PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 93–105 DOI: https://doi.org/10.2298/PIM1716093Y

f -KENMOTSU MANIFOLDS WITH THE SCHOUTEN–VAN KAMPEN CONNECTION

Ahmet Yıldız

Abstract. We study 3-dimensional f-Kenmotsu manifolds with the Schouten– van Kampen connection. With the help of such a connection, we study projectively flat, conharmonically flat, Ricci semisymmetric and semisymmetric 3-dimensional f-Kenmotsu manifolds. Finally, we give an example of 3dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection.

1. Introduction

The Schouten–van Kampen connection is one of the most natural connections adapted to a pair of complementary distributions on a differentiable manifold endowed with an affine connection [2,4,11]. Solov’ev investigated hyperdistributions in Riemannian manifolds using the Schouten–van Kampen connection [12–15]. Then Olszak studied the Schouten–van Kampen connection to an almost contact metric structure [8]. He characterized some classes of almost contact metric manifolds with the Schouten–van Kampen connection and found certain curvature properties of this connection on these manifolds.

On the other hand, let M be an almost contact manifold, i.e., M is a connected (2n+1)-dimensional differentiable manifold endowed with an almost contact metric structure (ϕ, ξ, η, g) [1]. Denote by Φ the fundamental 2-form of M, Φ(X, Y) = g(X, ϕY), X, Y ∈ χ(M), χ(M) being the Lie algebra of differentiable vector fields on M.

For further use, we recall the following definitions [1, 3, 10]. The manifold M and its structure (ϕ, ξ, η, g) is said to be:

i) normal, if the almost complex structure defined on the product manifold M × ℝ is integrable (equivalently [ϕ, ϕ] + 2dη ⊗ ξ = 0),

ii) almost cosymplectic, if dη = 0 and dΦ = 0, iii) cosymplectic, if it is normal and almost cosymplectic (equivalently, ∇ϕ = 0 , ∇ being covariant differentiation with respect to the Levi-Civita connection).

2010 Mathematics Subject Classification: 53C15, 53C25, 53C50.

Key words and phrases: Schouten-van Kampen connection, f-Kenmotsu manifolds, Riccisemisymmetric, semisymmetric, Einstein manifold, η-Einstein manifold.

Communicated by Stevan Pilipović.

The manifold M is called locally conformal, cosymplectic (respectively almost cosymplectic), if M has an open covering {U_{t}} endowed with differentiable functions σ_{t}: U_{i} → ℝ such that over each U_{t} the almost contact metric structure (ϕ_{t}, ξ_{t}, η_{t}, g_{t}) defined by

ϕ_{t} = ϕ, ξ_{t} = e^{σ_{t}} ξ, η_{t} = e^{−σ_{t}} η, g_{t} = e^{−2σ_{t}} g

is cosymplectic (respectively almost cosymplectic).

Also, Olszak and Rosca [9] studied normal locally conformal almost cosymplectic manifolds. They given a geometric interpretation of f-Kenmotsu manifolds and studied some curvature properties. Among others they proved that a Ricci symmetric f-Kenmotsu manifold is an Einstein manifold.

By an f-Kenmotsu manifold, we mean an almost contact metric manifold which is normal and locally conformal almost cosymplectic manifold.

In the present paper we study some curvature properties of 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection. The paper is organized as follows: after introduction, we give the Schouten–van Kampen connection and f-Kenmotsu manifolds. Then we adapt the Schouten–van Kampen connection on 3-dimensional f-Kenmotsu manifolds. In section 5, we study projectively flat 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection. In section 6, we consider conharmonically flat 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection. Section 7 is devoted to study Ricci semisymmetric 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection and we prove that if a 3-dimensional f-Kenmotsu manifold is Ricci semisymmetric, then it is an η-Einstein manifold. In section 8, we study semisymmetric 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection. Finally, we give an example of a 3-dimensional f-Kenmotsu manifold with the Schouten–van Kampen connection which verifies Theorem 5.1 and Theorem 6.1.

2. The Schouten–van Kampen connection

Let M be a connected pseudo-Riemannian manifold of an arbitrary signature (p, n − p), 0 ⩽ p ⩽ n, n = dimM ⩾ 2. By g and ∇ we denote the pseudo-Riemannian metric and Levi-Civita connection induced from the metric g on M respectively. Assume that H and V are two complementary, orthogonal distributions on M such that dimH = n−1, dimV = 1, and the distribution V is non-null. Thus T M = H ⊕ V, H ∩ V = {0} and H ⊥ V. Assume that ξ is a unit vector field and η is a linear form such that η(ξ) = 1, g(ξ, ξ) = ε = ±1 and

H = kerη, V = span{ξ}.

