CUBO, A Mathematical Journal Vol.22, No¯ 01, (71–84). April 2020 http: // doi. org/ 10. 4067/ S0719-06462020000100071 Certain results on the conharmonic curvature tensor of (κ, µ)-contact metric manifolds Divyashree G. 1 and Venkatesha 2 1 Department of Mathematics, Govt., Science College, Chitradurga-577501, Karnataka, India. 2 Department of Mathematics, Kuvempu University, Shankaraghatta - 577 451, Shimoga, Karnataka, India. gdivyashree9@gmail.com, vensmath@gmail.com ABSTRACT The paper presents a study of (κ, µ)-contact metric manifolds satisfying certain condi- tions on the conharmonic curvature tensor. RESUMEN El art́ıculo presenta un estudio de variedades (κ, µ)-contacto métricas satisfaciendo ciertas condiciones sobre el tensor de curvatura conharmónico. Keywords and Phrases: (κ, µ)-contact metric manifold, conharmonically flat, conharmonically locally φ-symmetric, φ-conharmonically semisymmetric, h-conharmonically semisymmetric. 2010 AMS Mathematics Subject Classification: 53C25, 53C50, 53D10 http://doi.org/10.4067/S0719-06462020000100071 72 Divyashree G. & Venkatesha CUBO 22, 1 (2020) 1 Introduction In 1995, Blair et al.[3] introduced the idea of a class of contact metric manifolds for which the characteristic vector field ξ belongs to the (κ, µ)-nullity distribution for some real numbers κ and µ and such type of manifolds are called (κ, µ)-contact metric manifold. The non-Sasakian (κ, µ)- contact metric manifolds have two classes, namely, the class consists of the unit tangent sphere bundles of spaces of constant curvature, equipped with the natural contact metric structure and the class contains all the three-dimensional unimodular Lie groups, except the commutative one admitting the structure of a left invariant (κ, µ)-contact metric manifold [3, 4, 9]. Boeckx [4] given a full classification of (κ, µ)-contact metric manifolds. (κ, µ)-contact metric manifolds have been studied by several authors in [5, 6, 13, 11] and others. A rank-four tensor N that remains invariant under conharmonic transformation for a (2n+1)- dimensional Riemannian manifold M is given by N(X, Y)Z = R(X, Y)Z − 1 2n − 1 [S(Y, Z)X − S(X, Z)Y (1.1) +g(Y, Z)QX − g(X, Z)QY], which is also of the form N(X, Y, Z, T) = R(X, Y, Z, T) − 1 2n − 1 [S(Y, Z)g(X, T) − S(X, Z)g(Y, T) (1.2) +g(Y, Z)g(QX, T) − g(X, Z)g(QY, T)], where R, S and Q represents the Riemannian curvature tensor, Ricci tensor and Ricci operator respectively. A manifold whose conharmonic curvature vanishes at every point of the manifold is called conharmonically flat manifold. Such a curvature tensor have been extensively studied by Siddiqui