1. Introduction

Friedmann and Schouten ^{[1, 2]} introduced the idea of semi-symmetric linear connection on a differentiable manifold. Hayden ^[3] introduced semi-symmetric metric connection on a Riemannian manifold and it was further developed by Yano ^[4]. Semi-symmetric connections play an important role in the study of Riemannian manifolds. There are various physical problems involving the semi-symmetric metric connection. For example, if a man is moving on the surface of the earth always facing one definite point, say Jaruselam or Mekka or the North pole, then this displacement is semi-symmetric and metric ^[1]. In 1975, Prvanovi´c ^[5] introduced the concept of semi-symmetric non-metric connection with the name pseudo-metric, which was further studied by Andonie ^{[6, 7]}. The study of semi-symmetric non-metric connection is much older than the nomenclature ’non-metric’ was introduced. In 1992, Agashe and Chafle ^[8] introduced a semi-symmetric connection satisfying on a Riemannian manifold, and called such a connection as semi-symmetric non-metric connection. Later, the curvature properties of the connection in an SP-Sasakian manifold were studied by Bhagwat Prasad ^[9], and many others.

On the other hand, in 1976, Sato ^[10] defined the notions of an almost paracontact Riemannian manifold. After that, T. Adati and K. Matsumoto ^[11] defined and studied para-Sasakian and SP-Sasakian manifolds which are regarded as a special kind of an almost contact Riemannian manifolds. Before Sato, in 1972, Kenmotsu ^[12] defined a class of almost contact Riemannian manifolds satisfying some special conditions. In 1995, Sinha and Sai Prasad ^[13] have defined a class of almost paracontact metric manifolds namely para Kenmotsu (briefly P-Kenmotsu) and special para Kenmotsu (briefly SP-Kenmotsu) manifolds.

In 1970, Pokhariyal and Mishra ^[14] have introduced new tensor fields, called W and E-tensor fields in a Riemannian manifold and studied their properties. In the present paper, we consider the W-curvature tensor of a semi-symmetric non-metric connection and obtained a relation connecting the curvature tensors of Mn with respect to semi-symmetric non-metric connection and the Riemannian connection. It is proved that in an SP-Kenmotsu manifold if the curvature tensor of the semi-symmetric non-metric connection vanishes then the manifold is protectively flat.

Let M_n be an n-dimensional differentiable manifold equipped with structure tensors (, ξ, η) where is a tensor of type (1, 1), ξ is a vector field, η is a 1-form such that

(1.1)

(1.2)

Then M_n is called an almost paracontact manifold.

Let g be the Riemannian metric in an n-dimensional almost paracontact manifold M_n such that

(1.3)

(1.4)

(1.5)

for all vector fields X and Y on M_n. Then the manifold M_n ^[10] is said to admit an almost paracontact Riemannian structure (, ξ, η, g) and the manifold is called an almost paracontact Riemannian manifold.

A manifold M_n with Riemannian metric ‘g’ admitting a tensor field of type (1, 1), a vector field ξ and 1-form η satisfying equations (1.1), (1.3) along with

(1.6)

(1.7)

(1.8)

is called a para Kenmotsu manifold or briefly P-Kenmotsu manifold ^[13], where is the covariant differentiation with respect to g.

It is known that ^[13] on a P-Kenmotsu manifold the following relations hold:

(1.9)

(1.10)

where R is the Riemannian curvature.

Let (M_n, g) be an n-dimensional Riemannian manifold admitting a tensor field of type (1, 1), a vector field ξ and 1-form η satisfying

(1.11)

(1.12)

is called a special para Kenmotsu manifold or briefly SP-Kenmotsu manifold ^[13].

A linear connection in a Riemannian manifold M_n is said to be semi-symmetric connection if its torsion tensor T satisfies

(1.13)

A semi-symmetric non-metric connection in an almost paracontact metric manifold with torsion tensor (1.13) is given by

(1.14)

where is a Riemannian connection with respect to metric g ^[8].

Apart from conformal curvature tensor, the projective curvature tensor is an other important tensor from the differential geometric point of view. The Weyl-projective curvature tensor W of type (1, 3) of a Riemannian manifold M_n with respect to the Riemannian connection is defined by ^[14]

(1.15)

for X, Y, Z where R is the curvature tensor and Ric is the Ricci tensor. If there exists a one-to-one correspondence between each coordinate neighbourhood of a Riemannian manifold M_n and a domain in Eucledian space such that any geodesic of the Riemannian manifold corresponds to a straight line in the Eucledian space, then M_n is said to be locally projectively flat. For M_n is locally projectively flat if and only if the projective curvature tensor W vanishes. For n = 2, the projective curvature tensor identically vanishes.

2. Curvature Tensor

The manifold M_n is considered to be an SP-Kenmotsu manifold. Let us denote the curvature tensor of the semi-symmetric non-metric connection by and the curvatre tensor of by R. By straight forward calculation, we get

(2.1)

As a consequence of equations (1.11) and (1.14), equation (2.1) reduces to

(2.2)

which is the relation between the curvature tensors of M_n with respect to the semi-symmetric non-metric connection and the Riemannian connection .

It is well known that a Riemannian manifold is of constant curvature if and only if it is projectively flat or conformally flat ^[15] and in general, the necessary and sufficient condition for a manifold with a symmetric linear connection to be projectively flat is that the projective curvature tensor with respect to it vanishes identically on a manifold ^[16].

As an example, if M_n is a Riemannian manifold with vanishing curvature tensor with respect to semi-symmetric non-metric connection, then M_n is projectively flat ^[8]. Analogus to this, we prove the following for an SP-Kenmotsu manifold which is Riemannian.

Theorem 2.1: If in an SP-Kenmotsu manifold M_n the curvature tensor of a semi-symmetric non-metric connection vanishes, then the manifold is projectively flat.

Proof: Since = 0, then equation (2.2) gives

(2.3)

On contracting the above equation, we get

(2.4)

Then, by (2.3) and (2.4), we get

(2.5)

or W = 0 from (1.15), proves that the manifold is projectively flat.

Theorem 2.2: If in an SP-Kenmotsu manifold the Ric tensor of a semi-symmetric non-metric connection vanishes, then the curvature tensor of is equal to the projective curvature tensor of the manifold M_n.

Proof: From equation (2.2), we have

(2.6)

On contracting the above equation, we get

(2.7)

Since = 0, we have

(2.8)

From equations (2.2) and (2.8), we have = W.

Theorem 2.3: In an SP-Kenmotsu manifold the projective curvature tensor of a semi-symmetric non-metric connection is equal to the projective curvature tensor of the manifold.

Proof: From equations (2.2) and (2.7), we get

(2.9)

The terms of the equation (2.9) can be rearranged as

(2.10)

which is ‘W = W, where ‘W is the Weyl projective curvature tensor with respect to the semi-symmetric non-metric connection.

Theorem 2.4: In an SP-Kenmotsu manifold with semi-symmetric non-metric connection we have

a)

b)

Proof: Using the Bianchi’s first identity with respect to the Riemannian connection equation (2.2) gives (a). From equation (2.6) we get (b).

Acknowledgement

The authors acknowledge Prof. Kalpana, Banaras Hindu University, Dr. B. Satyanarayana of Nagarjuna University and Dr. A. Kameswara Rao, G.V.P. College of Engineering for Women for their valuable suggestions in preparation of the manuscript. They are also thankful to the referee for his valuable comments in the improvement of this paper.

Competing Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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