Notes on quasi contact metric manifolds

Transcription

1 An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) Tomul LXII, 016, f., vol. 1 Notes on quasi contact metric manifolds Y.D. Chai J.H. Kim J.H. Park K. Sekigawa W.M. Shin Received: 11.III.014 / Revised: 6.VI.014 / Accepted: 14.VII.014 Abstract We study curvature identities on quasi contact metric manifolds based on the geometry of the corresponding quasi Kähler cones and we provide several results derived from the obtained curvature identities. Keywords Curvature identity Quasi contact metric manifold Mathematics Subject Classification (010) 53C5 53D15 1 Introduction An odd dimensional smooth manifold M is called an almost contact manifold if it admits a triple (φ, ξ, η) of a (1, 1) tensor field, a vector field ξ and a 1-form η satisfying φ = I + η ξ, φ(ξ) = 0, η φ = 0, (1.1) for any X X(M), X(M) denoting the Lie algebra of all smooth vector fields on M. It is well known that η(ξ) = 1. Further, an almost contact manifold M = (M, φ, ξ, η) equipped with a Riemannian metric g satisfying g(φx, φy ) = g(x, Y ) η(x)η(y ), η(x) = g(ξ, X), (1.) for any X, Y X(M) is called an almost contact metric manifold. Especially, an almost contact metric manifold M = (M, φ, ξ, η, g) satisfying Y.D. Chai, J.H. Kim, J.H. Park & W.M. Shin Department of Mathematics Sungkyunkwan University Suwon , Korea ydchai@skku.edu; mpkim@skku.edu; parkj@skku.edu; Jacarta1@skku.edu K. Sekigawa Department of Mathematics Niigata University Niigata , Japan sekigawa@math.sc.niigata-u.ac.jp dη(x, Y ) = g(x, φy ), (1.3) 349

2 Y.D. Chai, J.H. Kim, J.H. Park, K. Sekigawa, W.M. Shin for any X, Y X(M) is called a contact metric manifold. Now, let M = (M, φ, ξ, η, g) be a (n+1)-dimensional almost contact metric manifold and M = M R be the almost Hermitian structure ( J, ḡ) defined by JX = φx η(x) t, J = ξ, (1.4) ( t ḡ(x, Y ) = e t g(x, Y ), ḡ X, ) ( = 0, ḡ t t, ) = e t, t for any X, Y X(M) and t R. Remark 1.1 Let M = M R+ be the product manifold of M and a positive half line R + = {s R s > 0} be equipped with the following almost Hermitian structure ( J, g) defined by JX = φx + sη(x) s, J s = 1 s ξ, g(x, Y ) = s g(x, Y ), g(x, ( s ) = 0, g s, ) = 1, s (1.4 ) for any X, Y X(M) and s R +. Then, it is easily checked that (M, J, ḡ) is holomorphically isometric to ( M, J, g) under the diffeomorphism F : M M defined by F (p, t) = (p, e t ), for any (p, t) M. Taking account of Remark 1.1, we shall call M = (M, J, ḡ) the corresponding almost Hermitian cone to the almost contact metric manifold M = (M, φ, ξ, η, g). It is also checked that M = (M, φ, ξ, η, g) is a Sasakian manifold (resp. a contact metric manifold) if and only if the corresponding almost Hermitian cone M = (M, J, ḡ) is a Kähler manifold (resp. an almost Kähler manifold, see [7]). Gray and Hervella [3] introduced 16 classes of almost Hermitian manifolds. Among their classes, the classes of Kähler manifold K, almost Kähler manifolds AK, nearly Kähler manifolds N K and quasi Kähler manifolds QK have been studied extensively by many authors. The inclusion relations between the classes K, AK, N K and QK are as follows: K = N K AK and AK, N K QK. An almost contact metric manifold M = (M, φ, ξ, η, g) is called a quasi contact metric manifold if the corresponding almost Hermitian cone M = (M, J, ḡ) is a quasi Kähler manifold. Kim, Park and Sekigawa [4] discussed basic properties concerning a quasi contact metric manifold and gave a characterization of a quasi contact metric manifold and further a characterization of a contact metric manifold based on the result. It is well-known that quasi Kähler manifold is necessarily an almost Kähler manifold. Thus, a 3-dimensional quasi contact metric manifold is necessarily a contact metric manifold (see [5]). We refer to [5]. In this paper, we will work on the question Does there exist a (n + 1)(n )-dimensional quasi contact metric manifold which is not a contact metric manifold? which was raised on the previous paper [4]. 350

