In 1924, Friedmann and Schouten [4] introduced the idea of semi-symmetric connection on a differentiable manifold. A linear connection ∇̃ on a differentiable manifold M is said to be a semi-symmetric connection if the torsion tensor T of the connection ∇̃ satisfies T(X, Y) = u(Y)X − u(X)Y, where u is a 1-form and ξ₁ is a vector field defined by u(X) = g(X, ξ₁), for all vector fields X ∈ χ(M), χ(M) is the set of all differentiable vector fields on M.

In 1932, Hayden [9] introduced the idea of semi-symmetric metric connections on a differential manifold (M, g). A semi-symmetric connection ∇̃ is said to be a semi-symmetric metric connection if ∇̃g = 0.

After a long gap the study of a semi-symmetric connection ∇̄ satisfying

was initiated by Prvanović [14] with the name pseudo-metric semi-symmetric connection and was just followed by Andonie [16].

A semi-symmetric connection ∇̄ is said to be a semi-symmetric non-metric connection if it satisfies the condition (1.1).

In 1992, Agashe and Chafle [15] studied a semi-symmetric non-metric connection ∇̄, whose torsion tensor T̄ satisfies T̄(X, Y) = u(Y)X − u(X)Y and (∇̄_Xg)(Y, Z) = −u(Y)g(X, Z) − u(Z)g(X, Y) ≠ 0. They proved that the projective curvature tensor of the manifold with respect to these two connections are equal to each other. In 1992, Barua and Mukhopadhyay [5] studied a type of semi-symmetric connection ∇̄ which satisfies (∇̄_Xg)(Y, Z) = 2u(X)g(Y, Z) − u(Y)g(X, Z) − u(Z)g(X, Y). Since ∇̄ g ≠ 0, this is another type of semi-symmetric non-metric connection. However, the authors preferred the name semi-symmetric semimetric connection.

In 1994, Liang [24] studied another type of semi-symmetric non-metric connection ∇̄ for which we have (∇̄_Xg)(Y, Z) = 2u(X)g(Y, Z), where u is a non-zero 1-form and he called this a semi-symmetric recurrent metric connection.

The semi-symmetric non-metric connections was further developed by several authors such as De and Biswas [21], De and Kamilya [22], Liang [24], Singh et al. ([17, 18, 19]), Smaranda [7], Smaranda and Andonie [8], Barman ([1, 2, 3]) and many others.

A transformation of an n-dimensional differential manifold M, which transforms every geodesic circle of M into a geodesic circle, is called a concircular transformation ([12, 23]). A concircular transformation is always a conformal transformation [23]. Here geodesic circle means a curve in M whose first curvature is constant and whose second curvature is identically zero. Thus the geometry of concircular transformations, i.e., the concircular geometry, is a generalization of inversive geometry in the sense that the change of metric is more general than that induced by a circle preserving diffeomorphism (see also [6]). An interesing invariant of a concircular transformation is the concircular curvature tensor with respect to the Levi-Civita connection. It is defined by ([12, 13])

where X, Y, Z, U ∈ χ(M), R and r are the curvature tensor and the scalar curvature with respect to the Levi-Civita connection.

The concircular curvature tensor with respect to the semi-symmetric nonmetric connection is defined by

where R̄ and r̄ are the curvature tensor and the scalar curvature with respect to the semi-symmetric non-metric connection. Riemannian manifolds with vanishing concircular curvature tensor are of constant curvature. Thus the concircular curvature tensor is a measure of the failure of a Riemannian manifold to be of constant curvature.

In this paper we study a type of semi-symmetric non-metric connection due to Agashe and Chafle [15] on LP-Sasakian manifolds. The paper is organized as follows: After introduction in section 2, we give a brief account of LP-Sasakian manifolds. Section 3 deals with the semi-symmetric non-metric connection. The relation between the curvature tensor of an LP-Sasakian manifold with respect to the semi-symmetric non-metric connection and Levi-Civita connection have been studied in section 4. Section 5 is devoted to obtain ξ-concircularly flat LP-Sasakian manifold with respect to the semi-symmetric non-metric connection. Next Section, we deals with the LP-Sasakian manifolds admitting semi-symmetric non-metric connection ∇̄ satisfying · S̄ = 0, where S̄ denotes the Ricci tensor with respect to the semi-symmetric non-metric connection. Finally, we construct an example of a 5-dimensional LP-Sasakian manifold admitting the semi-symmetric non-metric connection to support the results obtained in Section 5.

