PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 113–128 DOI: https://doi.org/10.2298/PIM1817113M

A NEW CURVATURELIKE TENSOR FIELD IN AN ALMOST CONTACT RIEMANNIAN MANIFOLD II

Koji Matsumoto

Memories of Professor Mileva Prvanović

Abstract. In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)_{3}- curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)_{3}-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)_{3}-curvature tensor and we show that a Sasakian manifold with a flat (CHR)_{3}-curvature tensor is flat. Next, we introduce the notion of (CHR)_{3}-η-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)_{3}- η-Einstein manifold is η-Einstain. Moreover, we define the notion of (CHR)_{3}- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an almost contact Riemannian manifold and we get new invariant tensor fields (not the conformal curvature tensor) under this transformation. Finally, we prove that a conformally (CHR)_{3}-flat Sasakian manifold does not exist.

1. Almost contact Riemannian manifolds

A real (2n + 1)-dimensional differentiable Riemannian manifold (M^{2n+1} , g) is said to be an almost contact Riemannian manifold if it has a (1, 1)-tensor φ and a 1-form η which satisfy

φ^{2} = −I + η ⊗ ξ, η(φX) = 0, η(ξ) = 1, (1.1)

g(φX, φY) = g(X, Y) − η(X)η(Y) (1.2)

for any Y, X ∈ T M^{2n+1} , where ξ is defined by g(ξ, X) = η(X). From (1.1)_{3}, the vector field ξ is unit and we say this vector field the structure vector field of

2010 Mathematics Subject Classification: 53C40. Key words and phrases: curvaturelike tensor field, almost contact Riemannian manifold,

Sasakian manifold, (CHR)_{3}-curvature tensor.

The author was partially supported by Serbian Academy.

the almost contact Riemannian manifold. Next, in an almost contact Riemannian manifold M^{2n+1} , we define a 2-form F as F (X, Y) = g(φX, Y) for any X, Y ∈ T M^{2n+1} , where T M^{2n+1} denotes the tangent bundle of M^{2n+1} . Then the 2-form F is skew-symmetric and we say that this tensor field is the fundamental 2-form of this almost contact Riemannian manifold.

In an almost contact Riemannian manifold, a section which is given by X and φX for a unit vector field X is called a φ-section of X. The sectional curvature R(X, φX, φX, X) is said to be the φ-holomorphic sectional curvature of X, where R denotes the Riemannian curvature tensor with respect to g.

In a (2n + 1)-dimensional almost contact Riemannian manifold M^{2n+1} , we

define R^{∗}(X, Y, Z, W) which is called the second Riemannian curvature tensor as

R^{∗}(X, Y, Z, W) = R(X, Y, φZ, φW) for any X, Y, Z, W ∈ T M^{2n+1} .

An almost contact Riemannian manifold (M^{2n+1} , φ, g, ξ) is called a normal contact Riemannian or a Sasakian manifold if the structure vector field ξ and the fundamental 2-form F satisfies

∇_{X} ξ = φX, (∇_{X} F)(Y, Z) = η(Y)g(X, Z) − η(Z)g(X, Y)

for any X, Y, Z ∈ T M^{2n+1} .

In a Sasakian manifold, the Riemannian curvature tensor R and the Ricci tensor ρ with respect to g satisfy

R(X, Y, Z, ξ) = η(X)g(Y, Z) − η(Y)g(X, Z),
R(X, Y, φZ, φW) = R(φX, φY, Z, W) = R(X, Y, Z, W) + g(X, Z)g(Y, W)
− g(Y, Z)g(X, W) + F (X, W)F (Y, Z) − F (Y, W)F (X, Z),
R(φX, φY, φZ, φW) = R(X, Y, Z, W) + η(X)η(Z)g(Y, W) − η(X)η(W)g(Y, Z)

+ η(Y)η(W)g(X, Z) − η(Y)η(Z)g(X, W), (1.3)

ρ(φX, φY) = ρ(X, Y) − 2nη(X)η(Y),
ρ(φX, Y) + ρ(X, φY) = 0,  ρ(X, ξ) = 2nη(X)

for any X, Y, Z, W ∈ T M^{2n+1} .

A Sasakian manifold is said to be a Sasakian space form if it has a constant φ-holomorpic sectional curvature. Then the curvature tensor field R satisfies [2]

(1.4) R(X, Y, Z, W) = \frac{c + 3}{4} \{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\}
+\frac{c − 1}{4} \{ η(X)η(Z)g(Y, W) − η(Y)η(Z)g(X, W)
+ η(Y)η(W)g(X, Z) − η(X)η(W)g(Y, Z)
− F (Y, Z)F (X, W) + F (X, Z)F (Y, W)
− 2F (X, Y)F (Z, W)},

where c is a constant holomorphic sectional curvature.

A Sasakian space form with 0 holomorphic sectional curvature is called to be flat.

An almost contact Riemannian manifold M^{2n+1} is said to be η-Einstein if the Ricci tensor ρ with respect to g satisfies ρ(X, Y) = ag(X, Y) + bη(X)η(Y) for certain differentiable functions a and b on M^{2n+1} which are called the associated functions of ρ and any X, Y ∈ T M^{2n+1} . In particular, in an η-Einstein Sasakian manifold, associated functions a and b satisfy the following relation

(1.5)  a + b = 2n, τ = 2n(a + 1),

where τ is the scalar curvature with respect to g.

In a Sasakian manifold, the C-Bochner curvature tensor CB(X, Y, Z, W) is defined by [3]

(1.6)  CB(X, Y, Z, W) = R(X, Y, Z, W) + \frac{1}{2(n + 2)} \{ρ(X, Z)g(φY, φW)
−ρ(Y, Z)g(φX, φW) + ρ(Y, W)g(φX, φZ) − ρ(X, W)g(φY, φZ)
−ρ̃(X, Z)F (Y, W) + ρ̃(Y, Z)F (X, W) − ρ̃(Y, W)F (X, Z)
+ρ̃(X, W)F (Y, Z) − 2ρ̃(X, Y)F (Z, W) − 2ρ̃(Z, W)F (X, Y)}

−\frac{τ + 2n(2n + 3)}{4(n + 1)(n + 2)} \{F (X, W)F (Y, Z) − F (X, Z)F (Y, W) − 2F (X, Y)F (Z, W)\}

+\frac{τ − 3(2n + 4)}{4(n + 1)(n + 2)} \{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\}

+\frac{τ + 2n}{4(n + 1)(n + 2)} \{η(X)η(Z)g(Y, W) − η(Y)ηZ)g(X, W) − η(X)η(W)g(Y, Z)\},

where we put ρ̃(X, Y) = ρ(X, φY).

2. A curvaturelike tensor field in an almost contact Riemannian manifold

In this section, we define a new curvaturelike tensor field in an almost contact Riemanian manifold.