We can always choose such ξ and η at least locally (in a certain neighborhood of an arbitrarily chosen point of M). We also have η(X) = εg(X, ξ). Moreover, it holds that ∇_{X} ξ ∈ H.

f-KENMOTSU MANIFOLDS WITH THE SCHOUTEN–VAN KAMPEN CONNECTION 95

For any X ∈ T M, by X^{h} and X^{v} we denote the projections of X onto H and V, respectively. Thus, we have X = X^{h} + X^{v} with

(2.1)  X^{h} = X − η(X)ξ, X^{v} = η(X)ξ.

The Schouten–van Kampen connection ˜∇ associated to the Levi-Civita connection ∇ and adapted to the pair of the distributions (H, V) is defined by [2]

(2.2)  ˜∇_{X} Y = (∇_{X} Y^{h})^{h} + (∇_{X} Y^{v})^{v},

and the corresponding second fundamental form B is defined by B = ∇ − ˜∇. Note that condition (2.2) implies the parallelism of the distributions H and V with respect to the Schouten–van Kampen connection ˜∇.

From (2.1), one can compute

(∇_{X} Y^{h})^{h} = ∇_{X} Y − η(∇_{X} Y)ξ − η(Y)∇_{X} ξ,
(∇_{X} Y^{v})^{v} = (∇_{X} η)(Y)ξ + η(∇_{X} Y)ξ,

which enables us to express the Schouten–van Kampen connection with help of the Levi-Civita connection in the following way [12]

(2.3)  ˜∇_{X} Y = ∇_{X} Y − η(Y)∇_{X} ξ + (∇_{X} η)(Y)ξ.

Thus, the second fundamental form B and the torsion T̃ of ˜∇ are [12, 13]

B(X, Y) = η(Y)∇_{X} ξ − (∇_{X} η)(Y)ξ,
T̃(X, Y)= η(X)∇_{Y}ξ − η(Y)∇_{X} ξ + 2dη(X, Y)ξ.

With the help of the Schouten–van Kampen connection (2.3), many properties of some geometric objects connected with the distributions H, V can be characterized [12–14]. Probably, the most spectacular is the following statement: g, ξ and η are parallel with respect to ˜∇, that is, ˜∇ ξ = 0, ˜∇ g = 0, ˜∇ η = 0.

3. f-Kenmotsu manifolds

Let M be a real (2n + 1)-dimensional differentiable manifold endowed with an almost contact structure (ϕ, ξ, η, g) satisfying

(3.1)  \begin{matrix}ϕ^{2} = −I + η ⊗ ξ, η(ξ) = 1, \\ ϕξ = 0, η ∘ ϕ = 0, η(X) = g(X, ξ), \\ g(ϕX, ϕY) = g(X, Y) − η(X) η(Y), \end{matrix}

for any vector fields X, Y ∈ χ(M), where I is the identity of the tangent bundle T M, ϕ is a tensor field of (1, 1)-type, η is a 1-form, ξ is a vector field and g is a metric tensor field. We say that (M, ϕ, ξ, η, g) is a f-Kenmotsu manifold if the Levi-Civita connection of g satisfy [7]

(∇_{X} ϕ)(Y) = f{g(ϕX, Y)ξ − η(Y)ϕX},

where f ∈ C^{∞} (M) such that df ∧η = 0. If f = α = constant ≠ 0, then the manifold is an α-Kenmotsu manifold [5]. 1-Kenmotsu manifold is a Kenmotsu manifold [6]. If f = 0, then the manifold is cosymplectic [5]. An f-Kenmotsu manifold is said to be regular if f^{2} + f′ ≠ 0, where f′ = ξ(f).

For an f-Kenmotsu manifold from (3.1) it follows that

(3.2)  ∇_{X} ξ = f{X − η(X)ξ}.

Then using (3.2), we have

(3.3)  (∇_{X} η)(Y) = f{g(X, Y) − η(X) η(Y)}.

The condition df ∧ η = 0 holds if dimM ⩾ 5. This does not hold in general if dimM = 3 [9].

As is well known, in a 3-dimensional Riemannian manifold, we always have

R(X, Y)Z = g(Y, Z)QX − g(X, Z)QY + S(Y, Z)X − S(X, Z)Y
−\frac{τ}{2}\{g(Y, Z)X − g(X, Z)Y \} .

In a 3-dimensional f-Kenmotsu manifold M, we have [9]

(3.4)  R(X, Y)Z = ( \frac{τ}{2} + 2f^{2} + 2f′) \{g(Y, Z)X − g(X, Z)Y \}

− ( \frac{τ}{2} + 3f^{2} + 3f′) \{g(Y, Z) η(X)ξ − g(X, Z) η(Y)ξ

+ η(Y) η(Z)X − η(X) η(Z)Y },
(3.5)  S(X, Y) = ( \frac{τ}{2} + f^{2} + f′) g(X, Y) − ( \frac{τ}{2} + 3f^{2} + 3f′) η(X) η(Y),
QX = ( \frac{τ}{2} + f^{2} + f′) X − ( \frac{τ}{2} + 3f^{2} + 3f′) η(X)ξ,

where R denotes the curvature tensor, S is the Ricci tensor, Q is the Ricci operator and τ is the scalar curvature of M.