3 Notes on quasi contact metric manifolds 3 Preliminaries In this section, we prepare some basic terminologies and review the results from the previous paper [4]. Let M = (M, φ, ξ, η, g) be a (n+1)-dimensional almost contact metric manifold and h be the (1, 1) tensor field on M defined by h = 1 ξφ, (.1) here ξ denotes the Lie differentiation with respect to the characteristic vector field ξ. The tensor field h plays an important role in the geometry of almost contact metric manifolds. By the definition (.1), we may easily observe that the tensor field h satisfies the following equalities hξ = 0, trh = 0. (.) We denote by the Levi-Civita connection of g and R the curvature tensor defined by R(X, Y )Z = [ X, Y ]Z [X,Y ] Z, (.3) for any X, Y X(M). We denote by ρ and τ the Ricci tensor and the scalar curvature of M = (M, φ, ξ, η, g), respectively. Further, we denote by ρ and τ the Ricci -tensor and the scalar -curvature of M = (M, φ, ξ, η, g) defined respectively by for any X, Y, Z X(M), and ρ (X, Y ) = 1 tr(z R(X, φy )φz), (.4) where Q is the (1,1) tensor field on M defined by τ = trq, (.5) g(q X, Y ) = ρ (X, Y ), (.6) for any X, Y X(M). Recently, Kim, Park and Sekigawa [4] gave a following characterization of a contact metric manifold. Theorem.1 ([4], Theorem 3.) A contact metric manifold is characterized as an almost contact metric manifold M = (M, φ, ξ, η, g) satisfying the following conditions: h is symmetric, (.7) ( X φ)y + ( φx φ)φy = g(x, Y )ξ η(y )X η(x)η(y )ξ η(y )hx, (.8) for any X, Y X(M). They also proved the following. Theorem. ([4], Theorem 4.) A quasi contact metric manifold is characterized as an almost contact metric manifold M = (M, φ, ξ, η, g) satisfying the condition (.8) in Theorem.1 We here note that the equality (.8) is equivalent to the following on any almost contact metric manifold M = (M, φ, ξ, η, g): ( X φ)y + ( φx φ)φy = g(x, Y )ξ η(y )X + η(y ) φx ξ, (.9) for any X, Y X(M) (see [4], Lemma 4.1). 351

4 4 Y.D. Chai, J.H. Kim, J.H. Park, K. Sekigawa, W.M. Shin Remark.1 For any almost contact metric manifold M = (M, φ, ξ, η, g), the following equalities (.10), (.11) and (.1) are derived from (.8) or (.9): ( X η)y + ( φx η)φy + g(φx, Y )ξ = 0, (.10) for any X, Y X(M) (see [4], Proposition.6). ξ φ = 0, (.11) ξ ξ = 0, (.1) 3 Fundamental formulas In this section, we shall deduce several basic formulas on a quasi contact metric manifold based on the discussions in the previous sections. Let M = (M, φ, ξ, η, g) be a (n+1)-dimensional quasi contact metric manifold. Then, from (.1) and (.5), we see that the tensor field h takes the following form: hx = 1 ( φxξ + φ X ξ), (3.1) for any X X(M). From (3.1), taking account of (.1), we have Setting Y = ξ in (.9), we get hφx = 1 ( φ Xξ + φ φx ξ) = 1 ( Xξ ( φx φ)ξ). (3.) Operating φ to the both sides of (3.3), we have Thus, from (3.) and (3.4), we obtain On one hand, from (3.1) and (3.4), we obtain φ X ξ = η(x)ξ X + φx ξ. (3.3) ( φx φ)ξ = φx X ξ. (3.4) hφx = X ξ + φx. (3.5) φhx = 1 ( φ φxξ + φ X ξ) = 1 (( φxφ)ξ X ξ) = φx X ξ. (3.6) Thus, from (3.5) and (3.6), we have and hφ + φh = 0, (3.7) X ξ = φx φhx, (3.8) for any X X(M). From (3.7), we have η(hφx) = 0 for any X X(M), and hence, also η(hφ X) = η(hx) = 0, for any X X(M). Thus, we have η h = 0. (3.9) 35