In a differentiable manifold M, a (0, 4)-type tensor field T (X, Y, Z, W) is called curvaturelike if it satisfies

T (X, Y, Z, W) = −T (Y, X, Z, W), T (X, Y, Z, W) = T (Z, W, X, Y),
T (X, Y, Z, W) + T (X, Z, W, Y) + T (X, W, Y, Z) = 0

for any X, Y, Z, W ∈ T M [4].

In an almost contact Riemannian manifold (M^{2n+1} , φ, ξ, g), we define a (0,4)- type tensor field (CHR)_{3}(X, Y, Z, W) as

(2.1) 16(CHR)_{3}(X, Y, Z, W) = 3{R(X, Y, Z, W) + R(φX, φY, Z, W)

+ R(φZ, φW, X, Y) + R(φX, φY, φZ, φW)} − R(φW, φY, X, Z)
− R(φX, φZ, W, Y) − R(φY, φZ, X, W) − R(φX, φW, Y, Z)
+ R(φX, Z, φW, Y) + R(X, φZ, W, φY) + R(φX, W, Y, φZ)
+ R(X, φW, φY, Z)
+ η(X)P (Z, W, Y) − η(Y)P (Z, W, X)

+ η(Z)P (X, Y, W) − η(W)P (X, Y, Z)
+ η(X)η(W)Q(Y, Z) − η(X)η(Z)Q(Y, W)
+ η(Y)η(Z)Q(W, X) − η(Y)η(W)Q(Z, X),

where we put

(2.2) P (X, Y, Z) = 3{R(X, Y, Z, ξ) + R(φX, φY, Z, ξ)} + R(φX, φZ, Y, ξ)
+ R(φZ, φY, X, ξ) − R(X, φZ, φY, ξ) − R(φZ, Y, φX, ξ),
(2.3)  Q(X, Y)= 3R(ξ, X, Y, ξ) − R(ξ, φX, φY, ξ)

for any X, Y, Z, W ∈ T M^{2n+1} . Then, we can easily check the above tensor field is curvaturelike. We call this tensor field a (CHR)_{3}-curvature tensor in an almost contact Riemannian manifold or a contact holomorphic Riemannian curvature tensor of the second type.

About the tensors P (X, Y, Z) and Q(X, Y), we can easily see

(2.4) \begin{aligned}P (φX, φY, Z) & = P (X, Y, Z) − η(X)Q(Y, Z) + η(Y)Q(X, Z), \\ P (X, φY, φZ) & = 3\{R(X, φY, φZ, ξ) − R(φX, Y, φZ, ξ)\} \\ & − R(φX, Z, φY, ξ) + R(Z, Y, X, ξ) − R(X, Z, Y, ξ) \\ & + R(Z, φY, φX, ξ) + η(Y)Q(Z, X) \\ & − 2\{η(Z)S(X, Y) + η(Y)S(X, Z)\}, \\ P (φX, Y, φZ) & = 3\{R(φX, Y, φZ, ξ) − R(X, φY, φZ, ξ)\} + R(X, Z, Y, ξ) \\ & − R(Z, φY, φX, ξ) + R(φX, Z, φY, ξ) − R(Z, Y, X, ξ) \\ & − η(X)Q(Y, Z) + 2\{η(X)S(Y, Z) + η(Z)S(X, Y)\}, \\ P (X, Y, ξ) & = 0, P (ξ, X, Y) = Q(X, Y), P (X, ξ, Y) = −Q(X, Y), \end{aligned}

(2.5)  Q(φX, φY) = −Q(X, Y) + 2S(X, Y),  Q(ξ, X) = 0,

where we put S(X, Y) = R(ξ, X, Y, ξ) + R(ξ, φX, φY, ξ). In particular, by virtue of (1.3), the tensor fields P (X, Y, Z) and Q(X, Y) in a Sasakian manifold are respectively satisfied

P (X, Y, Z) = 2{η(X)g(Y, Z) − η(Y)g(X, Z)} = 2R(X, Y, Z, ξ),
Q(X, Y) = 2{g(X, Y) − η(X)η(Y)} = 2g(φX, φY)

for any X, Y, Z ∈ T M^{2n+1} .

An almost contact Riemannian manifold is said to be (CHR)_{3}-flat if the (CHR)_{3}- curvature tensor vanishes, identically.

In an almost contact Riemannian manifold, the (CHR)_{3}-curvature tensor satisfies the following equations

(CHR)_{3}(X, Y, Z, ξ)= 0,
(CHR)_{3}(X, Y, φZ, φW) = (CHR)_{3}(φX, φY, Z, W) = (CHR)_{3}(X, Y, Z, W),
(CHR)_{3}(X, φX, φX, X) = R(X, φX, φX, X) − 2η(X)R(X, φX, φX, ξ)
+ η(X)^{2}R(ξ, φXφX, ξ)

for any X, Y, Z, W ∈ T M^{2n+1} .

3. (CHR)_{3}-curvature tensor in a Sasakian manifold

In this section, we consider a (CHR)_{3}-flat Sasakian manifold and we prove this manifold is flat Sasakian.

By virtue of (1.3) and Bianchi identity [6], in a Sasakian manifold, the (CHR)_{3}- curvature tensor satisfies

(3.1) (CHR)_{3}(X, Y, Z, W) = R(X, Y, Z, W)

+ \frac{3}{4}\{g(X, Z)g(Y, W) − g(X, W)g(Y, Z)\}

+ \frac{1}{4}\{g(φX, W)g(φY, Z) − g(φX, Z)g(φY, W) − 2g(φX, Y)g(φZ, W)

+ η(X)η(Z)g(Y, W) − η(X)η(W)g(Y, Z)

+ η(Y)η(W)g(X, Z) − η(Y)η(Z)g(X, W)}.

Let us consider the (CHR)_{3}-flat Sasakian manifold, then we have from (3.1), the curvature tensor R is written as

R(X, Y, Z, W) = \frac{3}{4}\{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\}

− \frac{1}{4}\{g(φX, W)g(φY, Z) − g(φX, Z)g(φY, W) − 2g(φX, Y)g(φZ, W)

+ η(X)η(Z)g(Y, W) − η(X)η(W)g(Y, Z)

+ η(Y)η(W)g(X, Z) − η(Y)η(Z)g(X, W)}.

Comparing the above equation and (1.4), we obtain

Theorem 3.1. A (CHR)_{3}-flat Sasakian manifold is flat Sasakian.

4. The properties of the (CHR)_{3}-curvature tensor.

We put ρ(CHR)_{3}(X, Y) = ∑ _{i=1}^{2n+1} (CHR)_{3}(e_{i}, X, Y, e_{i}) for a local orthonormal frame (e_{1}, e_{2}, …, e_{2n+1} ) of an almost contact Riemannian manifold M^{2n+1} . Then we say ρ(CHR)_{3}(X, Y) a (CHR)_{3}-Ricci tensor.