and Ahsan [12], Ozgur [8], Avijit Sarkar et al. [10], Asghari and Taleshian [7] and many others. Our present work is organised in the following way: After introduction, section 2 includes basics related to (κ, µ)-contact metric manifold which will be used later. Section 3 deals with conharmonically flat (κ, µ)-contact metric manifolds. We proved that conharmonically locally φ- symmetric (κ, µ)-contact metric manifold is locally isometric to the Riemannian product En+1(0)× Sn(4) in section 4. Section 5 and 6 are devoted to the study of h-Conharmonically semisymmetric and φ-Conharmonically semisymmetric non-Sasakian (κ, µ)-contact metric manifolds respectively. Finally, we have shown that if the conharmonic curvature tensor on a (κ, µ)-contact metric manifold is divergent free then the Ricci tensor S is a Codazzi tensor. CUBO 22, 1 (2020) Certain results on the conharmonic curvature tensor of (κ, µ) . . . 73 2 Preliminaries A (2n+1)-dimensional differentiable manifold M2n+1 is called a contact manifold [1] if it carries a global 1-form η such that η ∧(dη)2n+1 6= 0 everywhere on M2n+1. It is well known that a contact metric manifold admits an almost contact metric structure (φ, ξ, η, g), where φ is a (1, 1)-tensor field, ξ is the characteristic vector field, and a Riemannian metric g such that φ2 = −I + η ⊗ ξ, g(X, ξ) = η(X), (2.1) η(ξ) = 1, g(X, Y) = g(φX, φY) + η(X)η(Y). (2.2) dη(X, Y) = g(X, φY), g(X, φY) = −g(Y, φX), (2.3) for all vector fields X, Y ∈ TM2n+1 and then we call a structure as contact metric structure. A manifold M2n+1 with such a structure is said to be contact metric manifold and it is denoted by (φ, ξ, η, g). φξ = 0, η ◦ φ = 0, dη(ξ, X) = 0. (2.4) We define a (1, 1)-tensor field h by h = 1 2 £ξφ, where £ξ is the Lie differentiation in the direction of ξ. Since the tensor field h is self-adjoint and anticommutes with φ, we have hξ = 0, φh + hφ = 0, trh = trφh = 0, (2.5) ∇Xξ = −φX − φhX, (2.6) (∇Xφ)Y = g(X, Y)ξ − η(Y)X, (2.7) where ∇ is the Levi-Civita connection and if X 6= 0 is an eigenvector of h corresponding to the eigenvalue λ, then φX is an eigenvector of h corresponding to the eigenvalue −λ. Blair et al. [3] studied the (κ, µ)-nullity condition and the (κ, µ)-nullity distribution N(κ, µ) of a contact metric manifold M is defined by [3] N(κ, µ) : p −→ Np(κ, µ) (2.8) = [Z ∈ TpM : R(X, Y)Z = (κI + µh){g(Y, Z)X − g(X, Z)Y}], for all X, Y ∈ TM2n+1. A contact metric manifold M2n+1 with ξ ∈ N(κ, µ) is called a (κ, µ)-contact metric manifold. In a (κ, µ)-contact metric manifold, we have R(X, Y)ξ = κ{η(Y)X − η(X)Y} + µ{η(Y)hX − η(X)hY}, (2.9) for all X, Y ∈ TM2n+1. In