From (3.4) and (3.5), we obtain

R(X, Y)ξ = −(f^{2} + f′){η(Y)X − η(X)Y }, (3.6)

S(X, ξ) = −2(f^{2} + f′) η(X). (3.7)

4. 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection

Let M be a 3-dimensional f-Kenmotsu manifold with the Schouten–van Kampen connection. Then using (3.2) and (3.3) in (2.3), we get

(4.1)  ˜∇_{X} Y = ∇_{X} Y + f(g(X, Y)ξ − η(Y)X).

Let R and R̃ be the curvature tensors of the Levi-Civita connection ∇ and the Schouten–van Kampen connection ˜∇,

R(X, Y) = [∇_{X} , ∇_{Y}] − ∇_{[X,Y]}, R̃(X, Y) = [ ˜∇_{X} , ˜∇_{Y} ] − ˜∇_{[X,Y]} .

Using (4.1), by direct calculations, we obtain the following formula connecting R and R̃ on a 3-dimensional f-Kenmotsu manifold M,

(4.2)  R̃(X, Y)Z = R(X, Y)Z + f^{2}{g(Y, Z)X − g(X, Z)Y }
+ f′{g(Y, Z) η(X)ξ − g(X, Z) η(Y)ξ
+ η(Y) η(Z)X − η(X) η(Z)Y }.

We will also consider the Riemann curvature (0, 4)-tensors R̃, R, the Ricci tensors S̃, S, the Ricci operators Q̃, Q and the scalar curvatures τ̃, τ of the connections ˜∇ and ∇ are given by

R̃(X, Y, Z, W) = R(X, Y, Z, W) + f^{2}{g(Y, Z) g(X, W) − g(X, Z) g(Y, W)} (4.3)
+ f′{g(Y, Z) η(X) η(W) − g(X, Z) η(Y) η(W)
+ g(X, W) η(Y) η(Z) − g(Y, W) η(X) η(Z)},

S̃(Y, Z) = S(Y, Z) + (2f^{2} + f′) g(Y, Z) + f′η(Y) η(Z)), (4.4)

Q̃X = QX + (2f^{2} + f′)X + f′η(X)ξ, (4.5)

τ̃ = τ + 6f^{2} + 4f′,

respectively, where

R̃(X, Y, Z, W) = g(R̃(X, Y)Z, W) and R(X, Y, Z, W) = g(R(X, Y)Z, W).

5. Projectively flat 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection

In this section, we study projectively flat 3-dimensional f-Kenmotsu manifolds with respect to the Schouten–van Kampen connection. In a 3-dimensional f-Kenmotsu manifold, the projective curvature tensor with respect to the Schouten– van Kampen connection is given by

(5.1)  P̃(X, Y)Z = R̃(X, Y)Z − \frac{1}{2}\{S̃(Y, Z)X − S̃(X, Z)Y \}.

If P̃ = 0, then the manifold M is called projectively flat manifold with respect to the Schouten–van Kampen connection.

Let M be a projectively flat manifold with respect to the Schouten–van Kampen connection. From (5.1), we have

(5.2)  R̃(X, Y)Z = \frac{1}{2}\{S̃(Y, Z)X − S̃(X, Z)Y \}.

Using (4.3) and (4.4) in (5.2), we get

g(R(X, Y)Z, W) + f^{2}{g(Y, Z) g(X, W) − g(X, Z) g(Y, W)} (5.3)

+ f′{g(Y, Z) η(X) η(W) − g(X, Z) η(Y) η(W)
+ g(X, W) η(Y) η(Z) − g(Y, W) η(X) η(Z)}
= \frac{1}{2}\{S(Y, Z) g(X, W) − S(X, Z) g(Y, W)
+ [2f^{2} + f′][g(Y, Z) g(X, W) − g(X, Z) g(Y, W)]
+ f′[η(Y) η(Z) g(X, W) − η(X) η(Z) g(Y, W)]}.

Now putting W = ξ in (5.3), we obtain

(f^{2} + f′){g(X, Z) η(Y) − g(Y, Z) η(X)} + (f^{2}+f′){g(Y, Z) η(X) − g(X, Z) η(Y)}

= \frac{1}{2}\{S(Y, Z) η(X) − S(X, Z) η(Y) + (2f^{2}+f′)[g(Y, Z) η(X) − g(X, Z) η(Y)]\} ,

which gives

(5.4) S(Y, Z) η(X) − S(X, Z) η(Y) + (2f^{2} + f′)[g(Y, Z) η(X) − g(X, Z) η(Y)] = 0.

Again putting X = ξ in (5.4), we get

(5.5)  S(Y, Z) = −(2f^{2} + f′) g(Y, Z) − f′η(Y) η(Z).

Thus M is an η-Einstein manifold with respect to the Levi-Civita connection.