By virtue of (2.1), (2.2) and (2.3), the local representations of (CHR)_{3}-curva-ture tensor, the tensor fields P and Q are respectively written by

(4.1) 16(CHR)_{3kjih} = 3(R_{kjih} + R^{∗}_{kjih} + R^{∗}_{ihkj} + R^{∗}_{kjml} φ_{i}^{m}φ_{h}^{l})

− R^{∗}_{hjki} − R^{∗}_{kihj} − R^{∗}_{jikh} − R^{∗}_{khji}

+ R_{tisj} φ_{k}^{t}φ_{h}^{s} + R_{kths} φ_{i}^{t}φ_{j} ^{s} + R_{thjs} φ_{k}^{t}φ_{i}^{s} + R_{ktsi} φ_{h}^{t}φ_{j} ^{s} + η_{k}P_{ihj} − η_{j} P_{kji} + η_{i}P_{kjh} − η_{h}P_{kji}

+ η_{k}η_{h}Q_{ji} − η_{k}η_{i}Q_{jh} + η_{j} η_{i}Q_{hk} − η_{j} η_{h}Q_{ik} ,

(4.2)  P_{jih} = 3(R_{jihl} + R^{∗}_{jihl} )ξ^{l} + (R^{∗}_{jhil} + R^{∗}_{hijl} )ξ^{l}

− (R_{jtsl} φ_{h}^{t}φ_{i}^{s} + R_{tisl} φ_{h}^{t}φ_{j} ^{s})ξ^{l},
(4.3)  Q_{ih} = (3R_{mihl} − R_{mtsl} φ_{i}^{t}φ_{h}^{s})ξ^{m}ξ^{l},

where we put R^{∗}_{kjih} = R_{tsih} φ_{k}^{t}φ_{j} ^{s} and the indices {k,j,...,h} run over the range {1,2,...,2n + 1}.

Moreover, equations (2.4) and (2.5) are respectively written as

(4.4)  \begin{aligned}P_{tsh} φ_{j} ^{t}φ_{i}^{s} & = P_{jih} − η_{j} Q_{ih} + η_{i}Q_{jh} , \\ P_{jts} φ_{i}^{t}φ_{h}^{s} & = 3(R_{jtsl} φ_{i}^{t}φ_{h}^{s}ξ^{l} − R_{tisl} φ_{j} ^{t}φ_{h}^{s})ξ^{l} \\ & − R_{thsl} φ_{j} ^{t}φ_{i}^{s}ξ^{l} + R_{hijl} ξ^{l} − R_{jhil} ξ^{l} \\ & + R_{htsl} φ_{i}^{t}φ_{j} ^{s}ξ^{l} + η_{i}Q_{hj} − 2(η_{i}S_{hj} + η_{h}S_{ij} ), \\ P_{tis} φ_{j} ^{t}φ_{h}^{s} & = 3(R_{tisl} φ_{j} ^{t}φ_{h}^{s} − R_{jtsl} φ_{i}^{t}φ_{h}^{s})ξ^{l} + R_{jhil} ξ^{l} \\ & − R_{htsl} φ_{i}^{t}φ_{j} ^{s}ξ^{l} + R_{thsl} φ_{j} ^{t}φ_{i}^{s}ξ^{l} − R_{hijl} ξ^{l} \\ & − η_{j} Q_{ih} + 2(η_{j} S_{ih} + η_{h}S_{ji} ), \\ P_{ihl} ξ^{l} & = 0, P_{lih} ξ^{l} = Q_{ih} , P_{ilh} ξ^{l} = −Q_{ih} , \end{aligned}

Q_{ts}φ_{i}^{t}φ_{h}^{s} = −Q_{ih} + 2S_{ih} , Q_{lh}ξ^{l} = 0, (4.5)

S_{ih} = {R_{mihl} + R_{mtsl} φ_{i}^{t}φ_{h}^{s}}ξ^{m}ξ^{l}. (4.6)

Since, we know ρ(CHR)_{3ji} = (CHR)_{3kjih} g^{kh}, we have from (4.1), (4.2), (4.3) and (4.4), we obtain

(4.7) 8ρ(CHR)_{3}(X, Y) = ρ(X, Y) + ρ(φX, φY)

− 3{ρ^{∗}(X, Y) + ρ^{∗}(Y, X)} − {ρ(X, ξ)η(Y) + ρ(Y, ξ)η(X)}

+ 3{ρ^{∗}(X, ξ)η(Y) + ρ^{∗}(Y, ξ)η(X)} + ρ(ξ, ξ)η(X)η(Y)

− {R(ξ, X, Y, ξ) + R(ξ, φX, φY, ξ)},

where we put

ρ(X, Y) = ∑_{i=1}^{2n+1} R(e_{i}, X, Y, e_{i}), ρ^{∗}(X, Y) = ∑_{i=1}^{2n+1} R^{∗}(e_{i}, X, Y, e_{i}).

In particular, the (CHR)_{3}-Ricci tensor in a Sasakian manifold is written by

(4.8) ρ(CHR)_{3}(X, Y) = ρ(X, Y) − \frac{1}{2}\{(3n − 1)g(X, Y) + (n + 1)η(X)η(Y)\}.

From the above equation, we can easily have ρ(CHR)_{3}(X, ξ) = 0.

Next, we put τ(CHR)_{3} = ∑ _{i=1}^{2n+1} ρ(CHR)_{3}(e_{i}, e_{i}) = ρ(CHR)_{3ji} g^{ji} which is called the (CHR)_{3}-scalar curvature of (CHR)_{3}-curvature tensor. Then from (4.7), we have 4τ(CHR)_{3} = τ − 2ρ(ξ, ξ) + 3τ^{∗}, where we put τ^{∗} = ∑ _{i=1}^{2n+1} ρ^{∗}(e_{i}, e_{i}). Then, in a Sasakian manifold, we have from (4.8)

(4.9)  τ(CHR)_{3} = τ − n(3n + 1).

Now, an almost contact Riemannian manifold M^{2n+1} is called (CHR)_{3}-η-Einstein if its (CHR)_{3}-Ricci tensor ρ(CHR)_{3}(X, Y) has the form

(4.10)  ρ(CHR)_{3}(X, Y) = αg(X, Y) + βη(X)η(Y)

for certain functions α and β which are called associated functions of ρ(CHR)_{3}. In particular, if our manifold is Sasakian, then we have from (4.8) and (4.10) the Ricci tensors is written as

ρ(X, Y) = ( α + \frac{3n − 1}{2} ) g(X, Y) + ( β + \frac{n + 1}{2} ) η(X)η(Y),

that is, our manifold is η-Einstein.

Conversely, if our manifold M^{2n+1} is η-Einstein, then we can easily see that Sasakian manifolds are (CHR)_{3}-η-Einstein. Moreover, by virtue of (1.5), we obtain α + β = 0 in a (CHR)_{3}-η-Einstein Sasakian manifold and the scalar curvature τ and the (CHR)_{3}-scalar curvature τ(CHR)_{3} are written as

(4.11)  τ = n(2α + 3n + 1), τ(CHR)_{3} = 2nα.

Thus we have

Theorem 4.1. A (2n + 1)-dimensional Sasakian manifold M^{2n+1} is (CHR)_{3}- η-Einstein if and only if M^{2n+1} is η-Einstein and the scalar curvatures τ and τ(CHR)_{3} are respectively written by (4.11) which includes only the associated function α.

