QUASI CONFORMALLY SYMMETRIC WEYL MANIFOLDS

The present paper deals with a type of non-quasi flat Weyl manifold which is called quasi conformally symmetric. Some necessary and sufficient conditions between concircularly symmetric, conformally symmetric and quasi conformally symmetric Weyl manifolds are obtained and a special condition on the Weyl manifold admitting a semi-symmetric non-metric connection is studied. AMS Subject Classification: 53C05, 53C25, 53C35


Introduction
The physical concepts in Weyl geometry has been widely reviewed in the review paper (see [9]). Herman Weyl first introduced a gauge invariant theory to unify gravity and electromagnetic theories in 1918. This theory is not acceptible as a unified theory since the electromagnetic potential does not couple to spinor being essential for the electromagnetic theory. This does not mean that Weyl geometry has a physical meaning as well in different theories and in the second part of the 20th century, the Weyl geometry has been studied in some research filds of physics such as quantum mechanics, particle physics, gravity and cosmology. Despite the unified theory of Weyl not being acceptable as physical theory, it introduced a useful theory in differential geometry. The mathematics of the theory is a generalization of the Riemannian geometry and the connection is an instructive example of non-metric connections.
A Weylian metric on a differentiable manifold M can be given by pairs (g, ϕ) of a non-degenerate symmetric differential 2-form g and a differential 1-form ϕ. The Weylian metric consists of the equivalance class of such pairs, with ∼ g, ∼ ϕ ∼ (g, ϕ) if and only if for a strictly positive real function λ > 0 on M. Choosing a representative means to gauge the Weylian metric; g is then the Riemannian component and ϕ the scale connection of the gauge. A change of representative (1) is called a Weyl(scale) transformation; it consists of a rescaling (i) and a scale gauge transformation (ii). A manifold with a Weylian metric (M, g, ϕ) will be called a Weyl manifold.
In the recent mathematical literature, a Weyl structure on a differentiable manifold M is spesified by a pair (g, ϕ) consisting of a conformal metric g and an affine, which is torsion-free, connection Γ, respectively its covariant derivative ∇. For any conformal metric g, there is a differential 1-form ϕ g such that ∇g + 2ϕ g ⊗ g = 0 (2) which is called weak compatibility of the affine connection with the metric. (2) can be expressed in local coordinates by where T k is a complementary covariant vector field (see [7]). Such a Weyl manifold will be denoted by W n (g ij , T k ). If T k = 0 or T k is gradient, a Riemannian manifold is obtained. One could also formulate above compatibility condition by where 1 denotes identity in Hom(V, V ) for every V = T x M , ϕ * g is the dual of ϕ g with respect to g and g Γ is the Levi-Civita connection of g. In local coordinates, (4) is given by where ijk's are the coefficients of the Levi-Civita connection (see [7]). The curvature tensor R h ijk of the symmetric connection Γ on the Weyl manifold is defined by In local coordinates, the conformal curvature tensor C (X, Y, Z, U ) and the concircular curvature tensor C (X, Y, Z, U ) of the symmetric connection Γ on the Weyl manifold are expressed by with the help of (6), where R h ijk , R ij and r denote the curvature tensor, the Ricci tensor and the scalar curvature tensor of Γ, respectively (see [5], [8]).
Let {e i } n i=1 be an orthonormal basis of the tangent space at any point of the manifold. Then, from (7) and (8), and where S (Y, Z) denotes the Ricci tensor of type (0, 2). From (10), it follows that n i=1 P (e i , e i ) = 0.
By using (7), and (8), the conformal curvature tensor C h ijk of the connection Γ is expressed in terms of the concircular curvature tensor C h ijk by the following equation:

Quasi Conformally Symmetric Weyl Manifolds
In 1968, Yano and Sawaki defined and studied a new curvature tensor called quasi conformal curvature tensor on a Riemannian manifold of dimension n which includes both the conformal and concircular curvature tensor (see [14]). The quasi conformal curvature tensor W (X, Y, Z, U ) of type (0, 4) on a Weyl manifold of dimension n (n > 3) is defined by where a, b are arbitrary constants not simultaneously zero, C and C are conformal curvature tensor and concircular curvature tensor of type (0, 4), respectively.
Substituting (9) and (10) in (12) yields In particular, if a = 1 and b = −1 n−2 , then the quasi conformal curvature tensor reduces to conformal curvature tensor. In a similar way, if a = 1 and b = 0, then the quasi conformal curvature reduces to concircular curvature tensor.
By substituting (7) and (8) in (12), the quasi conformal curvature tensor can be expressed by coordinates. An alternative definition of the quasi conformal curvature tensor is given in the form of rkm by using (8) and (10) in (14).
The study of symmetric Riemannian manifolds began with the work of E.Cartan. A Riemannian manifold (M n , g) is said to be locally symmetric due to Cartan (see [1]), if its curvature tensor R (Y, Z, U, V ) satisfies the condition (∇ X R) (Y, Z, U, V ) = 0, where ∇ denotes the operator of covariant differentiation with respect to the metric tensor g. This condition of local symmetry is equivalent to the fact that at every point P ∈ M n , the local geodesic symmetry F (P ) is an isometry. The class of symmetric Riemannian manifolds is very natural generalization of the class of the manifolds of constant curvature.
During the last five decades, the notion of locally symmetric manifolds has been weakened by many authors in several ways to a different extent such as recurrent manifold by A.G.Walker (see [13]), semi-symmetric manifold by Z.I.Szabo (see [10]), pseudo symmetric manifold by M.C.Chaki (see [2]), generalized pseudo symmetric manifold by M.C.Chaki (see [3]) and weakly symmetric manifold and weakly projectively symmetric manifold by L.Tamassy and T.Q.Binh (see [11]). On the other hand M.C.Chaki defined conformally symmetric manifold with B.Gupta in (see [4]).
The object of the present paper is to study a quasi conformally symmetric Weyl manifold which is defined by the following definition and is denoted by (W SW ) n : Definition 2.1. A non-flat Weyl manifold is said to be conformally symmetric, concircularly symmetric and quasi conformally symmetric, if the conformal curvature tensor C (Y, Z, U, V ), the concircular curvature tensor C (Y, Z, U, V ) and the quasi conformal curvature tensor W (Y, Z, U, V ) satisfy the conditions respectively, for all vector fields X, Y, Z, U, V ∈ χ (M ) which denotes the Lie algebra of all smooth vector fields on the manifold M and ∇ is the operator of covariant differentiation with respect to the Weylian metric g. Furthermore, in local coordinates, the same conditions are given by respectively, where ∇ denotes the covariant derivative with respect to the symmetric connection Γ.
The paper is organized as follows: Section 2 deals with some basic results of (W SW ) n . Section 3 gives some necessary and sufficient conditions on quasi conformally symmetric Weyl manifolds. Finally, in the last section a special condition is studied on a (W SW ) n admitting a semi symmetric non-metric connection.
Proof. Assume that a Weyl manifold is concircularly symmetric. By differentiating (11) covariantly with respect to ∇ and using ∇ l C h ijk = 0 or equivalently ∇ l C ij = 0 from Definition 2.1, By substituting the equations ∇ k g ij = 2T k g ij and ∇ k g ij = −2T k g ij by [7] into the formula (16), it is obtained as which states that a Weyl manifold is conformally symmetric. Then, by means of the covariant derivative of (12), it is seen that the manifold is quasi conformally symmetric.
Now, suppose that a Weyl manifold be quasi conformally symmetric. Then, by taking derivative of (12) covariantly and using Definition 2.1 , it is obtained as From (17), we have: Corollary 2.1. A (W SW ) n for which a + (n − 2) b = 0 and b = 0 is conformally symmetric.
Corollary 2.2. The conformal curvature tensor C (Y, Z, U, V ) and the concircular curvature tensor C (Y, Z, U, V ) of a (W SW ) n for which a+(n − 2) b = 0 and a = 0 satisfy the following condition: Corollary 2.3. A (W SW ) n for which a + (n − 2) b = 0 and b = 0 is concircularly symmetric (and conformally symmetric).

