WEYL MANIFOLDS WITH SEMI-SYMMETRIC CONNECTION

We define a semi-symmetric connection on a Weyl manifold and study projective curvature tensor and conformal curvature tensor after giving some properties of the curvature tensor with respect to semi-symmetric connection. KeywordsWeyl manifold, semi-symmetric connection, curvature tensor, group manifold.

In this paper, we define a semi-symmetric connection on a Weyl manifold and define the curvature tensor with respect to semi-symmetric connection.We give some theorems by means of a relation between curvature tensors with respect to semisymmetric connection and symmetric connection.After defining projective curvature tensor and conformal curvature tensor with respect to semi-symmetric connection,we obtain some theorems by using properties of these tensors.In the last section of the paper, with the help of [5] , we examine Weyl group manifolds.

THE CURVATURE TENSOR WITH RESPECT TO SEMI-SYMMETRIC
CONNECTION An n-dimensional manifold which has a symmetric connection and a conformal metric tensor g ij is said to be Weyl manifold , if the compatible condition is in the form of = 0.In this case, Weyl manifold is denoted by n W ( ij g , k T ).k T , which is a covariant vector, is called complementary vector of the manifold.If k T = 0 or k T is gradient, then Riemannian manifold is obtained.A generalized connection on a Weyl manifold is given by [6] i jk If Ω jk i is chosen by

1), then a semi-symmetric connection on a
Weyl manifold is defined by The torsion tensor T jk i with respect to semi-symmetric connection is defined by Analogous to the definition of the curvature tensor with respect to symmetric connection, we define the curvature tensor with respect to semi-symmetric connection by Remembering the definition of the curvature tensor with respect to symmetric connection and using (2.2) and (2.4), we have: The tensor S ij in (2.5) is defined by

Proof:
It is shown easily by using (2.6).
Theorem 2.1: The curvature tensor with respect to semi-symmetric connection has the following properties: Corollary 2.1: If S i is gradient on a Weyl manifold with semi-symmetric connection, then the followings hold: Theorem 2.2: If the curvature tensors with respect to semi-symmetric and symmetric connection coincide, then S i is gradient.
) is multiplied by g ij , S = 0 (2.12)By using (2.12) in (2.11), it is found that S ij = 0 .This shows that S i is gradient.

Definition 2.1:
A semi-symmetric connection is said to be local-flat, if the curvature tensor with respect to semi-symmetric connection.
Theorem 2.3: If the semi-symmetric connection defined on a Weyl manifold is localflat, then T k is gradient.Proof: From (2.7) and Definition 2.1 , (2.14) shows that T k is gradient.

THE PROJECTIVE CURVATURE TENSOR WITH RESPECT TO SEMI-SYMMETRIC CONNECTION A generalized connection on a non-Riemannian manifold is denoted by L
In this case the projective curvature tensor with respect to a generalized connection on a non-Riemannian manifold is defined in [ ] 7 by where and for the semisymmetric connection on a Weyl manifold, we define B jkl i with respect to semisymmetric connection as follows: where jk S = 1 2 By making contraction on the indices i and l in (3.2), we get: The equation by changing indices j and k in By using (3.2), (3.3) and (3.4) are used in (3.1), we obtain projective curvature tensor with respect to semi-symmetric connection on a Weyl manifold as follows: i jkl where jl H Theorem 3.1: Projective curvature tensor with respect to semi-symmetric connection has the following properties: Corallary 3.1: jk W = 0 if and only if S k is gradient.
Theorem 3.2: Projective curvature tensors with respect to semi-symmetric connection and symmetric connection are related with the following relation: Proof: By choosing S k = 0 in (3.5), the projective curvature tensor with respect to symmetric connection is obtained by where By virtue of (2.5),(3.5)and (3.7), (3.6) is obtained.
Definition 3.1: A semi-symmetric connection is said to be projectively-flat, if the projective curvature tensor with respect to semi-symmetric connection vanishes.Theorem 3.3: If a semi-symmetric connection is local-flat and S k is gradient, then the connection is also projectively flat.Proof: By using Definition 2.1 and [ ] ∇ k l S = 0, the required result is found.

THE CONFORMAL CURVATURE TENSOR WITH RESPECT TO SEMI-SYMMETRIC CONNECTION
Under conformal transformation, the metric tensor g ij and the semi-symmetric connection i jk Γ on a Weyl manifold are transformed as follows: where Consequently from (4.5), where W m m = g W mk mk .From (4.5), we have ij W defined by the following eguation by (4.7 On the other hand, if (4.4) is multiplied by By using (4.9) in (4.7), we obtain: Putting (4.9) and (4.10) in (4.4); we have: where C mijk is defined by where and Multiplying (4.12) by g mh ; we find the conformal curvature tensor with respect to semisymmetric connection in the following form: where and Theorem 4.1: The conformal curvature tensor with respect to semi-symmetric connection has the following properties:

Proof:
(i) With the help of (4.12), it is written the definition of C mijk .By adding this and (4.12), the required result is obtained.(ii) If the process in (i) is applied to the first two indices in (ii), the result is found.(iii) It is found by contraction on the indices h and i in (4.13).(iv) The same process is applied to the indices h and k in (4.13).(v) By changing cyclically the indices i,j and k , three equations are written.And by adding these equations, the required result is obtained.
Theorem 4.2: Conformal curvature tensors with respect to symmetric connection and semi-symmetric connection on a Weyl manifold coincide.
Proof: Conformal curvature tensor with respect to symmetric connection is defined with the help of (4.13) by where . Using (2.5) in (4.13) and with the help of (4.14), the required result is obtained.

2 λ
under the transformation of the metric tensor g ij in the form of ij g ˆ= ij g with λ is a point function.According to this transformation, if the quantity A changes by A ˆ= A k λ , then the quantity A is called a satellite of g ij with the weight of { } generalized covariant derivative of A.In this definition, A s ∇ denotes ordinary covariant derivative of A.
where Γ jk i is symmetric part and Ω jk i is anti-symmetric part , and the curvature tensor with respect to this connection is given by L jkl i = B jkl i with respect to symmetric part of the generalized connection.B jkl i is given with the following definition: is obtained by adding (3.5) and the relation by changing the indices j and k in (3.5).(ii) By contraction on the indices i and j in (3.5), the required result is obtained.(iii) It is easily seen by contraction on the indices i and l in (3.5).(iv) I.Bianchi identity for projective curvature tensor is shown by changing the indices i,j and k cyclically and using (3.5).

2 )
whereP k − * k P = T k and S k − * k S = Q k .With the help of (4.1) and (4.2), we define the change of curvature tensor under conformal transformation with respect to semi-symmetric connection by * h ijk

Definition 4 . 1 :
A semi-symmetric connection is called conformal-flat, if the conformal curvature tensor with respect to semi-symmetric connection vanishes.Definition 4.2: A manifold is called conformal-flat, if the conformal curvature tensor with respect to symmetric connection vanishes.Corallary 4.1: Semi-symmetric connection defined on a Weyl manifold is conformalflat if and only if the Weyl manifold is conformal-flat.Theorem 4.3: If semi-symmetric connection defined on a Weyl manifold is local-flat, then it is also conformal-flat.Proof: R mijk = 0 implies ij R =0 and R = 0.If the results are used in (4.13), we find h ijk C = 0 meaning the connection is conformal-flat.5.WEYL GROUP MANIFOLDS Definition 5.1:If the curvature tensor with respect to semi-symmetric connection and the tensor ij S vanish on a Weyl manifold, then that one is called Weyl group manifold.By above definition we have the following lemmas: Lemma 5.1: Weyl group manifold is local-flat.Lemma 5.2: i S is gradient on a Weyl group manifold.Theorem 5.1: Projective curvature tensors with respect to symmetric connection and semi-symmetric connection coincide on a Weyl group manifold.Proof: By using Definition 5.1 and Lemma 5.1 in (3.7), the required result is obtained.Corallary 5.1:Semi-symmetric connection defined on a Weyl group manifold is projectively-flat if and only if Weyl group manifold is projectively-flat.

Theorem 5 . 2 : 5 . 4 : 2 :
A Weyl group manifold is conformal-flat.Proof: If the conditions R ijk h = 0, ij S = 0, h ijk R = 0 are used in (4.14), the result is found.Theorem 5.3: A Weyl group manifold is projectively-flat.Proof: If the same conditions in the proof of Theorem 5.2 are used in (3.8),the required result is obtained.Theorem Conformal curvaure tensor and projective curvature tensor with respect to semi-symmetric connection coincide on a Weyl group manifold.Proof: Conformal curvature tensor, h ijk C , is defined in terms of the projective curvature tensor with respect to symmetric connection by the following relation: Semi-symmetric connection defined on a Weyl group manifold is conformal flat if and only if it is projectively-flat.
R mijk =R mijk +g mk S ij −g mj S ik +g ij S mk −g ik S mjThe tensor S ij is symmetric if and only if S i is gradient.