Corollary 4.1. In an η-(CHR)_{3}-Einstein Sasakian manifold M^{2n+1} , the scalar curvatures τ and τ(CHR)_{3} are constant if and only if one of its associated functions is constant.

5. (CHR)_{3}-space form

In this section, we define a notion of a (CHR)_{3}-space form in an almost contact Riemannian manifold. Then we consider a Sasakian (CHR)_{3}-space form. And in this manifold, we determine the (CHR)_{3}-curvature tensor by the structure tensors.

Definition 5.1. An almost contact Riemannian manifold is called a (CHR)_{3}- space form if its (CHR)_{3}-curvature tensor satisfies

(CHR)_{3}(X, φX, φX, X) = c∥X∥^{4}

for a certain constant c and any X ∈ T M^{2n+1} − {ξ}.

By virtue of (2.1), in an almost contact Riemannian manifold, we have

(CHR)_{3}(X, φX, φX, X) = R(X, φX, φX, X)

for any X ∈ T M^{2n+1} − {ξ}. This means that a Sasakian (CHR)_{3}-space form is a Sasakian space form. Thus we have from (1.4) and (3.1)

Theorem 5.1. In a (2n + 1)-dimensional Sasakian (CHR)_{3}-space form, the (CHR)_{3}-curvatuee tensor, (CHR)_{3}-Ricci tensor and the (CHR)_{3}-scalar curvature are respectively given by

(CHR)_{3}(X, Y, Z, W) = \frac{c}{4}\{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)

− F (X, W)F (Y, Z) + F (X, Z)F (Y, W) − 2F (X, Y)F (Z, W)

+ η(X)η(Z)g(Y, W) − η(X)η(W)g(Y, Z)

+ η(Y)η(W)g(X, Z) − η(Y)η(Z)g(X, W)},

ρ(CHR)_{3}(X, Y) = \frac{cn}{2}\{g(X, Y) − η(X)η(Y)\}, τ(CHR)_{3} = cn^{2}.

for any X, Y, Z, W ∈ T M^{2n+1} .

6. Conformal transformations of almost contact Riemannian manifolds

In an almost contact Riemannian manifold M^{2n+1} , we consider a following conformal transformation;

(6.1)  ḡ = e^{2f} g, ḡ^{−1} = e^{−2f} g^{−1}

for a certain positive differentiable function f on M^{2n+1} . Then it is well known [6] that

(6.2)  ¯∇_{X} Y = ∇_{X} Y + θ(X)Y + θ(Y)X − g(X, Y)U,

where ¯∇ (resp. ∇) means the covariant derivation with respect to ḡ (resp. g), θ = df and g(U, X) =^{def} θ(X). Moreover, between the Riemannian curvature tensors R̄(X, Y, Z, W) with respect to ḡ and R(X, Y, Z, W) with respect to g, we have the following relation [6]

e^{−2f} R̄(X, Y, Z, W) = R(X, Y, Z, W) + g(X, W)σ(Y, Z) + g(Y, Z)σ(X, W) (6.3)

− g(X, Z)σ(Y, W) − g(Y, W)σ(X, Z) ,

where we put

(6.4)  σ(X, Y) = (∇_{X} θ)(Y) − θ(X)θ(Y) + \frac{1}{2}g(X, Y)θ(U).

From (6.3) and (6.4), we obtain ρ̄(X, Y) = ρ(X, Y) + (2n − 1)σ(X, Y) + σg(X, Y), where ρ̄ denotes the Ricci tensor with respect to ḡ and

(6.5)  σ = ∑_{i=1}^{2n+1} σ(e_{i}, e_{i}) = σ_{ji} g^{ji} .

From the above two equations, we easily get e^{2f} τ̄ = τ + 4nσ.

Similarly with (2.1), (\overline{CHR})_{3}(X, Y, Z, W) with respect to ḡ is given by

16(\overline{CHR})_{3}(X, Y, Z, W) = 3\{R̄(X, Y, Z, W) + R̄(φX, φY, Z, W)

+ R̄(X, Y, φZ, φW) + R̄(φX, φY, φZ, φW)} − R̄(X, Z, φW, φY)
− R̄(φX, φZ, W, Y) − R̄(X, W, φY, φZ) − R̄(φX, φW, Y, Z)
+ R̄(φX, Z, φW, Y) + R̄(X, φZ, W, φY)
+ R̄(φX, W, Y, φZ) + R̄(X, φW, φY, Z)

A NEW CURVATURELIKE TENSOR FIELD… 121

+ η(X)P̄(Z, W, Y) − η(Y)P̄(Z, W, X) + η(Z)P̄(X, Y, W) − η(W)P̄(X, Y, Z) + η(X)η(W)Q̄(Y, Z) − η(X)η(Z)Q̄(Y, W)

+ η(Y)η(Z)Q̄(W, X) − η(Y)η(W)Q̄(Z, X),

where we put

(6.6) P̄(X, Y, Z) = 3{R̄(X, Y, Z, ξ) + R̄(φX, φY, Z, ξ)} + R̄(φX, φZ, Y, ξ)
+ R̄(φZ, φY, X, ξ) − R̄(X, φZ, φY, ξ) − R̄(φZ, Y, φX, ξ),
(6.7) Q̄(X, Y)= 3R̄(ξ, X, Y, ξ) − R̄(ξ, φX, φY, ξ).

for any X, Y, Z, W ∈ T M^{2n+1} .

Let us consider the relation between the tensor fields (CHR)_{3}(X, Y, Z, W) and (\overline{CHR})_{3}(X, Y, Z, W).

By virtue of (6.3), (6.6) and (6.7) are respectively written as (6.8) e^{−2f} P̄(X, Y, Z) = P (X, Y, Z) + η(X){3σ(Y, Z) − σ(φY, φZ)}

− η(Y){3σ(X, Z) − σ(φX, φZ)} + 2{g(Y, Z)σ(X, ξ) − g(X, Z)σ(Y, ξ)

+ g(φY, Z)σ(φX, ξ) − g(φX, Z)σ(φY, ξ) − 2g(φX, Y)σ(φZ, ξ)}
+ η(Z){η(Y)σ(X, ξ) − η(X)σ(Y, ξ)},

e^{−2f} Q̄(X, Y) = Q(X, Y) + 3σ(X, Y) − σ(φX, φY) + 2g(X, Y)σ(ξ, ξ) (6.9)

+ η(X)η(Y)σ(ξ, ξ) − 3{η(X)σ(Y, ξ) + η(Y)σ(X, ξ)} .

To calculate e^{−2f} (\overline{CHR})_{3}(X, Y, Z, W), we separate the following three parts; Part 1:

(6.10) (Ī) =^{put} 3{R̄(X, Y, Z, W) + R̄(φX, φY, Z, W) + R̄(X, Y, φZ, φW)
+ R̄(φX, φY, φZ, φW)} − R̄(X, Z, φW, φY) − R̄(φX, φZ, W, Y)
− R̄(X, W, φY, φZ) − R̄(φX, φW, Y, Z) + R̄(φX, Z, φW, Y)
+ R̄(X, φZ, W, φY) + R̄(φX, W, Y, φZ) + R̄(X, φW, φY, Z).