5 From (3.8) and (3.9), taking account of (.), we have η =( i η j )( i ξ j ) = (φ ij + φ aj h i a )(φ ij + φ b j h ib ) Notes on quasi contact metric manifolds 5 = φ ij φ ij + φ ij φ b j h ib + φ aj h i a φ ij + φ aj h i a φ b j h ib (3.10) = n + (g ib η i η b )h ib + (δ a i η a ξ i )h i a + (g ab η a η b )h i a h ib = n + h ij h ij = n + h. From (3.10), we see that η 0 everywhere on M, and also that, especially, if M = (M, φ, ξ, η, g) is a contact metric manifold, h = trh holds by virtue of Theorem.1, and hence, in this case, η = n + trh holds (see [8], (3.7)). Now, the equality (.9) is rewritten in terms of local coordinates as i φ j k + φ i a ( a φ b k )φ j b = g ij ξ k η j δ i k + η j φ i a a ξ k. From which, we have i φ j i + φ i a ( a φ b i )φ j b = η j (n + 1)η j + η j φ i a a ξ i Here, taking account of (.1), we get φ i a ( a φ b i )φ j b = φ i a a (φ b i φ j b ) φ i a φ b i a φ j b = 4nη j + η j φ i a a ξ i. (3.11) = φ i a a (ξ i η j ) + a φ j a ξ a η b a φ j b = φ i a ( a ξ i )η j + a φ j a. (3.1) Thus, from (3.11) and (3.1), we have i φ j i = nη j. (3.13) From (3.8), taking account of (3.7), we get div ξ = i ξ i = trφ tr(φh) = tr(φh) = tr(hφ) = tr(φh) = div ξ, and hence From (3.1), taking account of (.11) and (.1), we get ξ k k h j i = 1 ξk ( ( k φ j a ) a ξ i φ j a k a ξ i + ( k φ i a ) j η a + φ a i k j ξ a ) div ξ(= i ξ i ) = 0. (3.14) = 1 ξk φ j a a k ξ i 1 ξk φ j a R kab i ξ b + 1 ξk φ a i j k ξ a (3.15) + 1 ξk φ a i R kjb a ξ b = 1 φ j a ( a ξ k ) k ξ i 1 φ a i ( j ξ k ) k ξ a 1 ξk φ j a R kab i ξ b + 1 ξk φ a i R kjb a ξ b. 353

6 6 Y.D. Chai, J.H. Kim, J.H. Park, K. Sekigawa, W.M. Shin Here, taking account of (3.7) and (3.8), we get 1 φ j a ( a ξ k ) k ξ i = 1 φ j a (φ a k + φ b k h a b )(φ k i + φ c i h k c ) Similarly, we get = 1 φ j a ( δ a i + ξη a φ a k h c i φ k c h a i φ b k h a b h c i φ k c ) (3.16) = 1 φ j i + 1 φ a i h c a h j c. 1 φ a i ( j ξ k ) k ξ a = 1 φ j i 1 φ a i h c a h j c. (3.17) Thus, from (3.5), (3.15) and (3.17), we have ξ k k h j i = 1 ξk φ j a R kab i ξ b + 1 ξk φ a i R kjb a ξ b. (3.18) (3.18) is rewritten in index-free form as follows: ( ξ h)x = 1 R(ξ, φx)ξ + 1 φr(ξ, X)ξ. (3.18 ) On one hand, taking account of (.11), (.1), (3.7) and (3.8), we get and hence R(ξ, X)ξ = ξ ( X ξ) X ( ξ ξ) [ξ,x] ξ = ξ (φx + φhx) [ξ,x] ξ = ξ (φx + φhx) + (φ[ξ, X] + φh[ξ, X]) = φ ξ X φ( ξ h)x φh ξ X + φ ξ X φ X ξ + φh( ξ X X ξ) = φ( ξ h)x + φ(φx + φhx) + φh(φx + φhx) = φ( ξ h)x X + η(x)ξ + φ(φh)x + φ(hφ)x + φhφhx = φ( ξ h)x X + η(x)ξ + h X, ( ξ h)x η(( ξ h)x)ξ φx + φh X = φr(ξ, X)ξ. (3.19) Here, from (.1) and (3.9), we get η i (ξ k k h j i ) = ξ k η i ( k h j i ) = ξ k ( k η i )h j i = 0. (3.0) Thus, from (3.19) and (3.0), we have ( ξ h)x = φx h φx φr(x, ξ)ξ, (3.1) for any X X(M). Thus, from (3.18) and (3.1), we have also φx h φx φr(x, ξ)ξ = 1 R(ξ, φx)ξ + 1 φr(ξ, X)ξ, (3.) 354