Also, using (5.5) in (4.4), we obtain S̃(Y, Z) = 0. Hence the manifold M is a Ricci-flat manifold with respect to the Schouten–van Kampen connection. Then from (5.2) the manifold M is a flat manifold with respect to the Schouten–van Kampen connection.

Conversely, let M be a flat manifold with respect to the Schouten–van Kampen connection. Then we say that the manifold M is a Ricci-flat manifold with respect to the Schouten–van Kampen connection. Hence from (5.1), we get P̃(X, Y)Z = 0, that is, the manifold M is a projectively flat manifold with respect to the Schouten– van Kampen connection. Thus we have the following:

Theorem 5.1. Let M be a 3-dimensional f-Kenmotsu manifold with the Schouten–van Kampen connection. Then the following statements are equivalent:

i) M is projectively flat with respect to the Schouten–van Kampen connection, ii) M is Ricci flat with respect to the Schouten–van Kampen connection, iii) M is flat with respect to the Schouten–van Kampen connection.

6. Conharmonically flat 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection

In this section, we study conharmonically flat 3-dimensional f-Kenmotsu manifolds with respect to the Schouten–van Kampen connection. In a 3-dimensional f-Kenmotsu manifold the conharmonic curvature tensor with respect to the Schouten– van Kampen connection is given by

(6.1) K̃(X, Y)Z = R̃(X, Y)Z
− {S̃(Y, Z)X − S̃(X, Z)Y + g(Y, Z)Q̃X − g(X, Z)Q̃Y }.

If K̃ = 0, then the manifold M is called conharmonically flat manifold with respect to the Schouten–van Kampen connection.

Let M be a conharmonically flat manifold with respect to the Schouten–van Kampen connection. From (6.1), we have

(6.2)  R̃(X, Y)Z = S̃(Y, Z)X − S̃(X, Z)Y + g(Y, Z)Q̃X − g(X, Z)Q̃Y.

Using (4.3), (4.4) and (4.5) in (6.2), we get

R(X, Y)Z + f^{2}{g(Y, Z)X − g(X, Z)Y } (6.3)

+ f′{g(Y, Z) η(X)ξ − g(X, Z) η(Y)ξ + η(Y) η(Z)X − η(X) η(Z)Y }

= S(Y, Z)X − S(X, Z)Y

+ ( 4f^{2} + 2f′+ \frac{τ}{2} + f^{2} + f′) \{g(Y, Z)X − g(X, Z)Y \}

+ f′{η(Y) η(Z)X − η(X) η(Z)Y }

+ ( f′− \frac{τ}{2} − 3f^{2} − 3f′) \{g(Y, Z) η(X)ξ − g(X, Z) η(Y)ξ\}.

Now putting X = ξ in (6.3), we obtain

R(ξ, Y)Z + (f^{2} + f′){g(Y, Z)ξ − η(Z)Y } (6.4)

= S(Y, Z)ξ − S(ξ, Z)Y

+ ( 4f^{2} + 2f′+ \frac{τ}{2} + f^{2} + f′) \{g(Y, Z)ξ − η(Z)Y \}

+ f′{η(Y) η(Z)ξ − η(Z)Y }

+ ( f′− \frac{τ}{2} − 3f^{2} − 3f′) \{g(Y, Z)ξ − η(Z) η(Y)ξ\}.

Using (3.4) and (3.7) in (6.4), we get

(6.5) S(Y, Z)ξ− S(ξ, Z)Y + ( 4f^{2} + 2f′+ \frac{τ}{2} + f^{2} + f′) \{g(Y, Z)ξ − η(Z)Y \}

+ f′{η(Y) η(Z)ξ − η(Z)Y }

+ ( f′− \frac{τ}{2} − 3f^{2} − 3f′) \{g(Y, Z)ξ − η(Z) η(Y)ξ\} = 0.

Taking the inner product with ξ in (6.5), we have

S(Y, Z) + 2(f^{2} + f′) η(Y) η(Z) + (2f^{2} + f′){g(Y, Z) − η(Y) η(Z)} = 0,

which gives

(6.6)  S(Y, Z) = −(2f^{2} + f′) g(Y, Z) − f′η(Y) η(Z).

Thus M is an η-Einstein manifold with respect to the Levi-Civita connection.

Using (6.6) in (4.4), we obtain S̃(Y, Z) = 0. Hence the manifold M is a Ricciflat manifold with respect to the Schouten–van Kampen connection. Then from (6.2) the manifold M is a flat manifold with respect to the Schouten–van Kampen connection.

Conversely, let M be a flat manifold with respect to the Schouten–van Kampen connection. Then we say that the manifold M is a Ricci-flat manifold with respect to the Schouten–van Kampen connection. Hence from (6.1), we get K̃(X, Y)Z = 0. i.e., the manifold M is a conharmonically flat manifold with respect to the Schouten–van Kampen connection. Thus we have the following:

Theorem 6.1. Let M be a 3-dimensional f-Kenmotsu manifold with the Schouten–van Kampen connection. Then the following statements are equivalent:

i) M is conharmonically flat with respect to the Schouten–van Kampen connection, ii) M is Ricci flat with respect to the Schouten–van Kampen connection, iii) M is flat with respect to the Schouten–van Kampen connection.