Part 2:

(6.11)  (\overline{II}) =^{put} η(X)P̄(Z, W, Y) − η(Y)P̄(Z, W, X)
+ η(Z)P̄(X, Y, W) − η(W)P̄(X, Y, Z).

Part 3:

(6.12)  (\overline{III}) =^{put} η(X)η(W)Q̄(Y, Z) − η(X)η(Z)Q̄(Y, W)x
+ η(Y)η(Z)Q̄(W, X) − η(Y)η(W)Q̄(Z, X).

By virtue of (6.3), (6.8) and (6.9), we obtain

e^{−2f} (Ī) = (I) + 2g(X, W){σ(Y, Z) + σ(φY, φZ)} + 2g(Y, Z){σ(X, W) + σ(φX, φW)} − 2g(X, Z){σ(Y, W) + σ(φY, φW)} − 2g(Y, W){σ(X, Z) + σ(φX, φZ)} + 2g(φX, W){σ(φY, Z) − σ(Y, φZ)} + 2g(φY, Z){σ(φX, W) − σ(X, φW)} − 2g(φX, Z){σ(φY, W) − σ(Y, φW)} − 2g(φY, W){σ(φX, Z) − σ(X, φZ)} − 4g(φX, Y){σ(φZ, W) − σ(Z, φW)} − 4g(φZ, W){σ(φX, Y) − σ(X, φY)} + η(X)η(W){σ(Y, Z) − 3σ(φY, φZ)} − η(Y)η(W){σ(X, Z) − 3σ(φX, φZ)} − η(X)η(Z){σ(Y, W) − 3σ(φY, φW)}

+ η(Y)η(Z){σ(X, W) − 3σ(φX, φW)}.

e^{−2f} (\overline{II}) = (II) + 2η(X)η(Z)\{3σ(Y, W) − σ(φY, φW)\} − 2η(X)η(W){3σ(Y, Z) − σ(φY, φZ)} − 2η(Y)η(Z){3σ(X, W) − σ(φX, φW)}

+ 2η(Y)η(W){3σ(X, Z) − σ(φX, φZ)} − 2g(φY, W){η(X)σ(φZ, ξ)

− η(Z)σ(φX, ξ)} + 2g(φY, Z){η(X)σ(φW, ξ) − η(W)σ(φX, ξ)}

+ 2g(φX, W){η(Y)σ(φZ, ξ) − η(Z)σ(φY, ξ)} − 2g(φX, Z){η(Y)σ(φW, ξ)

− η(W)σ(φY, ξ)} − 4g(φZ, W){η(X)σ(φY, ξ) − η(Y)σ(φX, ξ)}
− 4g(φX, Y){η(Z)σ(φW, ξ) − η(W)σ(φX, ξ)} + 2g(Y, W){η(X)σ(Z, ξ)
+ η(Z)σ(X, ξ)} − 2g(Y, Z){η(X)σ(W, ξ) + η(W)σ(X, ξ)}

−2g(X, W){η(Y)σ(Z, ξ)+η(Z)σ(Y, ξ)}+2g(X, Z){η(Y)σ(W, ξ)+η(W)σ(Y, ξ)},

e^{−2f} (\overline{III}) = (III) + η(X)η(W)\{3σ(Y, Z) − σ(φY, φZ)\} − η(X)η(Z){3σ(Y, W) − σ(φY, φW)} + η(Y)η(Z){3σ(X, W) − σ(φX, φW)}

− η(Y)η(W){3σ(X, Z) − σ(φX, φZ)} + 2σ(ξ, ξ){η(X)η(W)g(Y, Z)

− η(X)η(Z)g(Y, W) + η(Y)η(Z)g(X, W) − η(Y)η(W)g(X, Z)},

where (Ī) denotes the geometric equation with respect to ḡ of (I) with respect to g and etc.. By virtue of the above three equations, we have (6.13) 8e^{−2f} (\overline{CHR})_{3}(X, Y, Z, W) = 8(CHR)_{3}(X, Y, Z, W)

+ g(φX, φW)D(Y, Z) − g(φX, φZ)D(Y, W) + g(φY, φZ)D(X, W)

− g(φY, φW)D(X, Z) + g(φX, W){A(Y, Z) + B(Y, Z)}

+ g(φY, Z){A(X, W) + B(X, W)} − g(φX, Z){A(Y, W) + B(Y, W)} − g(φY, W){A(X, Z) + B(X, Z)} − 2g(φX, Y){A(Z, W) + B(Z, W)} − 2g(φZ, W){A(X, Y) + B(X, Y)} + g(Y, W)C(X, Z) − g(Y, Z)C(X, W) + g(X, Z)C(Y, W) − g(X, W)C(Y, Z) + σ(ξ, ξ){η(X)η(W)g(Y, Z)

− η(X)η(Z)g(Y, W) + η(Y)η(Z)g(X, W) − η(Y)η(W)g(X, Z)},

where we put

A(X, Y)= σ(φX, Y) − σ(X, φY),
B(X, Y) = η(X)σ(φY, ξ) − η(Y)σ(φX, ξ),
C(X, Y)= η(X)σ(Y, ξ) + η(Y)σ(X, ξ),

A NEW CURVATURELIKE TENSOR FIELD… 123

D(X, Y) = σ(X, Y) + σ(φX, φY).

7. An invariant tensor field under a conformal transformation The equation (6.13) is written locally by

(7.1) 8e^{−2f} (\overline{CHR})_{3kjih} = 8(CHR)_{3kjih} + (g_{kh} − η_{k}η_{h})D_{ji}

+ (g_{ji} − η_{j} η_{i})D_{kh} − (g_{ki} − η_{k}η_{i})D_{jh} − (g_{jh} − η_{j} η_{h})D_{ki}

+ φ_{kh}(A_{ji} + B_{ji} ) + φ_{ji} (A_{kh} + B_{kh})
− φ_{ki}(A_{jh} + B_{jh}) − φ_{jh} (A_{ki} + B_{ki}) − 2φ_{kj} (A_{ih} + B_{ih})

− 2φ_{ih} (A_{kj} + B_{kj} ) + g_{jh} C_{ki} − g_{ji} C_{kh} + g_{ki}C_{jh} − g_{kh}C_{ji}

+ σ_{ml} ξ^{m}ξ^{l}(η_{k}η_{h}g_{ji} − η_{k}η_{i}g_{jh} + η_{j} η_{i}g_{kh} − η_{j} η_{h}g_{ki}).

