Semi-Symmetric Metric Connection on Homothetic Kenmotsu Manifolds

The object of the paper is to study homothetic Kenmotsu manifold with respect to semisymmetric metric connection. We discuss locally φ-symmetric homothetic Kenmotsu manifold and ξ-projectively flat homothetic Kenmotsu manifold with respect to semisymmetric metric connection. Finally, we construct an example of 3-dimensional homothetic Kenmotsu manifold to verify some results.

In 1924, Friedman and Schouten [6] introduced the idea of semi-symmetric linear connection on a differentiable manifold. In this continuation, Hayden [7] introduced a semi-symmetric metric connection on a Riemannian manifold. A linear connection ∇ in an -dimensional differentiable manifold is said be a semi-symmetric connection [6] if its torsion tensor , = ∇ − ∇ − [ , ] satisfies the following condition , = − ( ) , (1.1) where is a 1-form associates with vector field as = ( , ), (1.2) for all vector field , ∈ , ( ) is the set of all differentiable vector field on .
In addition to (1.1), a semi-symmetric connection ∇ is called semi-symmetric metric connection [7], if it satisfies ∇ , = 0, (1.3) A relation between the semi-symmetric metric connection ∇ and the Levi-Civita connection ∇ on , is given by [8] We also have Let be the curvature tensor of semi-symmetric metric connection ∇ given by where is the Riemannian curvature tensor of connection ∇, Θ is a tensor field of type (0, 2) and is a tensor field of type (1,1) is given by The semi-symmetric metric connection in a Kenmotsu manifold was studied by Tripathi [9], Bagewadi et al. [10], De and Pathak [11]. In a recent paper Barman [12] studied the semi-symmetric metric connection in Lorentzian -Sasakian manifold. The calculation used to find the results of semi-symmetric metric connection was also used by Eshghi [13] and Moniruzzaman et al. [14] in tensor notation form.

Homothetic Kenmotsu Manifolds
A 2 + 1 -dimensonal differentiable manifold 2 +1 is said to be an almost contact metric manifold [15] if it admits a 1, 1 tensor field , a vector field , a 1-form and a Riemannian metric which satisfies where is the Ricci tensor of the Levi-Civita connection and is the Ricci operator.

Curvature Tensor of A Homothetic Kenmotsu Manifold with respect to the Semi Symmetric Metric Connection
Using ( 3) which is the relation between curvature tensors connection ∇ and ∇. Taking the inner product of (3.3) with , we get ′ , , , = ′ , , , Thus we can state the following theorem: Theorem 3.1. In a homothetic Kenmotsu manifold 2 +1 with respect to the semisymmetric metric connection ∇ (i): the curvature tensor is given by Taking the cyclic permutation of , and in (3.3), we get , = ,

7) and
, = ,   (3.14) Using the fact ′ , , , − ′ , , , = 0 in the subtraction of (3.14) from Thus in the view of (3.11), (3.13) and (3.15), we can state the following: Thus from above, we can state the following: Theorem 3.5. If the Ricci tensor of a semi-symmetric metric connection ∇ in a homothetic Kenmotsu manifold vanishes, then the manifold 2 +1 is -Einstein manifold. Thus we can state the following: Theorem 4.1. A 2 + 1 -dimensional homothetic Kenmotsu manifold is locallysymmetric with respect to the semi-symmetric metric connection if and only if the manifold is also locally -symmetric with respect to the Levi-Civita connection.

-Projectively Flat Homothetic Kenmotsu Manifolds with respect to the Semi-Symmetric Metric Connection
Definition 5.1. The pseudo projective curvature tensor in an almost metric manifold 2 +1 with respect to Levi-Civita connection is defined as follows [17] , = , − The projective curvature tensor is an important tensor in differential geometry. In a Riemannian manifold 2 +1 , if there exist a one-to-one correspondence between each coordinate neighborhood of 2 +1 and a domain in Euclidean space such that any geodesic of the Riemannian manifold corresponds to a straight line in the Euclidean space then 2 +1 is said to be locally projectively flat. For recent developments on projective curvature tensor, we refer [18,19]. where and are the Riemannian curvature tensor and Ricci tensor with respect to the semi-symmetric metric connection ∇ respectively. This notion was first defined by Tripathi and Dwivedi [20]. A manifold 2 +1 is called horizontal -projectively flat manifold if (5.2) holds for , orthogonal to . Using If we consider , orthogonal to , then (5.5) reduces to , = 0, we called such a manifold a horizontal -projectively flat manifold. Hence we state the following: On taking , orthogonal to in (5.7), we get , = ( , ) . (5.8) Hence we state the following theorem: Theorem 5.4. A 2 + 1 -dimensional homothetic Kenmotsu manifold is horizontalprojective flat with respect to the semi-symmetric metric connection if and only if the manifold is -projectively flat with respect to the Levi-Civita connection.

Example
In this section we consider coordinate space 3 (with coordinate , , ) and calculate the components of curvature tensor with respect to semi-symmetric metric connection, then we verified the theorem (3.2) and (3.5) Let us consider the 3-dimensional manifold = ( , , ) ∈ 3 where ( , , ) are the standard coordinate of 3 . Let 1 , 2 , 3 are linearly independent vector fields at each point of and given by where is constant. Let be the Riemannian metric defined by Using above Koszul formula for the Riemannian metric , we can easily calculate (6.6) From above it can be easily seen that the manifold ( , , , ) satisfies the condition where is constant. Hence the structure ( , , , ) is homothetic Kenmotsu structure and the manifold equipped with homothetic Kenmotsu structure is a homothetic Kenmotsu manifold. By the use of (6.6) in (1.4), we obtain ∇ 1 1 = − 1 3 , ∇ 1 2 = 0, ∇ 1 3 = − − 1 1 , ∇ 2 1 = 0, ∇ 2 2 = − 1 3 , ∇ 2 3 = − − 1 2 , ∇ 3 1 = 0, ∇ 3 2 = 0, ∇ 3 3 = 0. (6.7) Therefore, the manifold is a homothetic Kenmotsu manifold with respect to the semisymmetric metric connection. By using the equation (6.6), we can obtain the component of the curvature tensor with respect to Levi-Civita connection as follows: Also the curvature tensor with respect to semi-symmetric metric connection can be obtained by using (6.7) as follows: From the above results of the curvature tensor it follows that 2 ∇ , = 2 ∇ , = 0. Therefore, this example supports the theorem (4.1).
Using the expression of the curvature tensors with respect to the semi-symmetric metric connection we can obtain the Ricci tensors as follows: Let and are any two vector fields given by = 1 1 + 2 2 + 3 3 and = 1 1 + 2 2 + 3 3 .
Using (6.11) and above relations, we get , 3 = 0. Therefore, the manifold is -projectively flat on a homothetic Kenmotsu manifold with respect to the semi-symmetric metric connection which verifies the Theorem 5.1.

Conclusion
It has been seen that at each point of a differentiable manifold there is an -dimentional tangent space and any two tangent space and at point and respectively are isomorphic. However, in order to obtain a definite isomorphism relating and it is necessary to introduce some additional structure on the manifold called a connection which connect the different points on the manifold. In this paper we find the curvature tensor, Ricci tensor, scalar curvature tensor and other properties of homothetic Kenmotsu manifold with respect to semi-symmetric metric connection. The result of this paper will be helpful for researchers to develop the new connection on homothetic Kenmotsu manifold. The given example of homothetic Kenmotsu manifold of this paper will be used in the study of Relativity and Quantum mechanics.