7. Ricci semisymmetric 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection

A f-Kenmotsu manifold with the Schouten–van Kampen connection is called Ricci semisymmetric if R̃(X, Y) · S̃ = 0, where R̃(X, Y) is treated as a derivation of the tensor algebra for any tangent vectors X, Y. Then

(7.1)  S̃(R̃(X, Y)Z, W) + S̃(Z, R̃(X, Y)W) = 0.

Using (4.3) and (4.4) in (7.1), we get

S(R(X, Y)Z, W) + S(Z, R(X, Y)W) + f′{η(R(X, Y)Z) η(W)

+ f′η(R(X, Y)W) η(Z)} + f^{2}{S(X, W) g(Y, Z) − S(Y, W) g(X, Z)

+ S(X, Z) g(Y, W) − S(Y, Z) g(X, W)}

− f′(f^{2} + f′){g(Y, Z) η(X) η(W) − g(X, Z) η(Y) η(W) + g(Y, W) η(X) η(Z)

− g(X, W) η(Y) η(Z)} + f′{S(X, W) η(Y) η(Z) − S(Y, W) η(X) η(Z)

+ S(X, Z) η(Y) η(W) − S(Y, Z) η(X) η(W)} = 0.

Let M be Ricci semisymmetric with respect to the Levi-Civita connection. Then we have

f′{η(R(X, Y)Z) η(W) + f′η(R(X, Y)W) η(Z)} + f^{2}{S(X, W) g(Y, Z) (7.2)

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

− f′(f^{2} + f′){g(Y, Z) η(X) η(W) − g(X, Z) η(Y) η(W) + g(Y, W) η(X) η(Z)

− g(X, W) η(Y) η(Z)} + f′{S(X, W) η(Y) η(Z) − S(Y, W) η(X) η(Z)

+ S(X, Z) η(Y) η(W) − S(Y, Z) η(X) η(W)} = 0.

Putting W = ξ in (7.2), we obtain

f′η(R(X, Y)Z) + f^{2}{S(X, ξ) g(Y, Z) − S(Y, ξ) g(X, Z)

+ S(X, Z) η(Y) − S(Y, Z) η(X)}

− f′(f^{2} + f′){g(Y, Z) η(X) − g(X, Z) η(Y)} + f′{S(X, ξ) η(Y) η(Z)

− S(Y, ξ) η(X) η(Z) + S(X, Z) η(Y) − S(Y, Z) η(X)} = 0.

After some calculations, we get

(7.3)  2(f^{2} + f′)^{2} {g(Y, Z) η(X) − g(X, Z) η(Y)}
− (f^{2} + f′) {S(Y, Z) η(X) − S(X, Z) η(Y)} = 0.

Again putting X = ξ in (7.3), we have

2(f^{2} + f′)^{2}{g(Y, Z) − η(Y) η(Z)} − (f^{2} + f′){S(Y, Z) + 2(f^{2} + f′) η(Y) η(Z)} = 0,

which gives

(7.4)  (f^{2} + f′){S(Y, Z) + 4(f^{2} + f′) η(Y) η(Z) − 2(f^{2} + f′) g(Y, Z)} = 0.

Let f^{2} + f′ ≠ 0, then from (7.4), we get

(7.5)  S(Y, Z) = 2(f^{2} + f′) g(Y, Z) − 4(f^{2} + f′) η(Y) η(Z).

Hence the manifold is an η-Einstein manifold with respect to the Levi-Civita connection.

Using (7.5) in (4.4), we obtain

S̃(Y, Z) = (4f^{2} + 3f′) g(Y, Z) − (4f^{2} + 3f′) η(Y) η(Z).

Thus we have the following:

Theorem 7.1. Let M be a Ricci semisymmetric 3-dimensional regular f-Kenmotsu manifold with the Schouten–van Kampen connection. If M is a Ricci semisymmetric 3-dimensional f-Kenmotsu manifold with respect to the Levi-Civita connection, then M is an η-Einstein manifold with respect to the Schouten–van Kampen connection.

8. Semisymmetric 3-dimensional f-Kenmotsu manifolds with the Schouten–van Kampen connection

In this section, we study a semisymmetric regular 3-dimensional f-Kenmotsu manifold with the Schouten–van Kampen connection. If a 3-dimensional f-Ken-motsu manifold with the Schouten–van Kampen connection is semisymmetric then we can write

(R̃(X, Y) · R̃)(Z, U)W = 0,

which gives

(8.1)  R̃(X, Y)R̃(Z, U)W− R̃(R̃(X, Y)Z, U)W
− R̃(Z, R̃(X, Y)U)W − R̃(Z, U)R̃(X, Y)W = 0.