a (κ, µ)-contact metric manifold, the following relations hold [3, 11]: 74 Divyashree G. & Venkatesha CUBO 22, 1 (2020) h2 = (κ − 1)φ2, (2.10) (∇Xφ)Y = g(X + hX, Y)ξ − η(Y)(X + hX), (2.11) R(ξ, X)Y = κ[g(X, Y)ξ − η(Y)X] + µ[g(hX, Y)ξ − η(Y)hX], (2.12) S(X, ξ) = 2nκη(X), (2.13) S(X, Y) = [2(n − 1) − nµ]g(X, Y) + [2(n − 1) + µ]g(hX, Y) (2.14) +[2(1 − n) + n(2κ + µ)]η(X)η(Y), QX = [2(n − 1) − nµ]X + [2(n − 1) + µ]hX (2.15) +[2(n − 1) + n(2κ + µ)], S(φX, φY) = S(X, Y) − 2nkη(X)η(Y) − 2(2n − 2 + µ)g(hX, Y), (2.16) g(QX, Y) = S(X, Y). (2.17) From (2.6), we have (∇Xη)Y = g(X + hX, φY), (2.18) (∇Xh)Y = {(1 − κ)g(X, φY) + g(X, hφY)}ξ + η(Y){h(φX + φhX)} (2.19) −µη(X)φhY, where S is the Ricci tensor of type (0, 2), Q is the Ricci operator and r is the scalar curvature of the manifold. It is well known that in a Sasakian manifold, the Ricci operator Q commutes with φ. But in a (κ, µ)-contact metric manifold Q does not commute with φ. In general, in a (κ, µ)-contact metric manifold Blair et al.[3] proved the following: Proposition 1. Let Mn be a (κ, µ)-contact metric manifold, then the relation Qφ − φQ = 2[2(n − 1) + µ]hφ, holds. From the definition of η-Einstein manifold, it follows easily that Qφ = φQ. Hence from Proposition 2.1 we obtain either µ = −2(n−1), or the manifold is Sasakian. Using µ = −2(n−1), from (2.14) we obtain that the manifold is an η-Einstein manifold. Therefore Yildiz and De [13] proved the following: Proposition 2. In a non-Sasakian (κ, µ)-contact metric manifold, the following conditions are equivalent: (i) η-Einstein manifold, (ii) Qφ = φQ. CUBO 22, 1 (2020) Certain results on the conharmonic curvature tensor of (κ, µ) . . . 75 For n = 1, from Proposition 2.1 and Proposition 2.2, Yildiz and De [13] obtained the following: Corolary 1. A 3-dimensional non-Sasakian (κ, µ)-contact η-Einstein manifold is an N(k)-contact metric manifold. Lemma 2.1. [2]:Let M2n+1 (φ, ξ, η, g) be a contact metric manifold with R(X, Y)ξ = 0 for all vector fields X, Y tangent to M2n+1. Then M2n+1 is locally isometric to the Riemannian product En+1(0) × Sn(4). 3 Conharmonically flat (κ, µ)-contact metric manifolds From (1.2), for a (2n+1)-dimensional conharmonically flat (κ, µ)-contact metric manifold, we have R(X, Y, Z, T) = 1 2n − 1 [S(Y, Z)g(X, T) − S(X, Z)g(Y, T) + g(Y, Z)g(QX, T) (3.1) −g(X, Z)g(QY, T)]. Substituting Z = ξ in (3.1) and using (2.1), (2.9) and (2.13), we obtain κ[η(Y)g(X, T) − η(X)g(Y, T)] + µ[η(Y)g(hX, T) − η(X)g(hY, T)] (3.2) = 1 2n − 1 [2nκη(Y)g(X, T) − 2nκη(X)g(Y, T) + η(Y)g(QX, T) − η(X)g(QY, T)]. Again, by taking Y = ξ and using (2.1), (2.2), (2.5) and (2.13), (3.2) becomes S(X, T) = −κg(X, T) + (2n + 1)κη(X)η(T) + (2n − 1)µg(hX, T). (3.3) From the equation (3.3), it follows that if µ = 0, then the manifold is an η-Einstein manifold. Conversely, if the manifold is η-Einstein, then we can write S(X, T) = a1g(X, T) + b1η(X)η(T). (3.4) On equating (3.3) and (3.4), we find a1g(X, T) + b1η(X)η(T) = −κg(X, T) + (2n + 1)κη(X)η(T) + (2n − 1)µg(hX, T).