Using (7.1), we get 4ρ(\overline{CHR})_{3}(X, Y) = 4ρ(CHR)_{3}(X, Y) + (n + 2)D(X, Y), (7.2)

−(n + 2)C(X, Y) + {σ − σ(ξ, ξ)}g(X, Y) − {σ − (n + 3)σ(ξ, ξ)}η(X)η(Y), e^{2f} τ(\overline{CHR})_{3} = τ(CHR)_{3} + (n + 1)\{σ − σ(ξ, ξ)\}. (7.3) Since, we have from (7.2) and (7.3)

(7.4)  D(X, Y) = \frac{1}{n + 2} \{4ρ(\overline{CHR})_{3}(X, Y) − \frac{τ(\overline{CHR})_{3}}{n + 1} ḡ(X, Y)\}

− \frac{1}{n + 2} \{4ρ(CHR)_{3}(X, Y) − \frac{τ(CHR)_{3}}{n + 1}g(X, Y)\}

+ C(X, Y) + \frac{σ − (n + 3)σ(ξ, ξ)}{n + 2} η(X)η(Y).

The above equation and (7.1) give us (7.5) g(φX, φY)D(Z, W)

= \frac{e^{−2f}}{n + 2} \{4ρ(\overline{CHR})_{3}(Z, W) − \frac{τ(\overline{CHR})_{3}}{n + 1} ḡ(Z, W)\} ḡ(φX, φY)

− \frac{1}{n + 2} \{4ρ(CHR)_{3}(Z, W) − \frac{τ(CHR)_{3}}{n + 1}g(Z, W)\}g(φX, φY)

+ g(φX, φY)C(Z, W) + \frac{σ − (n + 3)σ(ξ, ξ)}{n + 2} g(φX, φY)η(Z)η(W).

By virtue of (7.5) and (6.13), we have (7.6) e^{−2f} T̄(X, Y, Z, W) − T (X, Y, Z, W)

= g(φX, W){A(Y, Z) + B(Y, Z)} + g(φY, Z){A(X, W) + B(X, W)} − g(φX, Z){A(Y, W) + B(Y, W)} − g(φY, W){A(X, Z) + B(X, Z)} − 2g(φX, Y){A(Z, W) + B(Z, W)} − 2g(φZ, W){A(X, Y) + B(X, Y)}

+ \frac{σ − σ(ξ, ξ)}{n + 2} \{g(X, W)η(Y)η(Z) + g(Y, Z)η(X)η(W)

− g(X, Z)η(Y)η(W) − g(Y, W)η(X)η(Z)},

where we put

(7.7) T (X, Y, Z, W)= 8(CHR)_{3}(X, Y, Z, W) − \frac{1}{n + 2} \{E(Y, Z)g(φX, φW)
+E(X, W) g(φY, φZ) − E(Y, W)g(φX, φZ) − E(X, Z)g(φY, φW)},
(7.8)  E(X, Y) = 4ρ(CHR)_{3}(X, Y) − \frac{τ(CHR)_{3}}{n + 1}g(X, Y).

Equation (7.6) is locally written by

(7.6′) e^{−2f} T̄_{kjih} − T_{kjih} = φ_{kh}(A_{ji} + B_{ji} ) + φ_{ji} (A_{kh} + B_{kh})

− φ_{ki}(A_{jh} + B_{jh}) − φ_{jh} (A_{ki} + B_{ki}) − 2φ_{kj} (A_{ih} + B_{ih})

− 2φ_{ih} (A_{kj} + B_{kj} ) + \frac{σ − σ(ξ, ξ)}{n + 2} (g_{kh}η_{j} η_{i} + g_{ji} η_{k}η_{h}

− g_{ki}η_{j} η_{h} − g_{jh} η_{k}η_{i}).

Contracting (7.6′) with φ^{kh}, we obtain

(7.9)  A_{ji} + B_{ji} = \frac{φ^{ts}}{2(n + 1)} (e^{−2f} T̄_{tjis} − T_{tjis} ) − \frac{σ − σ(ξ, ξ)}{n + 1} φ_{ji} .

Substituting (7.9) into (7.6′), we have

(7.10) e^{−2f} \{ T̄_{kjih} − \frac{φ̄^{ts}}{2(n + 1)} (T̄_{tjis} φ̄_{kh} + T̄_{tkhs} φ̄_{ji} − T̄_{tjhs} φ̄_{ki} − T̄_{tkis} φ̄_{jh}

− 2T̄_{tkjs} φ̄_{ih} − 2T̄_{tihs} φ̄_{kj} ) + \frac{2τ(\overline{CHR})_{3}}{(n + 1)^{2}}(φ̄_{kh} φ̄_{ji} − φ̄_{ki} φ̄_{jh} − 2φ̄_{kj} φ̄_{ih} )\}

− \{T_{kjih} − \frac{φ^{ts}}{2(n + 1)} (T_{tjis} φ_{kh} + T_{tkhs} φ_{ji} − T_{tjhs} φ_{ki} − T_{tkis} φ_{jh}

− 2T_{tkjs} φ_{ih} − 2T_{tihs} φ_{kj} ) + \frac{2τ(CHR)_{3}}{(n + 1)^{2}}(φ_{kh}φ_{ji} − φ_{ki}φ_{jh} − 2φ_{kj} φ_{ih})\}

= \frac{σ − σ(ξ, ξ)}{n + 2} (g_{kh}η_{j} η_{i} + g_{ji} η_{k}η_{h} − g_{ki}η_{j} η_{h} − g_{jh} η_{k}η_{i}),

where we put φ̄_{ji} = φ_{j} ^{l} ḡ_{li} and φ̄^{ji} = φ_{l}^{i} ḡ^{lj}. The right hand side of (7.10) is written by

(7.11)  e^{−2f} \frac{τ(\overline{CHR})_{3}}{(n + 1)(n + 2)} (ḡ_{kh} η̄_{j} η_{i} + ḡ_{ji} η̄_{k}η_{h} − ḡ_{ki} η̄_{j} η_{h} − ḡ_{jh} η̄_{k}η_{i})
−\frac{τ(CHR)_{3}}{(n + 1)(n + 2)} (g_{kh}η_{j} η_{i} + g_{ji} η_{k}η_{h} − g_{ki}η_{j} η_{h} − g_{jh} η_{k}η_{i}),

where we put η̄_{i} = ξ^{l} ḡ_{li}. Thus we have

Theorem 7.1. In an almost contact Riemannian manifold, the tensor field S(X, Y)Z is invariant under the conformal transformation (6.1), where the tensor field S_{kji} ^{h} is defined by

A NEW CURVATURELIKE TENSOR FIELD… 125

(7.12) S_{kji} ^{l}g_{lh} = T_{kjih} − \frac{φ^{ts}}{2(n + 1)} (T_{tjis} φ_{kh} + T_{tkhs} φ_{ji} − T_{tjhs} φ_{ki} − T_{tkis} φ_{jh}

− 2T_{tkjs} φ_{ih} − 2T_{tihs} φ_{kj} ) + \frac{2τ(CHR)_{3}}{(n + 1)^{2}}(φ_{kh}φ_{ji} − φ_{ki}φ_{jh} − 2φ_{kj} φ_{ih})

− \frac{τ(CHR)_{3}}{(n + 1)(n + 2)} (g_{kh}η_{j} η_{i} + g_{ji} η_{k}η_{h} − g_{ki}η_{j} η_{h} − g_{jh} η_{k}η_{i}).