Using (4.2) in (8.1), we have

R̃(X, Y)R(Z, U)W − R(R̃(X, Y)Z, U)W
− R(Z, R̃(X, Y)U)W − R(Z, U)R̃(X, Y)W = 0,

which gives

(8.2)  (R̃(X, Y) · R)(Z, U)W = 0.

Again using (4.2) in (8.2), we obtain

R(X, Y)R(Z, U)W − R(R(X, Y)Z, U)W − R(Z, R(X, Y)U)W (8.3)

− R(Z, U)R(X, Y)W + f^{2}{g(R(Z, U)W, Y)X − g(R(Z, U)W, X)Y

− g(Y, Z)R(X, U)W + g(X, Z)R(Y, U)W − g(Y, U)R(Z, X)W

+ g(X, U)R(Z, Y)W − g(Y, W)R(Z, U)X + g(X, W)R(Z, U)Y }

+ f′{g(R(Z, U)W, Y) η(X)ξ−g(R(Z, U)W, X) η(Y)ξ + η(R(Z, U)W) η(Y)X

− η(R(Z, U)W) η(X)Y − g(Y, Z) η(R(X, U)W)ξ + g(X, Z) η(R(Y, U)W)ξ

− η(Y) η(Z)R(X, U)W + η(X) η(Z)R(Y, U)W − g(Y, U) η(R(Z, X)W)ξ + g(X, U) η(R(Z, Y)W)ξ − η(Y) η(U)R(Z, X)W + η(X) η(U)R(Z, Y)W

− g(Y, W) η(R(Z, U)X)ξ + g(X, W) η(R(Z, U)Y)ξ

− η(Y) η(W)R(Z, U)X + η(X) η(W)R(Z, U)Y } = 0.

Now from (8.3), we can say:

If 0 ≠ f = constant (say f = α), then f′ = 0. Hence we get R·R = −α^{2}Q(g, R). Therefore the manifold M is a pseudosymmetric α-Kenmotsu manifold.

If f is not constant, then using X = ξ in (8.3), we get

R(ξ, Y)R(Z, U)W − R(R(ξ, Y)Z, U)W − R(Z, R(ξ, Y)U)W (8.4)

− R(Z, U)R(ξ, Y)W + f^{2}{g(R(Z, U)W, Y)ξ − g(R(Z, U)W, ξ)Y

− g(Y, Z)R(ξ, U)W + g(ξ, Z)R(Y, U)W − g(Y, U)R(Z, ξ)W

+ g(ξ, U)R(Z, Y)W − g(Y, W)R(Z, U)ξ + g(ξ, W)R(Z, U)Y }

+ f′{g(R(Z, U)W, Y)ξ − g(R(Z, U)W, ξ) η(Y)ξ + η(R(Z, U)W) η(Y)ξ

− η(R(Z, U)W)Y − g(Y, Z) η(R(ξ, U)W)ξ + g(ξ, Z) η(R(Y, U)W)ξ

− η(Y) η(Z)R(ξ, U)W + η(Z)R(Y, U)W − g(Y, U) η(R(Z, ξ)W)ξ

+ g(ξ, U) η(R(Z, Y)W)ξ − η(Y) η(U)R(Z, ξ)W + η(U)R(Z, Y)W

− g(Y, W) η(R(Z, U)ξ)ξ + g(ξ, W) η(R(Z, U)Y)ξ

− η(Y) η(W)R(Z, U)ξ + η(W)R(Z, U)Y } = 0.

Taking the inner product with ξ in (8.4), we obtain

η(R(ξ, Y)R(Z, U)W) − η(R(R(ξ, Y)Z, U)W) − η(R(Z, R(ξ, Y)U)W) (8.5)

− η(R(Z, U)R(ξ, Y)W) + f^{2}{g(R(Z, U)W, Y) − g(R(Z, U)W, ξ) η(Y)

− g(Y, Z) η(R(ξ, U)W) + g(ξ, Z) η(R(Y, U)W) − g(Y, U) η(R(Z, ξ)W)

+ g(ξ, U) η(R(Z, Y)W) − g(Y, W) η(R(Z, U)ξ) + g(ξ, W) η(R(Z, U)Y)}

+ f′{g(R(Z, U)W, Y) − g(R(Z, U)W, ξ) η(Y) + η(R(Z, U)W) η(Y)

− η(R(Z, U)W) η(Y) − g(Y, Z) η(R(ξ, U)W) + g(ξ, Z) η(R(Y, U)W)

− η(Y) η(Z) η(R(ξ, U)W) + η(Z) η(R(Y, U)W) − g(Y, U) η(R(Z, ξ)W)

+ g(ξ, U) η(R(Z, Y)W) − η(Y) η(U) η(R(Z, ξ)W) + η(U) η(R(Z, Y)W)

− g(Y, W) η(R(Z, U)ξ) + g(ξ, W) η(R(Z, U)Y)

− η(Y) η(W) η(R(Z, U)ξ) + η(W) η(R(Z, U)Y)} = 0.

Let {e_{i}} (1 ⩽ i ⩽ 3) be an orthonormal basis of the tangent space at any point of M. Then the sum for 1 ⩽ i ⩽ 3 of the relation (8.5) for Y = Z = e_{i} gives

η(R(ξ, e_{i})R(e_{i}, U)W) − η(R(R(ξ, e_{i})e_{i}, U)W) − η(R(e_{i}, R(ξ, e_{i})U)W)

− η(R(e_{i}, U)R(ξ, e_{i})W) + f^{2}{g(R(e_{i}, U)W, e_{i}) − g(R(e_{i}, U)W, ξ) η(e_{i})

− g(e_{i}, e_{i}) η(R(ξ, U)W) + g(ξ, e_{i}) η(R(e_{i}, U)W) − g(e_{i}, U) η(R(e_{i}, ξ)W)

+ g(ξ, U) η(R(e_{i}, e_{i})W) − g(e_{i}, W) η(R(e_{i}, U)ξ) + g(ξ, W) η(R(e_{i}, U)e_{i})} + f′{g(R(e_{i}, U)W, e_{i}) − g(R(e_{i}, U)W, ξ) η(e_{i}) + η(R(e_{i}, U)W) η(e_{i})

− η(R(e_{i}, U)W) η(e_{i}) − g(e_{i}, e_{i}) η(R(ξ, U)W) + g(ξ, e_{i}) η(R(e_{i}, U)W)

− η(e_{i}) η(e_{i}) η(R(ξ, U)W) + η(e_{i}) η(R(e_{i}, U)W) − g(e_{i}, U) η(R(e_{i}, ξ)W)

+ g(ξ, U) η(R(e_{i}, e_{i})W) − η(e_{i}) η(U) η(R(e_{i}, ξ)W) + η(U) η(R(e_{i}, e_{i})W)

− g(e_{i}, W) η(R(e_{i}, U)ξ) + g(ξ, W) η(R(e_{i}, U)e_{i})

− η(e_{i}) η(W) η(R(e_{i}, U)ξ) + η(W) η(R(e_{i}, U)e_{i})} = 0.

After some calculations, we obtain

2(f^{2} + f′){S(U, W) − 2g(R(ξ, W)U, ξ)}
− f^{2}{S(U, W) − 2g(R(ξ, W)U, ξ) − 2(f^{2} + f′) η(U) η(W)}
− f′{S(U, W) − 2g(R(ξ, W)U, ξ) − 2(f^{2} + f′) η(U) η(W)} = 0,

which gives

(8.6)  (f^{2} + f′){S(U, W) − 2g(R(ξ, W)U, ξ) + 2(f^{2} + f′) η(U) η(W)} = 0.

Let f^{2} + f′ ≠ 0. Then from (8.6), we get

(8.7)  S(U, W) − 2g(R(ξ, W)U, ξ) + 2(f^{2} + f′) η(U) η(W) = 0.

Using (3.6) in (8.7), we obtain S(U, W) = −2(f^{2} + f′) g(U, W).

Thus we have the following:

Theorem 8.1. Let M be a 3-dimensional regular f-Kenmotsu manifold with the Schouten–van Kampen connection. If M is semisymmetric with respect to the Schouten–van Kampen connection, then:

i) If 0 ≠ f = α = constant, then the manifold M is a pseudosymmetric α-Kenmotsu manifold, or,

ii) If f is not constant, then the manifold M is an Einstein manifold.

9. An example of a 3-dimensional f-Kenmotsu manifold with the Schouten–van Kampen connection

We consider the 3-dimensional manifold M = {(x, y, z) ∈ ℝ^{3}, z ≠ 0}, where (x, y, z) are the standard coordinates in ℝ^{3}. The vector fields

e_{1} = z^{2} \frac{∂}{∂x}, e_{2} = z^{2} \frac{∂}{∂y}, e_{3} = \frac{∂}{∂z}

are linearly independent at each point of M. Let g be the Riemannian metric defined by

g(e_{1}, e_{3}) = g(e_{2}, e_{3}) = g(e_{1}, e_{2}) = 0, g(e_{1}, e_{1}) = g(e_{2}, e_{2}) = g(e_{3}, e_{3}) = 1.

Let η be the 1-form defined by η(Z) = g(Z, e_{3}) for any Z ∈ χ(M). Let ϕ be the (1, 1) tensor field defined by ϕ(e_{1}) = −e_{2}, ϕ(e_{2}) = e_{1}, ϕ(e_{3}) = 0. Then using linearity of ϕ and g we have

η(e_{3}) = 1, ϕ^{2}Z = −Z + η(Z)e_{3}, g(ϕZ, ϕW) = g(Z, W) − η(Z) η(W),

for any Z, W ∈ χ(M). Now, by direct computations we obtain

[e_{1}, e_{2}] = 0, [e_{2}, e_{3}] = −\frac{2}{z}e_{2}, [e_{1}, e_{3}] = −\frac{2}{z}e_{1}.