(3.5) Now, in (3.5) replacing T by φX and using (2.3), we get (2n − 1)µg(hX, φX) = 0, (3.6) for all X. Consequently, µ = 0. Hence, an n-dimensional conharmonically flat (κ, µ)-contact metric manifold is an η-Einstein man- ifold if and only if µ = 0. But from (2.14), it follows that a (κ, µ)-contact metric manifold is 76 Divyashree G. & Venkatesha CUBO 22, 1 (2020) η-Einstein if and only if {2(n − 1) + µ} = 0. If we consider a (2n + 1)-dimensional (n > 1) con- harmonically flat η-Einstein (κ, µ)-contact metric manifold, then n = 1, which contradicts the fact that n > 1. Hence, the theorem can be stated as follows: Theorem 3.1. An (2n + 1)-dimensional (n > 1) conharmonically flat (κ, µ)-contact metric man- ifold cannot be an η-Einstein manifold. 4 Conharmonically locally φ-symmetric (κ, µ)-contact met- ric manifolds Definition 4.1. An (2n + 1)-dimensional (n > 1) (κ, µ)-contact metric manifold M2n+1 is said to be conharmonically locally φ-symmetric if it satisfies φ2((∇WN)(X, Y)Z) = 0, (4.1) for all X, Y, Z, W orthogonal to ξ. Taking covariant differentiation of (1.1), we have (∇WN)(X, Y)Z = (∇WR)(X, Y)Z − 1 2n − 1 [(∇WS)(Y, Z)X − (∇WS)(X, Z)Y (4.2) +g(Y, Z)(∇WQ)(X) − g(X, Z)(∇WQ)(Y)], where ∇ denotes the Levi-Civita connection on the manifold. Differentiating equations (2.8), (2.14) and (2.15) covariantly with respect to W, we obtain (∇WR)(X, Y)Z = Wκ{g(Y, Z)X − g(X, Z)Y} + Wµ{g(Y, Z)hX − g(X, Z)hY} (4.3) +µ[g(Y, Z)({(1 − κ)g(W, φX) + g(W, hφX)}ξ +η(X){h(φW + φhW)} − µη(W)φhX) −g(X, Z)({(1 − κ)g(W, φY) + g(W, hφY)}ξ +η(Y){h(φW + φhW)} − µη(W)φhY)], (∇WS)(Y, Z)X = {2(1 − n) + n(2κ + µ)}[g(W, φY)η(Z)X (4.4) +g(hW, φY)η(Z)X + g(W, φZ)η(Y)X + g(hW, φZ)η(Y)X] +(2(n − 1) + µ)[{(1 − κ)g(W, φY)η(Z)X + g(W, hφY)η(Z)X +g(h(φW + φhW), Z)η(Y)X} − µg(φhY, Z)η(W)X] CUBO 22, 1 (2020) Certain results on the conharmonic curvature tensor of (κ, µ) . . . 77 and (∇WQ)(X) = {2(n − 1) + µ}[{(1 − κ)g(W, φX) + g(W, hφX)}ξ (4.5) +η(X){h(φW + φhX)} − µη(W)φhX] +{2(n − 1) + n(2κ + µ)}g(W, φX)ξ +{2(n − 1) + n(2κ + µ)}g(hW, φX)ξ −{2(n − 1) + n(2κ + µ)}η(X)φW −{2(n − 1) + n(2κ + µ)}η(X)φhW. Now, considering equations (4.3), (4.4) and (4.5) in (4.2) and also taking X, Y, Z, W orthogonal to ξ, we get (∇WN)(X, Y)Z = Wκ[g(Y, Z)X − g(X, Z)Y] + Wκ[g(Y, Z)hX − g(X, Z)hY] (4.6) +µ[(1 − κ)g(Y, Z)g(W, φX)ξ + (1 − κ)g(Y, Z)g(W, hφX)ξ −(1 − κ)g(X, Z)g(W, φY)ξ − (1 − κ)g(X, Z)g(W, hφY)ξ] − 1 2n − 1 [{2(n − 1) + µ}{(1 − κ)[g(Y, Z)g(W, φX)ξ −g(X, Z)g(W, φY)ξ] + g(Y, Z)g(W, hφX)ξ −g(X, Z)g(W, hφY)ξ} +{2(n − 1) + n(2κ + µ)}[g(Y, Z)g(W, φX)ξ +g(Y, Z)g(hW, φX)ξ − g(X, Z)g(W, φY)ξ −g(X, Z)g(hW, φY)ξ]]. Applying φ2 on both sides of (4.6), one can obtain φ2((∇WN)(X, Y)Z) = φ 2 {Wκ[g(Y, Z)X − g(X, Z)Y] + Wκ[g(Y, Z)hX (4.7) −g(X, Z)hY] + µ[(1 − κ)g(Y, Z)g(W, φX)ξ +(1 − κ)g(Y, Z)g(W, hφX)ξ − (1 − κ)g(X, Z)g(W, φY)ξ −(1 − κ)g(X, Z)g(W, hφY)ξ] − 1 2n − 1 [{2(n − 1) + µ}{(1 − κ)[g(Y, Z)g(W, φX)ξ −g(X, Z)g(W, φY)ξ] + g(Y, Z)g(W, hφX)ξ −g(X, Z)g(W, hφY)ξ} +{2(n − 1) + n(2κ + µ)}{g(Y, Z)g(W, φX)ξ +g(Y, Z)g(hW, φX)ξ − g(X, Z)g(W, φY)ξ −g(X, Z)g(hW, φY)ξ}]}. 78 Divyashree G. & Venkatesha CUBO 22, 1 (2020) From (4.1) and using (2.1), (4.7) becomes (Wκ)[g(X, Z)Y − g(Y, Z)X] + (Wκ)[g(Y, Z)η(X) − g(X, Z)η(Y)]ξ (4.8) +(Wµ)[g(X, Z)hY − g(Y, Z)hX] = 0. Again, considering X, Y orthogonal to ξ, one can get (Wκ)[g(X, Z)Y − g(Y, Z)X] + (Wµ)[g(X, Z)hY − g(Y, Z)hX] = 0. (4.9) By taking inner product of (4.9) with V, we have (Wκ)[g(X, Z)g(Y, V) − g(Y, Z)g(X, V)] + (Wµ)[g(X, Z)g(hY, V) (4.10) −g(Y, Z)g(hX, V)] = 0. On contraction, the above equation yields −2n(Wκ)g(Y, Z) + (Wµ)g(Z, hY) = 0. (4.11) Setting Y = ξ in (4.11) and using (2.5), we get 2n(Wκ)η(Z) = 0. (4.12) If we assume that κ = 0 in (4.11) then either µ = 0 or g(Z, hY) = 0. Further, if κ = 0 = µ in (2.9), then we get R(X, Y)ξ = 0 for all X, Y and in the light of Lemma 2.1, the manifold under consideration is locally isometric to the Riemannian product En+1 × Sn(4). So from Lemma 2.1, we can state the theorem as follows: Theorem 4.2. Let M2n+1 (φ, ξ, η, g) be a conharmonically locally φ-symmetric (κ, µ)-contact metric manifold. Then the manifold is locally isometric to the Riemannian product En+1(0)×Sn(4). 5 h-Conharmonically semisymmetric non-Sasakian (κ, µ)-contact metric manifolds Definition 5.1. A Riemannian manifold (M2n+1, g) is said to be h-conharmonically semisym- metric if it satisfies N(X, Y) · h = 0. (5.1) The following lemma which was proved in [3] is helpful to state our theorem. CUBO 22, 1 (2020) Certain results on the conharmonic curvature tensor of (κ, µ) . . . 