7 Notes on quasi contact metric manifolds 7 for any X X(M). From (3.1), taking account of (3.7), we have also φ X φ h X φ R(X, ξ)ξ = 1 φr(ξ, φx)ξ+ 1 φ R(ξ, X)ξ, and hence X+η(X)ξ+h X+R(X, ξ)ξ η(r(x, ξ)ξ)ξ = 1 φr(ξ, φx)ξ 1 R(ξ, X)ξ + 1 η(r(ξ, X)ξ)ξ, we get X + η(x)ξ + h X + R(X, ξ)ξ = 1 φr(ξ, φx)ξ 1 R(ξ, X)ξ, (3.3) for any X X(M). Here, (3.3) is rewritten in terms of local coordinates as (3.3) δ j i + η j ξ i + h a i h j a + R jab i ξ a ξ b = 1 φ b i R ack b φ j c ξ a ξ k 1 R ajb i ξ a ξ b, and hence n + trh + ρ ab ξ a ξ b = 1 ρ akξ a ξ k + 1 ρ abξ a ξ b = 0, namely ρ(ξ, ξ) = n trh. (3.4) Remark 3.1 From (3.18 ), we have the following equality: g(( ξ h)x, Y ) g(( ξ h)y, X) = 1 R(ξ, φx, ξ, Y ) 1 g(r(ξ, X)ξ, φy ) for any X X(M). (3.5) is rewritten in terms of local coordinates as: From (3.5), we have further + 1 R(ξ, φy, ξ, X) + 1 g(r(ξ, Y )ξ, φx) = 0, (3.5) ξ k ( k (h ij h ji )) = 0. (3.5 ) ξ H = 0, (3.6) where H is the skew-symmetric part of H, where H(X, Y ) = g(hx, Y ) for any X X(M). (3.6) is trivial in the case where M = (M, φ, ξ, η, g) is a contact metric manifold. 4 Curvature identities and results In this section, we shall establish several curvature identities on a quasi contact metric manifold based on [4] the formulas obtained and also the ones in the previous sections of the present paper and further provide several results concerning the question introduced in 1. Let M = (M, φ, ξ, η, g) be a (n+1)-dimensional quasi contact metric manifold and M = (M R, J, ḡ) be the corresponding quasi Kähler cone endowed with the quasi Kähler structure ( J, ḡ) defined by (1.4). First, we recall the following curvature identity on a quasi Kähler manifold M = (M, J, ḡ) (see []): R λµνκ + J λ J β µ J γ ν J κ Rβγ J λ J β µ Rβνκ J γ ν J κ Rλµγ + J λ J γ ν Rµγκ + J β µ J κ Rλβν (4.1) + J λ J κ Rµν + J β µ J γ ν Rλβγκ = ḡ κ ( λ J µ ) J ν ḡ ν ( µ J λ ) J κ, 355