In particular, in a Sasakian manifold, we have from (4.8) and (4.9)

(7.13)  E_{ji} = 4ρ_{ji} − \frac{1}{n + 1} (τ + 3n^{2} + 3n − 2)g_{ji} − 2(n + 1)η_{j} η_{i}.

By virtue of (3.1) and (7.13), equation (7.7) is given by

(7.14) T_{kjih} = 8\{R_{kjih} + \frac{1}{4}(φ_{kh}φ_{ji} − φ_{ki}φ_{jh} − 2φ_{kj} φ_{ih})\}

− \frac{4}{n + 2} \{ρ_{ji} (g_{kh} − η_{j} η_{h}) + ρ_{kh}(g_{ji} − η_{j} η_{i}) − ρ_{jh} (g_{ki}η_{k}η_{i})

− ρ_{ki}(g_{jh} − η_{j} η_{h})\} + \frac{2\{τ − 2(3n + 4)\}}{(n + 1)(n + 2)} (g_{kh}g_{ji} − g_{ki}g_{jh})

− \frac{τ + n(3n + 5)}{(n + 1)(n + 2)} (g_{kh}η_{j} η_{i} + g_{ji} η_{k}η_{h} − g_{ki}η_{j} η_{h} − g_{jh} η_{k}η_{i}).

Moreover, in a Sasakian manifold, using the Bianchi identity, we obtain

(7.15)  R_{tjis} φ^{ts} = −(2n − 1)φ_{ji} − ρ̃_{ji} ,

where we put ρ̃_{ji} = ρ_{jl} φ_{i}^{l}. Using this, we get

(7.16)  φ^{ts}T_{tjis} = \frac{2\{τ − (6n^{3} + 13n^{2} + 3n − 2)\}}{(n + 1)(n + 2)} φ_{ji} − \frac{8(n + 1)}{n + 2} ρ̃_{ji} .

By virtue of (4.9), (7.14), (7.15) and (7.16), the tensor field S(X, Y, Z, W) in a Sasakian manifold satisfies

(7.17) \frac{1}{2}S(X, Y, Z, W) = 4R(X, Y, Z, W) − \frac{2}{n + 2} \{ρ(Y, Z)g(φX, φW)

+ ρ(X, W)g(φZ, φZ) − ρ(Y, W)g(φX, φZ) − ρ(X, Z)g(φY, φW)}

+ \frac{2}{n + 2} \{ρ̃(Y, Z)φ(X, W) + ρ̃(X, W)φ(Y, Z) − ρ̃(Y, W)φ(X, Z)

− ρ̃(X, Z)φ(Y, W) − 2ρ̃(X, Y)φ(Z, W) − 2ρ̃(Z, W)φ(X, W)}

+ \frac{τ − 2(3n + 4)}{(n + 1)(n + 2)} \{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\}

+ \frac{\{τ + 2n(2n + 3)\}}{(n + 1)(n + 2)} \{φ(X, W)φ(Y, Z) − φ(X, Z)φ(Y, W)

− 2φ(X, Y)φ(Z, W)\} − \frac{τ + 2n}{(n + 1)(n + 2)} \{(g(X, W)η(Y)η(Z)

+ g(Y, Z)η(X)η(W) − g(X, Z)η(Y)η(W) − g(Y, W)η(X)η(Z)}.

Thus comparing the above equation and (1.6), we have

Theorem 7.2. In a Sasakian manifold, the tensor field \frac{1}{8}S(X, Y, Z, W) is a C-Bochner curvature tensor.

We call the tensor field \frac{1}{8}S(X, Y, Z, W) the conformal (CHR)_{3}-curvature tensor or the C-Bochner curvature tensor in an almost contact Riemannian manifold. And an almost contact Riemannian manifold M^{2n+1} is called conformally (CHR)_{3}-flat if the tensor field S(X, Y)Z vanishes for any X, Y, Z ∈ T M^{2n+1} , identically.

8. A conformally (CHR)_{3}-flat almost contact Riemannian manifold

Let an almost contact Riemannian manifold M^{2n+1} be conformally (CHR)_{3}- flat. Then we have from (7.12)

(8.1) T_{kjih} = \frac{φ^{ts}}{2(n + 1)} (T_{tjis} φ_{kh} + T_{tkhs} φ_{ji} − T_{tjhs} φ_{ki} − T_{tkis} φ_{jh}

− 2T_{tkjs} φ_{ih} − 2T_{tihs} φ_{kj} ) − \frac{2τ(CHR)_{3}}{(n + 1)^{2}}(φ_{kh}φ_{ji} − φ_{ki}φ_{jh} − 2φ_{kj} φ_{ih})

+ \frac{τ(CHR)_{3}}{(n + 1)(n + 2)} (g_{kh}η_{j} η_{i} + g_{ji} η_{k}η_{h} − g_{ki}η_{j} η_{h} − g_{jh} η_{k}η_{i}).

By virtue of (7.7) and (7.8), we have

(8.2) T_{kjih} = 8(CHR)_{3kjih} − \frac{4}{n + 2} \{(g_{kh} − η_{k}η_{h})ρ(CHR)_{3ji}

+ (g_{ji} − η_{j} η_{i})ρ(CHR)_{3kh} − (g_{ki} − η_{k}η_{i})ρ(CHR)_{3jh}

− (g_{jh} − η_{j} η_{h})ρ(CHR)_{3ki} \} + \frac{2τ(CHR)_{3}}{(n + 1)(n + 2)} (g_{kh}g_{ji} − g_{ki}g_{jh})

− \frac{τ(CHR)_{3}}{(n + 1)(n + 2)} (g_{ji} η_{k}η_{h} + g_{kh}η_{j} η_{i} − g_{jh} η_{k}η_{i} − g_{ki}η_{j} η_{h}).

Contracting (8.2) with φ^{kh}, we have

(8.3) T_{tjis} φ^{ts} = 8(CHR)_{3tjis} φ^{ts} + \frac{4}{n + 2} \{ρ(CHR)_{3jl} φ_{i}^{l} − ρ(CHR)_{3il} φ_{j} ^{l}\}

+\frac{2τ(CHR)_{3}}{(n + 1)(n + 2)} φ_{ji} .