79 Lemma 5.1. [3]: Let M2n+1(φ, ξ, η, g) be a contact metric manifold with ξ belonging to the (κ, µ)-nullity distribution. Then for any vector fields X, Y, Z, R(X, Y)hZ − hR(X, Y)Z = {κ[g(hY, Z)η(X) − g(hX, Z)η(Y)] (5.2) +µ(κ − 1)[g(X, Z)η(Y) − g(Y, Z)η(X)]}ξ +κ{g(Y, φZ)φhX − g(X, φZ)φhY + g(Z, φhY)φX −g(Z, φhX)φY + η(Z)[η(X)hY − η(Y)hX]} −µ{η(Y)[(1 − κ)η(Z)X + µη(X)hZ] −η(X)[(1 − κ)η(Z)Y + µη(Y)hZ] + 2g(X, φY)φhZ}. Let M2n+1 be h-conharmonically semisymmetric non-Sasakian (κ, µ)-contact metric manifold. The condition N(X, Y) · h = 0 can be expressed as follows, (N(X, Y) · h)Z = N(X, Y)hZ − hN(X, Y)Z = 0, (5.3) for any vector fields X, Y, Z. With the help of (1.1) and (5.2), (5.3) can be written as [κ{g(hY, Z)η(X) − g(hX, Z)η(Y)} + µ(κ − 1){g(X, Z)η(Y) − g(Y, Z)η(X)}]ξ (5.4) +κ{g(Y, φZ)φhX − g(X, φZ)φhY + g(Z, φhY)φX − g(Z, φhX)φY + η(Z)[η(X)hY −η(Y)hX]} − µ{η(Y)[(1 − κ)η(Z)X + µη(X)hZ] − η(X)[(1 − κ)η(Z)Y + µη(Y)hZ] +2g(X, φY)φhZ} − 1 2n − 1 [S(Y, hZ)X − S(X, hZ)Y + g(Y, hZ)QX − g(X, hZ)QY −S(Y, Z)hX + S(X, Z)hY − g(Y, Z)QhX + g(X, Z)QhY] = 0. By taking inner product of (5.4) with T, we get [κ{g(hY, Z)η(X) − g(hX, Z)η(Y)} + µ(κ − 1){g(X, Z)η(Y) − g(Y, Z)η(X)}]η(T) (5.5) +κ{g(Y, φZ)g(φhX, T) − g(X, φZ)g(φhY, W) + g(Z, φhY)g(φX, T) −g(Z, φhX)g(φY, W) + η(Z)[η(X)g(hY, W) − η(Y)g(hX, T)]} −µ{η(Y)[(1 − κ)η(Z)g(X, T) + µη(X)g(hZ, T)] − η(X)[(1 − κ)η(Z)g(Y, T) +µη(Y)g(hZ, T)] + 2g(X, φY)g(φhZ, T)} − 1 2n − 1 [S(Y, hZ)g(X, T) −S(X, hZ)g(Y, T) + g(Y, hZ)S(X, T) − g(X, hZ)S(Y, T) − S(Y, Z)g(hX, T) +S(X, Z)g(hY, T) − g(Y, Z)S(hX, T) + g(X, Z)S(hY, T)] = 0. Setting Y = T = ξ in (5.5) and using (2.2) and (2.5), we get 1 2n − 1 S(X, hZ) = −µ(1 − κ)g(X, Z) + [2(1 − µ) + (1 − κ)]η(X)η(Z) (5.6) +[κ − 2(2n + 1)κ 2(n − 1) g(X, hZ)]. 80 Divyashree G. & Venkatesha CUBO 22, 1 (2020) Replacing X by hX in the above equation and using (2.10), we have S(X, Z) = −κg(X, Z) + κη(X)η(Z) − 2µ(n − 1)g(hX, Z). (5.7) If we consider µ = 0 in (5.7) then it is an η-Einstein manifold. Using (2.14) in (5.7) and simplifying, we finally obtain S(X, Z) = n1g(X, Z) + n2η(X)η(Z), (5.8) where n1 = −κ[2(n−1)+µ]+µ(2n−1)[2(n−1)+nµ] [2(n−1)+µ]+µ(2n−1) and n2 = κ[2(n−1)+µ]+µ(2n−1)[2(1−n)+n(2κ+µ)] [2(n−1)+µ]+µ(2n−1) . Thus from (5.8), we can conclude the following theorem: Theorem 5.2. Let M2n+1(φ, ξ, η, g) be a non-Sasakian (κ, µ)-contact metric manifold. If M is h-conharmonically semisymmetric, then the manifold is an η-Einstein manifold with constant coefficients. From Proposition 2.2 and Theorem 5.5 we can state the following: Corolary 2. If M2n+1 is a h-conharmonically semisymmetric (κ, µ)-contact metric manifold then the Ricci operator Q commutes with φ i.e., Qφ = φQ. 6 φ-Conharmonically semisymmetric non-Sasakian (κ, µ)-contact metric manifolds Definition 6.1. A Riemannian manifold (M2n+1, g) is said to be φ-conharmonically semisym- metric if N(X, Y) · φ = 0. (6.1) Now we need the following lemma: Lemma 6.1. [3]: Let M2n+1(φ, ξ, η, g) be