8 8 Y.D. Chai, J.H. Kim, J.H. Park, K. Sekigawa, W.M. Shin where we suppose that the Latin indices 1 λ, µ, ν,, β, γ, n + = and a, b, c, i, j, k, l, run over the range 1,,, n+1. Setting λ = i, µ = j, ν = k, κ = l in (4.1), and taking account of the formulas (see [8], (.6)), we have R ijkl +φ i a φ j b φ k c φ l d R abcd +φ i a φ k c R ajcl +φ j b φ l d R ibkd φ i a φ j b R abkl φ k c φ l d R ijcd + φ i a φ l d R ajkd + φ j b φ k c R ibcl + 4φ il φ jk 4φ jl φ ik 4g il g jk + 4g jl g ik + 4g il η j η k + 4g jk η i η l 4g jl η i η k 4g ik η j η l = ( i φ j a ) a φ kl + ( j φ i a ) a φ kl η k ( i φ jl j φ il ) + η l ( i φ jk j φ ik ) η j i φ kl + η i j φ kl. (4.) Similarly, setting λ =, µ = j, ν = k, κ = l in (4.1), we have ξ a φ j b φ k c φ l d R abcd + ξ a φ k c R ajcl ξ a φ j b R abkl + ξ a φ l d R ajkd = ( j ξ a ) a φ kl +η k l η j η l j η k +φ j a a φ kl +4η k φ jl 4η l φ jk. (4.3) Furthermore, setting λ = i, µ =, ν = k, κ = in (4.1), we have φ i a ξ b φ k c ξ d R abcd + ξ b ξ d R ibkd + 4g ik 4η i η k = φ ak i ξ a + φ i a a η k + ( i ξ a ) a η k. (4.4) Transvecting (4.) with g il and taking account of (3.11), we have ρ jk +(g ad ξ a ξ d )φ j b φ k c R abcd +φ la φ k c R ajcl +φ j b φ id R ibkd φ la φ j b R abkl φ k c φ id R ijcd +(g ad ξ a ξ d )R ajkd +ρ bc φ j b φ k c 8(n 1)(g jk η j η k ) (4.5) = ( i φ j a ) a φ ki + ( j φ i a ) a φ k i η k i φ j i ξ i j φ ik η j i φ k i + η i j φ k i. Here, taking account of (3.8), we get ξ i j φ ik + η i j φ k i = ξ i j φ ki + η i j φ k i = 4η i j φ k i = 4( j η i )φ k i = 4( j ξ i )φ ki = 4φ ki (φ j i + φ a i h j a ) = 4g kj 4η k η j + 4(g ka η k η a )h j a. (4.6) Thus, taking account of (3.13), the right-hand side of (4.5) reduces to ( i φ a j ) a φ ki + ( j φ a i ) a φ i k + 4(n 1)η j η k + 4g jk + 4h jk. (4.7) We have also (g ad ξ a ξ d )φ b j φ c k R abcd = ρ bc φ b j φ c k φ b j φ c k R abcd ξ a ξ d, φ la φ k c R ajcl = 1 φla φ k c (R jalc R jlac ) = 1 φla φ k c R jcal = 1 φ k c R jca l φ l a = ρ jk, (4.8) φ j b φ id R ibkd = 1 φ j b φ id (R bidk R bdik ) = 1 φ j b φ id R bkid 356

9 Notes on quasi contact metric manifolds 9 = 1 φ j b R kbd i φ i d = ρ kj, φ la φ b j R abkl = 1 φ j b φ la (R balk R blak ) = 1 φ j b φ la R bkal = 1 φ j b R l bka φ a l = ρ kj, φ c k φ id R ijcd = 1 φ k c φ id (R jidc R jdic ) = 1 φ k c φ id R jcid = 1 φ k c R d jci φ i a = ρ jk. Thus, from (4.5) (4.8), taking account of (.4), we have ρ jk φ b j φ c k R abcd ξ a ξ d ξ a ξ d R ajkd + ρ bc φ b j φ c k ρ jk ρ kj 8(n 1)(g jk η j η k ) = ( i φ a j ) a φ ki (4.9) + ( j φ a i ) a φ i k + 8nη j η k + 4(g jk η j η k ) + 4h jk. Thus, transvecting (4.9) with g jk, we have τ ρ ab ξ a ξ b + τ ρ bc ξ b ξ c τ τ 16n(n 1) = ( i φ j a ) a φ i j + ( j φ i a ) a φ ji + 8n + 8n, (4.10) and hence, from (4.10) taking account of (3.4), we have 4(τ τ ) 8n + 4trh 16n = ( i φ jk j φ ik ) k φ ij. (4.11) Here, we assume especially that M is a contact metric manifold. Then, dφ = 0 holds, where Φ(X, Y ) = g(x, φy ). Then, in this case, the right hand side of (4.11) reduces to ( i φ ja j φ ia ) a φ ij = ( i φ ja + j φ ai ) a φ ij = ( a φ ij ) a φ ij = φ. Thus, (4.11) reduces to τ τ + 4n trh = 1 ( φ 4n). (4.11 ) This is nothing but the Olszak result [6]. Therefore, we see that (4.11) is a generalization of the Olszak formula. Theorem 4.1 Let M = (M, φ, ξ, η, g) be a (n + 1)-dimensional quasi contact metric manifold with dφ = 0. Then, we have the equality (4.11). Further, the equality τ τ + 4n trh = 0 holds if and only if M is Sasakian. Proof. From direct calculation, we have 0 q( i φ jk g ij η k + η j g ik )( i φ jk g ij ξ k + ξ j g ik ) = φ (g ij i φ jk )ξ k + (g ik i φ jk )ξ j (g ij i φ jk )η k + (n + 1) (g ij ξ j g ik η k ) + (g ik i φ jk )η j (g ij ξ k η j g ik ) + (n + 1) (4.1) = φ + ( i φ k i )ξ k + ( i φ j i )ξ j + ( i φ k i )ξ k + ( k φ j k )ξ j + 4n + = φ + 4( n) + 4n = φ 4n. 357

