Z)Y], (5.2) for all X, Y, Z and W orthogonal to ξ. Taking covariant differentiation of (2.8) with respect to W, we get (∇WR̃)(X, Y)Z = dr(W) 2 [g(Y, Z)X − g(X, Z)Y] − dr(W) 2 [g(Y, Z)η(X)ξ (5.3) −g(X, Z)η(Y)ξ − η(Y)η(Z)X − η(X)η(Z)Y] − r − 6 2 [g(Y, Z)(∇Wη)(X)ξ −g(X, Z)(∇Wη)(Y)ξ + (∇Wη)(Y)η(Z)X + η(Y)(∇Wη)(Z)X −(∇Wη)(X)η(Z)Y − η(X)(∇Wη)(Z)Y]. Again taking X, Y, Z and W orthogonal to ξ, we obtain (∇WR̃)(X, Y)Z = dr(W) 2 [g(Y, Z)X − g(X, Z)Y] − r − 6 2 [g(Y, Z)g(φX, W)ξ (5.4) − g(X, Z)g(φY, W)ξ]. From above equation it follows that φ2((∇WR̃)(X, Y)Z) = dr(W) 2 [g(X, Z)Y − g(Y, Z)X]. (5.5) Taking covariant differentiation of (5.1) with respect to W, we get (∇W ˜̃ C)(X, Y)Z = (∇WR̃)(X, Y)Z − dr(W) 6 [g(Y, Z)X − g(X, Z)Y], (5.6) from which it follows that φ2((∇W ˜̃ C)(X, Y)Z) = φ2((∇WR̃)(X, Y)Z) (5.7) − dr(W) 6 [g(Y, Z)φ2X − g(X, Z)φ2Y]. By virtue of (2.1), (5.2), (5.5) in (5.7), yields R(X, Y)Z = [ r 6 − ( B(W) A(W) + dr(W) 3A(W) )] [g(Y, Z)X − g(X, Z)Y]. (5.8) Since in a locally generalized concircularly φ-recurrent K-contact manifold A(W) 6= 0. On con- tracting above equation over W, we get R(X, Y)Z = µ[g(Y, Z)X − g(X, Z)Y], (5.9) where µ = r 6 − ( B(ei) A(ei) + dr(ei) 3A(ei) ) is a scalar. Then by Schur’s theorem [7] µ will be constant on the manifold. Thus we have the following result: Theorem 5.1. A three dimensional locally generalized concircularly φ-recurrent K-contact mani- fold is a space of constant curvature. 8 Venkatesha & R.T. Naveen Kumar CUBO 19, 2 (2017) References [1] Archana Singh, J.P. Singh and Rajesh Kumar, On a type of semi generalized recurrent P- Sasakian manifolds, FACTA UNIVERSITATIS, Ser. Math. Inform, 31 (1), (2016), 213-225. [2] D.E. Blair, Contact manifolds in Riemannian geometry. Lecture Notes in Math., No. 509, Springer, 1976. [3] U.C. De and N. Guha, On generalized recurrent manifold, J. Nat. Acad. Math., 9 (1991), 85-92. [4] U.C. 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Introduction Preliminaries On three dimensional Ricci -recurrent K-contact manifold On three dimensional semi-generalized recurrent K-contact manifolds On three dimensional locally generalized concircularly -recurrent K-contact manifolds