a contact metric manifold with ξ belonging to the (κ, µ)-nullity distribution. Then for any vector fields X, Y, Z, R(X, Y)φZ − φR(X, Y)Z = {(1 − κ)[g(φY, Z)η(X) − g(φX, Z)η(Y)] (6.2) +(1 − µ)[g(φhY, Z)η(X) − g(φhX, Z)η(Y)]}ξ −g(Y + hY, Z)(φX + φhX) + g(X + hX, Z)(φY +φhY) − g(φY + φhY, Z)(X + hX) + g(φX +φhX, Z)(Y + hY) − η(Z){(1 − κ)[η(X)φY −η(Y)φX] + (1 − µ)[η(X)φhY − η(Y)φhX)]}. CUBO 22, 1 (2020) Certain results on the conharmonic curvature tensor of (κ, µ) . . . 81 Let M2n+1 be a (2n+1)-dimensional φ-conharmonically semisymmetric non-Sasakian (κ, µ)- contact metric manifold. The condition N(X, Y) · φ = 0 turns into, (N(X, Y) · φ)Z = N(X, Y)φZ − φN(X, Y)Z = 0, (6.3) for any vector fields X, Y, Z. In view of (1.1) and (6.2), (6.3) becomes {(1 − κ)[g(φY, Z)η(X) − g(φX, Z)η(Y)] + (1 − µ)[g(φhY, Z)η(X) − g(φhX, Z)η(Y)]}ξ (6.4) −g(Y + hY, Z)(φX + φhX) + g(X + hX, Z)(φY + φhY) − g(φY + φhY, Z)(X + hX) +g(φX + φhX, Z)(Y + hY) − η(Z){(1 − κ)[η(X)φY − η(Y)φX] + (1 − µ)[η(X)φhY −η(Y)φhX)]} − 1 2n − 1 [S(Y, φZ)X − S(X, φZ)Y + g(Y, φZ)QX − g(X, φZ)QY −S(Y, Z)φX + S(X, Z)φY − g(Y, Z)QφX + g(X, Z)QφY] = 0. Taking inner product of (6.4) with T, we get {(1 − κ)[g(φY, Z)η(X) − g(φX, Z)η(Y)] + (1 − µ)[g(φhY, Z)η(X) (6.5) −g(φhX, Z)η(Y)]}η(T) − g(Y, Z)g(φX, T) − g(hY, Z)g(φX, T) − g(Y, Z)g(φhX, T) −g(hY, Z)g(φhX, T) + g(X, Z)g(φY, T) + g(hX, Z)g(φY, T) + g(X, Z)g(φhY, T) +g(hX, Z)g(φhY, T) − g(φY, Z)g(X, T) − g(φY, Z)g(hX, T) − g(φhY, Z)g(X, T) −g(φhY, Z)g(hX, T) + g(φX, Z)g(Y, T) + g(φhX, Z)g(Y, T) + g(φX, Z)g(hY, T) +g(φhX, Z)g(hY, T) − η(Z){(1 − κ)[η(X)g(φY, T) − η(Y)g(φX, T)] +(1 − µ)[η(X)g(φhY, T) − η(Y)g(φhX, T)]} − 1 2n − 1 [S(Y, φZ)g(X, T) −S(X, φZ)g(Y, T) + g(Y, φZ)g(QX, T) − g(X, φZ)g(QY, T) − S(Y, Z)g(φX, T) +S(X, Z)g(φY, T) − g(Y, Z)g(QφX, T) + g(X, Z)g(QφY, T)] = 0. Treating Y = T = ξ in (6.5) and using (2.1), (2.2), (2.4), (2.5) and (2.13), we have 1 2n − 1 S(X, φZ) = {(κ − 2) + 2(2n + 1)κ 2n − 1 }g(X, φZ) − µg(φX, hZ). (6.6) Substituting X by φX in (6.6) and using (2.1), (2.2) and (2.16), one can get S(X, Z) = [(κ − 2)(2n − 1) + 2nκ]g(X, Z) − [(κ − 2)(2n − 1)]η(X)η(Z) (6.7) +[µ(κ − 1)(2n − 1) + 2{2(n − 1) + µ}]g(hX, Z). Making use of (2.14), (6.7) yields S(X, Z) = n3g(X, Z) + n4η(X)η(Z), (6.8) 82 Divyashree G. & Venkatesha CUBO 22, 1 (2020) where n3 = {(κ−2)(2n−1)+2nκ}{2(n−1)+µ}−{µ(κ−1)(2n−1)+2[2(n−1)+µ]}{2(n−1)−nµ} [2(n−1)+µ]−[µ(κ−1)(2n−1)+2{2(n−1)+µ}] and n4 = [(2−κ)(2n−1)][2(n−1)+µ]−{µ(κ−1)(2n−1)+2[2(n−1)+µ]}[2(1−n)+2n(2κ+µ)] [2(n−1)+µ]−[µ(κ−1)(2n−1)+2{2(n−1)+µ}] . Hence from (6.8), the theorem can be stated as follows: Theorem 6.2. If a (2n + 1)-dimensional non-Sasakian (κ, µ)-contact metric manifold M2n+1 is φ-conharmonically semisymmetric then the manifold is an η-Einstein manifold with constant coefficients. Similarly, from Proposition 2.2 and Theorem 6.6, we get the following statement: Corolary 3. If M2n+1 is a φ-conharmonically semisymmetric (κ, µ)-contact metric manifold then the Ricci operator Q commutes with φ i.e., Qφ = φQ. 7 (κ, µ)-contact metric manifold with divergent free conhar- monic curvature tensor In this section, we study divergent free conharmonic curvature tensor on (κ, µ)-contact metric manifold. Let M2n+1(φ, ξ, η, g) (n > 1) be a (κ, µ)-contact metric manifold satisfying the following condition (DivN)(X, Y)Z = 0. (7.1) In view of (7.1), (1.1) leads to (DivR)(X, Y)Z = 1 2n − 1 [(∇XS)(Y, Z) − (∇YS)(X, Z) + g(Y, Z)dr(X) (7.2) −g(X, Z)dr(Y)]. The above equation simplifies to, 2(n − 1) (2n − 1) [(∇XS)(Y, Z) − (∇YS)(X, Z)] − 1 (2n − 1) [g(Y, Z)dr(X) (7.3) −g(X, Z)dr(Y)] = 0. On contracting and taking summation over i, 1 ≤ i ≤ n in (7.3), we get 2(3n − 1)dr(Y) = 0, (7.4) which implies dr(Y) = 0, (7.5) CUBO 22, 1 (2020) Certain results on the conharmonic curvature tensor of (κ, µ) . . . 83 since 2(3n − 1) 6= 0. Further, considering (7.5) in (7.3), we obtain (∇XS)(Y, Z) − (∇YS)(X, Z) = 0, (7.6) which gives (∇XQ)Y = (∇YQ)X. (7.7) Thus, we can state: Theorem 7.1. Let M2n+1(φ, ξ, η, g) (n > 1) be a (κ, µ)-contact metric manifold. If the manifold has divergent free conharmonic curvature tensor then the Ricci tensor S is a Codazzi tensor. 84 Divyashree G. & Venkatesha CUBO 22, 1 (2020) References [1] Blair D. E, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer- Verlag, 1976. [2] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 29, 319-324, 1977. [3] D.E. Blair, T. Koufogiorgos and B.J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math., 19, 189-214, 1995. [4] E. Boeckx, A full classification of contact metric (κ, µ)-spaces, Illinois J. of Math. 44, 212-219, 2000. [5] A. Ghosh, R. Sharma and J.T. Cho, Contact metric manifolds with η-parallel torsion tensor, Ann. Glob. Anal. Geom. 34, 287-299, 2008. [6] Jun J. B. 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De, A classification of (κ, µ)-contact metric manifolds, Commun. Korean Math. Soc., 2, 327-339, 2012. Introduction Preliminaries Conharmonically flat (,)-contact metric manifolds Conharmonically locally -symmetric (,)-contact metric manifolds h-Conharmonically semisymmetric non-Sasakian (,)-contact metric manifolds -Conharmonically semisymmetric non-Sasakian (,)-contact metric manifolds (,)-contact metric manifold with divergent free conharmonic curvature tensor