10 10 Y.D. Chai, J.H. Kim, J.H. Park, K. Sekigawa, W.M. Shin Thus, from (4.11 ) and (4.1), taking account of the result (see [1], Theorem 6.3), we have immediately the required conclusion. Let M = (M, φ, ξ, η, g) be a (n+1)-dimensional quasi contact metric manifold. Then, from (3.10), it follows immediately that the characteristic vector field ξ is never parallel. We here assume that the characteristic vector field ξ of M is a Killing vector field. Then we have i η j + j η i = 0. (4.13) From (4.13), we have easily Thus, transvecting (4.14) with ξ j, we have i i η j + ρ j k η k = 0. (4.14) η + ρ jk ξ i ξ k = 0. (4.15) Thus, from (4.15), taking account of (3.10) and (3.4), we have Thus, we have the following. h + trh = 0. (4.16) Theorem 4. Let M = (M, φ, ξ, η, g) be a (n + 1)-dimensional quasi contact metric manifold with the Killing characteristic vector field ξ. Then, the tensor field h is skewsymmetric. Proof. We set h + = 1 (h + t h), h = 1 (h t h), (4.17) where t h is the transpose of h with respect to the metric g. Then, from (4.17), we immediately get h = h + + h, t h + = h +, t h = h (4.18) Further, we get h = tr(h( t h)) = tr(( t h)h), tr(h + h ) = tr(h h + ) = 0. (4.19) Thus, from (4.17), (4.18) and (4.19), we get and h = tr(h( t h)) = tr((h + + h )(h + h )) = tr(h + ) tr(h ), (4.0) tr(h ) = trh + + trh. (4.1) Thus, from (4.16), taking account of (4.0) and (4.1), we have trh + = 0, (4.) and hence h + = 0 since h + is symmetric with respect to the metric g and therefore, h is skew-symmetric. This complete the proof of Theorem

11 Notes on quasi contact metric manifolds 11 Now it is well-known that a contact metric manifold M = (M, φ, ξ, η, g) is Sasakian if and only if R(X, Y )ξ = η(y )X η(x)y, (4.3) holds for any X, Y X(M) (see [1], Proposition 7.6). We have the following which is a generalization of the fact. Theorem 4.3 A quasi contact metric manifold M = (M, φ, ξ, η, g) is Sasakian if and only if the equality (4.3) holds for any X, Y X(M). Proof. The equality (4.3) is rewritten in terms of local coordinates R ijl k ξ l = η j δ i k η i δ j k. (4.3 ) Transvecting (4.) with ξ l and taking account of (4.3 ), we have η i g jk η j g ik = i φ jk j φ ik + η i g kj η j g ki + η i h jk η j h ik, and hence i φ jk j φ ik + η i h jk η j h ik = 0. (4.4) Thus, transvecting (4.4) with ξ k and taking account of (.), (3.8) and (3.9), we have Here, taking account of (3.8) and (3.9), we get ξ k i φ jk ξ k j φ ik = 0. (4.5) ξ k i φ jk = φ jk i ξ k = φ jk (φ i k + φ a k h i a ) = g ji η j η i + (g ja η j η a )h i a = g ji η j η i + h ij. (4.6) Thus, from (4.5) and (4.6), we have finally g ji η j η i + h ij (g ij η i η j ) h ji = 0, and hence h ij h ji = 0. (4.7) Thus, it follows that M is a contact metric manifold by virtue of Theorem 4.1, and therefore, M is Sasakian. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( ). References 1. Blair, D.E. Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, 03. Birkhäuser Boston, Inc., Boston, MA, 00.. Gray, A. Curvature identities for Hermitian and almost Hermitian manifolds, Thoku Math. J., 8 (1976), Gray, A.; Hervella, L.M. The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 13 (1980), Kim, J.H.; Park, J.H.; Sekigawa, K. A generalization of contact metric manifolds, Balkan J. Geom. Appl., 19 (014), Jin, J.E.; Park, J.H.; Sekigawa, K. Notes on some classes of 3-dimensional contact metric manifolds, Balkan J. Geom. Appl., 17 (01), Olszak, Z. On contact metric manifolds, Tôhoku Math. J., 31 (1979), Tashiro, Y. On contact structure of hypersurfaces in complex manifolds. I, II, Tôhoku Math. J., 15 (1963), 6 78, Park J.H., Sekigawa K., Shin W.M. Curvature identities on contact metric manifolds and its applications, arxiv: