Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions

We study Riemannian manifolds M admitting a semi-symmetric metric connection ∇̃ such that the vector field U is a parallel unit vector field with respect to the Levi-Civita connection ∇. We prove that R · R̃ = 0 if and only if M is semisymmetric; if R̃ ·R = 0 or R · R̃− R̃ ·R = 0 or M is semisymmetric and R̃ · R̃ = 0, then M is conformally flat and quasi-Einstein. Here R and R̃ denote the curvature tensors of ∇ and ∇̃, respectively.


INTRODUCTION
Let ∇ be a linear connection in an n-dimensional differentiable manifold M. The torsion tensor T is given by The connection ∇ is symmetric if its torsion tensor T vanishes, otherwise it is non-symmetric.If there is a Riemannian metric g in M such that ∇g = 0, then the connection ∇ is a metric connection, otherwise it is non-metric [19].It is well known that a linear connection is symmetric and metric if and only if it is the Levi-Civita connection.
Hayden [9] introduced a metric connection ∇ with a non-zero torsion on a Riemannian manifold.Such a connection is called a Hayden connection.In [8,13], Friedmann and Schouten introduced the idea of a semisymmetric linear connection in a differentiable manifold.A linear connection is said to be a semi-symmetric connection if its torsion tensor T is of the form where the 1-form ω is defined by ω(X) = g(X,U), and U is a vector field.In [12], Pak showed that a Hayden connection with the torsion tensor of the form ( 1) is a semi-symmetric metric connection.In [18], Yano considered a semi-symmetric metric connection and studied some of its properties.He proved that in order that a Riemannian manifold admits a semi-symmetric metric connection whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian manifold be conformally flat.For some properties of Riemannian manifolds with a semi-symmetric metric connection see also [1,4,10,16,17].
In [14,15], Szabó studied semisymmetric Riemannian manifolds, that is Riemannian manifolds satisfying the condition R • R = 0.It is well known that locally symmetric manifolds (i.e.Riemannian manifolds satisfying the condition ∇R = 0) are trivially semisymmetric.But the converse statement is not true.According to Szabó, many geometrists have studied semisymmetric Riemannian manifolds.
Motivated by the studies of the above authors, in this paper we consider Riemannian manifolds (M, g) admitting a semi-symmetric metric connection such that U is a unit parallel vector field with respect to the Levi-Civita connection ∇.We investigate the conditions where R and R denote the curvature tensors of ∇ and ∇, respectively.
The paper is organized as follows.In Section 2 and Section 3, we give the necessary notions and results which will be used in the next sections.In Section 4, we prove that then M is conformally flat and quasi-Einstein.

PRELIMINARIES
An n-dimensional Riemannian manifold (M, g), n > 2, is said to be an Einstein manifold if its Ricci tensor S satisfies the condition S = r n g, where r denotes the scalar curvature of M.
where a, b are scalars of which b = 0 and D is a non-zero 1-form, then M is called a quasi-Einstein manifold [3].For a (0, k)-tensor field T , k ≥ 1, on (M, g) we define the tensor R • T (see [5]) by If R • R = 0, then M is called semisymmetric [14].In addition, if E is a symmetric (0, 2)-tensor field, then we define the (0, k + 2)-tensor Q(E, T ) (see [5]) by where The Weyl tensor of a Riemannian manifold (M, g) is defined by where r denotes the scalar curvature of M. For n ≥ 4, if C = 0, the manifold is called conformally flat [19].Now we give the Lemmas which will be used in Section 4.
Lemma 2.1.[6] Let (M, g), n ≥ 3, be a semi-Riemannian manifold.Let at a point x ∈ M be given a nonzero symmetric (0, 2)-tensor E and a generalized curvature tensor B such that at x the following condition is satisfied: Q(E, B) = 0.Moreover, let V be a vector at x such that the scalar ρ = a(V ) is non-zero, where a is a covector defined by a(X)

SEMI-SYMMETRIC METRIC CONNECTION
If ∇ is the Levi-Civita connection of a Riemannian manifold M, then we have and X,Y,U are vector fields on M. Let R and R denote the Riemannian curvature tensor of ∇ and ∇, respectively.Then we know that [18] R(X,Y, Z,W where Now assume that U is a parallel unit vector field with respect to the Levi-Civita connection, i.e., ∇U = 0 and So θ is a symmetric (0, 2)-tensor field.Hence equation ( 5) can be written as where ∧ is the Kulkarni-Nomizu product, which is defined by Since U is a parallel unit vector field, it is easy to see that R is a generalized curvature tensor and it is trivial that R(X,Y )U = 0. Hence by a contraction we find S(Y,U) = ω(QY ) = 0, where S denotes the Ricci tensor of ∇ and Q is the Ricci operator defined by g(QX,Y ) = S(X,Y ).It is easy to see that we also have the following relations: θ 2 (X,Y ) : Using ( 7), (9), and (10), we get C = C, where S, C, and r denote the Ricci tensor, Weyl tensor, and the scalar curvature of M with respect to semisymmetric connection ∇.

MAIN RESULTS
The tensors R • R and Q(θ , T ) are defined in the same way with (3) and (4).Let (R • R) hi jklm and ( R • R) hi jklm denote the local components of the tensors R • R and R • R, respectively.Hence, we have the following proposition: Proposition 4.1.Let (M, g) be a Riemannian manifold admitting a semi-symmetric metric connection.If U is a parallel unit vector field with respect to the Levi-Civita connection ∇, then Proof.Since U is parallel, we have R • θ = 0.So from (7) we get Applying ( 5) in (3) and using (4), we obtain This completes the proof of the proposition.
As an immediate consequence of Proposition 4.1 we have the following theorem: Theorem 4.2.Let (M, g) be a Riemannian manifold admitting a semi-symmetric metric connection and U be a parallel unit vector field with respect to the Levi-Civita connection ∇.Then R • R = 0 if and only if M is semisymmetric.
Theorem 4.3.Let (M, g) be a semisymmetric n > 3 dimensional Riemannian manifold admitting a semisymmetric metric connection.If U is a parallel unit vector field with respect to the Levi-Civita connection ∇ and R • R = 0, then M is a conformally flat quasi-Einstein manifold.
Proof.Since the condition R • R = 0 holds on M, from Proposition 4.1 we have So we have two possibilities: Suppose that (15) holds at a point x.Thus we have where z ∈ T * x M and ρ ∈ R. Because of non-zero coefficient of g, this relation does not occur.Thus the case ( 16) must be fulfilled at x.By virtue of Lemma 2.1, (14) gives us So from Lemma 2.2 we obtain C = 0, which gives us that M is conformally flat.Moreover, contracting (14) with g i j , we get where λ = r n−1 : M → R is a function.So by virtue of (2), M is quasi-Einstein.Thus the proof of the theorem is completed.Theorem 4.4.Let (M, g) be a Riemannian manifold admitting a semi-symmetric metric connection.If U is a parallel unit vector field with respect to the Levi-Civita connection ∇ and R • R − R • R = 0, then M is a conformally flat quasi-Einstein manifold.
Proof.Using ( 11) and ( 12), we get Using the same method in the proof of Theorem 4.3, we obtain that M is conformally flat and quasi-Einstein.So we get the result as required.
Theorem 4.5.Let (M, g) be an n > 3 dimensional semisymmetric Riemannian manifold admitting a semisymmetric metric connection.If U is a parallel vector field with respect to the Levi-Civita connection ∇ and R • R = 0, then M is conformally flat and quasi-Einstein.
Proof.From (5) we have Using (8), equation ( 17) is reduced to We suppose that ( R • R) hi jklm = 0 and M is semisymmetric.Using the same method in the proof of Theorem 4.3, we obtain that M is conformally flat and quasi-Einstein.This proves the theorem.
The following example shows that there is a Riemannian manifold with a semi-symmetric metric connection having a parallel vector field associated to the 1-form satisfying R Example.Let M 2m+1 be a (2m + 1)-dimensional almost contact manifold endowed with an almost contact structure (ϕ, ξ , η), that is, ϕ is a (1,1)-tensor field, ξ is a vector field, and η is a 1-form such that Let g be a compatible Riemannian metric with (ϕ, ξ , η), that is, for all X,Y ∈ χ(M).Then M 2m+1 becomes an almost contact metric manifold equipped with an almost contact metric structure (ϕ, ξ , η, g).An almost contact metric manifold is cosymplectic [2] if ∇ X ϕ = 0. From the formula ∇ X ϕ = 0 it follows that ∇ X ξ = 0, ∇ X η = 0, and R(X,Y )ξ = 0.