Some Necessary and Sufficient Conditions On a Quasi Conformally Symmetric Weyl Manifold
The quasi conformal curvature tensor W (X, Y, Z, U ) contains both the concircular curvature tensor C (X, Y, Z, U ) and the conformal curvature tensor C (X, Y, Z, U ). Therefore, in this section, we will improve some necessary and sufficient conditions for a (W SW ) n to be concircularly symmetric and conformally symmetric, respectively.
Theorem 3.1. A necessary and sufficient condition for a conformally symmetric Weyl manifold to be concircularly symmetric is that it is concircularly Ricci symmetric.
Proof. Now, assume that a Weyl manifold is conformally symmetric. Then, by taking covariantly derivative of (7) with respect to ∇, using ∇ l C h ijk = 0 from Definition 2.1 and remembering that R ij = R (ij) + R [ij] , it is obtained that Under consideration of a concircularly Ricci symmetric Weyl manifold, ∇ l C ij = 0 implies ∇ l R ij = (∇ l + 2T l ) r n g ij by (10) and therefore ∇ l R [ij] = 0. So, (19) reduces to which is equivalent to ∇ l C h ijk = 0 is obtained by (8). This means that the Weyl manifold is concircularly symmetric.
Conversely, a conformally symmetric Weyl manifold which is concircularly symmetric is automatically concircularly Ricci symmetric.
By using Theorem 2.1 and Theorem 3.1, we have: Corollary 3.1. A conformally symmetric Weyl manifold which is concircularly Ricci symmetric is quasi conformally symmetric .
Theorem 3.2. A necessary and sufficient condition for a (W SW ) n for which a + (n − 2) b = 0 and b = 0 to be concircularly symmetric is that it is concircularly Ricci symmetric.
Proof. Suppose that a (W SW ) n for which a + (n − 2) b = 0 and b = 0 is concircularly symmetric. In this case, the manifold is automatically concircularly Ricci symmetric.
Conversely, if a (W SW ) n for which a + (n − 2) b = 0 and b = 0 which is conformally symmetric by Corollary 2.1 is concircularly Ricci symmetric, then the manifold is concircularly symmetric by Theorem 3.1. Theorem 3.3. A necessary and sufficient condition for a (W SW ) n for which a + (n − 2) b = 0 for a ∈ R, b = 0 to be conformally symmetric (concircularly symmetric) is that it is concircularly symmetric (conformally symmetric).
Proof. Firstly, we have to say that a ∈ R is considered as the union of a = 0 and a = 0. Let's consider a (W SW ) n for which a + (n − 2) b = 0 for a ∈ R, b = 0. Under consideration of conformally symmetric Weyl manifold, (W SW ) n is obtained as concircularly symmetric by Corollary 2.2, Corollary 2.4 and Theorem 3.1.
Conversely, under consideration of concircularly symmetric Weyl manifold, (W SW ) n for which a+(n − 2) b = 0 for a ∈ R, b = 0 is obtained as conformally symmetric by taking derivative of (11) covariantly with respect to ∇.

A (W SW ) n Admitting a Semi-Symmetric Non-Metric Connection With a Special Condition
A generalized connection ∇ on the Weyl manifold is defined by V.Murgescu [6] as follows: where a jkh = g jr Ω r kh +g rk Ω r jh +g rh Ω r jk and Γ i jk 's are the coefficients of the symmetric connection ∇ defined in Section 1.
By choosing (20), the following equation denotes the coefficients Γ i jk 's of a semi symmetric non-metric connection ∇ on the Weyl manifold: (see [12]) where S i = −2a i and a i is an arbitrary covariant vector field. The curvature tensor R mijk of the semi symmetric non-metric connection ∇ on the Weyl manifold is defined by where S ij = S i,j −S i S j + 1 2 g ij g kr S k S r and S i,j denotes the covariant derivative of S i with respect to the symmetric connection ∇. By means of (22), the Ricci tensor R ij and the scalar curvature r of the connection ∇ are obtained as r = r + 2 (n − 1) S.
Theorem 4.1. The quasi conformal curvature tensor of a Weyl manifold with a vanishing curvature tensor with respect to a semi symmetric non-metric connection ∇ is of the form W (X, Y, Z, U ) = [a + (n − 2) b] C (X, Y, Z, U ) . (25) Proof. Suppose that the curvature tensor R mijk with respect to ∇ vanishes on the manifold, i.e., R mijk = 0. Then the curvature tensor R mijk of the connection ∇ is in the form of Furthermore, from (26), we have R h hjk = 0 which means that R kj is symmetric. By using this fact and (26) in (7), the conformal curvature tensor C mijk of the connection ∇ is obtained as C mijk = 0.
By taking covariant differentiation of (25), under consideration of a quasi conformally symmetric Weyl manifold, it is obtained that [a + (n − 2) b] (∇ X C) (Y, Z, U, V ) = 0 which leads us the following: Corollary 4.1. A (W SW ) n for which a + (n − 2) b = 0 for any a, b with a vanishing curvature tensor with respect to a semi symmetric non-metric connection ∇ is concircularly symmetric.