Substituting (8.3) into (8.1), we obtain

(8.4) 4(CHR)_{3kjih} = \frac{2}{n + 2} \{(g_{kh} − η_{k}η_{h})ρ(CHR)_{3ji} + (g_{ji} − η_{j} η_{i})ρ(CHR)_{3kh}

− (g_{ki} − η_{k}η_{i})ρ(CHR)_{3jh} − (g_{jh} − η_{j} η_{h})ρ(CHR)_{3ki} }

+ \frac{2φ^{ts}}{n + 1} \{(CHR)_{3tjis} φ_{kh} + (CHR)_{3tkhs} φ_{ji} − (CHR)_{3tjhs} φ_{ki}

− (CHR)_{3tkis} φ_{jh} − 2(CHR)_{3tkjs} φ_{ih} − 2(CHR)_{3tihs} φ_{kj} }

+ \frac{1}{(n + 1)(n + 2)} \{(ρ(CHR)_{3jl} φ_{i}^{l} − ρ(CHR)_{3il} φ_{j} ^{l})φ_{kh}

+ (ρ(CHR)_{3kl} φ_{h}^{l} − ρ(CHR)_{3hl} φ_{k}^{l})φ_{ji} − (ρ(CHR)_{3jl} φ_{h}^{l} − ρ(CHR)_{3hl} φ_{j} ^{l})φ_{ki}

− (ρ(CHR)_{3kl} φ_{i}^{l} − ρ(CHR)_{3il} φ_{k}^{l})φ_{jh} − 2(ρ(CHR)_{3kl} φ_{j} ^{l} − ρ(CHR)_{3jl} φ_{k}^{l})φ_{ih}

− 2(ρ(CHR)_{3il} φ_{h}^{l} − ρ(CHR)_{3hl} φ_{i}^{l})φ_{kj} \} − \frac{τ(CHR)_{3}}{(n + 1)(n + 2)} (g_{kh}g_{ji} − g_{ki}g_{jh}

− η_{k}η_{h}g_{ji} − η_{j} η_{i}g_{kh} + η_{k}η_{i}g_{jh} + η_{j} η_{h}g_{ki} + φ_{kh}φ_{ji} − φ_{ki}φ_{jh} − 2φ_{kj} φ_{ih}).

Next, contraction of (8.4) with g^{kh} gives us

(8.5) 6(2n + 3)ρ(CHR)_{3ji} = (2n + 5){ρ(CHR)_{3jl} ξ^{l}η_{i} + ρ(CHR)_{3il} ξ^{l}η_{j} }

+ 6(n + 2)φ^{ts}{(CHR)_{3tjls} φ_{i}^{l} + (CHR)_{3tils} φ_{j} ^{l}} − 6ρ(CHR)_{3ml} φ_{j} ^{m}φ_{i}^{l}.

Contracting the above equation with φ_{b}^{j} φ_{a}^{i} and changing the indices, we have

(8.6) (n + 2)φ^{ts}{(CHR)_{3tils} φ_{j} ^{l} + (CHR)_{3tjls} φ_{i}^{l}}

= (2n + 3)ρ(CHR)_{3ml} φ_{j} ^{m}φ_{i}^{l} + ρ(CHR)_{3ji} + ρ(CHR)_{3}(ξ, ξ)η_{j} η_{i}
− (n + 2)φ^{ts}ξ^{m}{φ_{j} ^{l}η_{i} + φ_{i}^{l}η_{j} } − {ρ(CHR)_{3jl} ξ^{l}η_{i} + ρ(CHR)_{3il} ξ^{l}η_{j} }.

Substituting (8.6) into (8.5), we get

12(n + 1){ρ(CHR)_{3ji} − ρ(CHR)_{3ml} φ_{j} ^{m}φ_{i}^{l}}

= (2n − 1){ρ(CHR)_{3jl} ξ^{l}η_{i} + ρ(CHR)_{3il} ξ^{l}η_{j} }
+ 6ρ(CHR)_{3}(ξ, ξ)η_{j} η_{i} − 6(n + 2)φ^{ts}(CHR)_{3tlms} ξ^{m}(φ_{j} ^{l}η_{i} + φ_{i}^{l}η_{j}).

From the above equation, we can easily get

(8.7) ρ(CHR)_{3}(X, Y) − ρ(CHR)_{3}(φX, φY) + ρ(CHR)_{3}(ξ, ξ)η(X)η(Y)

− {ρ(CHR)_{3}(ξ, X)η(Y) + ρ(CHR)_{3}(ξ, Y)η(X)} = 0.

Next, we have from (8.7)

(8.8)  τ(CHR)_{3} = \frac{n + 4}{3(n + 2)} ρ(CHR)_{3}(ξ, ξ) + (CHR)_{3tmls} φ^{ts}φ^{ml} .

Thus we have

Theorem 8.1. Let M^{2n+1} be a conformally (CHR)_{3}-flat almost contact Riemannian manifold. Then the (CHR)_{3}-curvature tensor, the (CHR)_{3}-Ricci tensor ρ(CHR)_{3} and the (CHR)_{3}-scalar curvature τ(CHR)_{3} satisfy (8.4), (8.7) and (8.8), respectively.

By virtue of (7.17), in a conformally (CHR)_{3}-flat Sasakian manifold, the Riemannian curvature tensor satisfies

(8.9) R(X, Y, Z, W) = \frac{1}{2(n + 2)} \{ρ(Y, Z)g(φX, φW) + ρ(X, W)g(φY, φZ)

− ρ(Y, W)g(φX, φZ) − ρ(X, Z)g(φY, φW)}

− \frac{1}{2(n + 2)} \{ρ̃(Y, Z)F (X, W) + ρ̃(X, W)F (Y, Z) − ρ̃(Y, W)F (X, Z)

− ρ̃(Y, W)F (X, Z) − ρ̃(X, Z)F (X, Y) − 2ρ̃(X, Y)F (Z, W) − 2ρ̃(Z, W)F (X, Y)}

− \frac{τ − 2(3n + 4)}{4(n + 1)(n + 2)} \{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\}

+ \frac{τ + 2n(2n + 3)}{4(n + 1)(n + 2)} \{F (X, W)F (Y, Z) − F (X, Z)F (Y, W) − 2F (X, Y)F (Z, W)\}

+ \frac{τ + 2n}{4(n + 1)(n + 2)} \{η(X)η(W)g(Y, Z) + η(Y)η(Z)g(X, W)

− η(X)η(Z)g(Y, W) − η(Y)η(W)g(X, Z)}.

Contracting (8.9) with g^{kh}, we get g_{ji} = (2n + 1)η_{j} η_{i}. Thus we have

Theorem 8.2. There does not exist any conformally (CHR)_{3}-flat Sasakian manifold.

From the above theorem, we can easily know

Remark 8.1. There does not exist any C-Bochner flat Sasakian manifold.

Acknowledgments. The author is thankful to the referees for their valuable comments and suggestions.

References

1. K. Matsumoto, A new curvatuelike tensor field in an almost contact Riemannian manifold, to appear

2. K. Matsumoto, I. Mihai, Ricci tensor of C-totally real submanifolds in Sasakian space forms, Nihonkai Math. J. 13 (2002), 191–198.

3. M. Matsumoto, G. Chuman, On the C-Bochner curvature tensor, TRU Math. 5 (1969), 21–30.

4. M. Prvanović, Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor, J. Geom. 103 (2012), 89–101.

5. S. Sasaki, Almost Contact Manifolds, Lecture Notes, Tôkoku University, 1965.

6. K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pargamon Press, 1965.

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