The Riemannian connection ∇ of the metric tensor g is given by Koszul’s formula which is

(9.1)  2g(∇_{X} Y, Z) = Xg(Y, Z) + Y g(Z, X) − Zg(X, Y)
− g(X, [Y, Z]) − g(Y, [X, Z]) + g(Z, [X, Y]).

Using (9.1), we have

2g(∇_{e_{1}} e_{3}, e_{1}) = 2g( − \frac{2}{z}e_{1}, e_{1}) , 2g(∇_{e_{1}} e_{3}, e_{2}) = 0 and 2g(∇_{e_{1}} e_{3}, e_{3}) = 0.

Hence ∇_{e_{1}} e_{3} = −\frac{2}{z}e_{1}. Similarly, ∇_{e_{2}} e_{3} = −\frac{2}{z}e_{2} and ∇_{e_{3}} e_{3} = 0. (9.1) further yields

(9.2)  \begin{aligned}∇_{e_{1}} e_{2} & = 0, & ∇_{e_{2}} e_{2} & = \frac{2}{z}e_{3}, & ∇_{e_{3}} e_{2} & = 0, \\ ∇_{e_{1}} e_{1} & = \frac{2}{z}e_{3}, & ∇_{e_{2}} e_{1} & = 0, & ∇_{e_{3}} e_{1} & = 0. \end{aligned}

From (9.2), we see that the manifold satisfies ∇_{X} ξ = f{X − η(X)ξ} for ξ = e_{3}, where f = −\frac{2}{z}. Hence we conclude that M is an f-Kenmotsu manifold. Also f^{2} + f′ ≠ 0. Hence M is a regular f-Kenmotsu manifold [16].

It is known that

(9.3)  R(X, Y)Z = ∇_{X} ∇_{Y}Z − ∇_{Y}∇_{X} Z − ∇_{[X,Y]}Z.

With the help of the above formula and using (9.3), it can be easily verified that

(9.4)  \begin{aligned}R(e_{1}, e_{2})e_{3} & = 0, & R(e_{2}, e_{3})e_{3} = −\frac{6}{z^{2}} e_{2}, \\ R(e_{1}, e_{3})e_{3} & = −\frac{6}{z^{2}} e_{1}, & R(e_{1}, e_{2})e_{2} = −\frac{4}{z^{2}} e_{1}, \\ R(e_{3}, e_{2})e_{2} & = −\frac{6}{z^{2}} e_{3}, & R(e_{1}, e_{3})e_{2} = 0, \\ R(e_{1}, e_{2})e_{1} & = \frac{4}{z^{2}} e_{2}, & R(e_{2}, e_{3})e_{1} = 0, \\ R(e_{1}, e_{3})e_{1} & = \frac{6}{z^{2}} e_{3}. \end{aligned}

Now the Schouten–van Kampen connection on M is given by

(9.5)  \begin{aligned}˜∇_{e_{1}} e_{3} & = ( − \frac{2}{z} − f) e_{1}, & ˜∇_{e_{2}} e_{3} = ( − \frac{2}{z} − f) e_{2}, \\ ˜∇_{e_{3}} e_{3} & = −f(e_{3} − ξ), & ˜∇_{e_{1}} e_{2} = 0, \\ ˜∇_{e_{2}} e_{2} & = \frac{2}{z}(e_{3} − ξ), & ˜∇_{e_{3}} e_{2} = 0, \\ ˜∇_{e_{1}} e_{1} & = \frac{2}{z}(e_{3} − ξ), & ˜∇_{e_{2}} e_{1} = 0 \\ ˜∇_{e_{3}} e_{1} & = 0. \end{aligned}

From (9.5), we can see that ˜∇_{e_{i}} e_{j} = 0 (1 ⩽ i, j ⩽ 3) for ξ = e_{3} and f = −\frac{2}{z}. Hence M is a 3-dimensional f-Kenmotsu manifold with respect to the Schouten–van Kampen connection. Also using (9.4), it can be seen that R̃ = 0. Thus the manifold M is a flat manifold with respect to the Schouten–van Kampen connection. Since a flat manifold is a Ricci-flat manifold with respect to the Schouten–van Kampen connection, the manifold M is both a projectively flat and a conharmonically flat 3-dimensional f-Kenmotsu manifold with respect to the Schouten–van Kampen connection. So, from Theorems 5.1 and 6.1, M is an η-Einstein manifold with respect to the Levi-Civita connection.

Acknowledgement. The author is grateful to the referees for their comments and valuable suggestions for improvement of this work.

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Education Faculty (Received 10 12 2015) Department of Mathematics (Revised 08 03 2017) Inonu University Malatya Turkey

 a.yildiz@inonu.edu.tr