An n-dimensional differentiable manifold M with structure (φ, ξ, η, g) is said to be a Lorentzian almost Paracontact manifold (briefly, LAP-manifold) ([10, 11]), if it admits a (1, 1)- tensor field φ, a contravariant vector field ξ, a 1-form η and a Lorentzian metric g which satisfy

where ∇ denotes the covariant differentiation with respect to Lorentzian metric g and for any vector field X and Y ∈ χ(M), χ(M) is the set of all differentiable vector fields on M and the tensor field Φ(X, Y) is a symmetric (0, 2)-tensor field [10].

An LAP-manifold with structure (φ, ξ, η, g) satisfying the relation [10]

is called a normal Lorentzian paracontact manifold or Lorentzian para-Sasakian manifold (briefly LP-Sasakian manifold). Also, since the vector field η is closed in an LP-Sasakian manifold, we have ([10, 11])

Also in an LP-Sasakian manifold, the following relations holds [10]:

for any vector field X, Y, Z ∈ χ(M), S denotes the Ricci tensor of M with respect to the Levi-Civita connection.

Let M be an n-dimensional differential manifold with Lorentzian metric g. If ∇̄ is the semi-symmetric non-metric connection on a differential manifold M, a linear connection ∇̄ is given by [15]

Then R̄ and R are related by [15]

for all vector fields X, Y, Z ∈ χ(M), χ(M) is the set of all differentiable vector fields on M, where β is a (0, 2) tensor field denoted by

From (3.1) yields

In this section we obtain the expressions of the curvature tensor, Ricci tensor and scalar curvature of M with respect to the semi-symmetric non-metric connection defined by (3.1).

Analogous to the definitions of the curvature tensor R of M with respect to the Levi-Civita connection ∇, we define the curvature tensor R̄ of M with respect to the semi-symmetric non-metric connection ∇̄ given by

where X, Y, Z ∈ χ(M), the set of all differentiable vector fields on M.

Combining (2.8) and (3.3), we get

Using (4.1) in (3.2) [20], we have

From (4.2), implies that

Putting X = ξ in (4.2) and using (2.1) and (2.11) [20], we obtain

Again putting Z = ξ in (4.2) and using (2.1) and (2.10) [20], we get

Combining (2.9) and (4.2), we have

Taking a frame field from (4.2) [20], we obtain

where S̄ denotes the Ricci tensor with respect to the semi-symmetric non-metric connection.

From (4.7), implies that

Putting Z = ξ in (4.8) and using (2.13) [20], we get

Combining (2.8) and (3.1), we have

Again taking a frame field from (4.7) [20], we obtain

where α = trace of φ and r̄ denotes the scalar curvature with respect to the semi-symmetric non-metric connection.

From [20] and the above discussions we can state the following:

Proposition 4.1

For an LP-Sasakian manifold M with respect to the semi-symmetric non-metric connection ∇̄,

• (1) the curvature tensor R̄ is given by R̄(X, Y)Z = R(X, Y)Z + g(φX, Z)Y − η(X)η(Z)Y − g(φY, Z)X + η(Y)η(Z)X,

• (2) the Ricci tensor S̄ is given by S̄(Y, Z) = S(Y, Z) − (n − 1)g(φY, Z) + (n − 1)η(Y)η(Z),

• (3) the scalar curvature r̄ is given by r̄ = r − (n − 1)(α + 1),

• (4) R̄(X, Y)Z = −R̄(Y, X)Z,

• (5) the Ricci tensor S̄ is symmetric.

Definition 5.1

A LP-Sasakian manifold M with respect to the semi-symmetric non-metric connection is said to be ξ-concircularly flat if

for all vector fields X, Y ∈ χ(M), χ(M) is the set of all differentiable vector fields on M.

Theorem 5.1

An n-dimensional LP-Sasakian manifold with respect to the semi-symmetric non-metric connection is ξ-concircularly flat if and only if the manifold with respect to the Levi-Civita connection is also ξ-concircularly flat provided trace of φ = α = n − 1.

Theorem 5.2

If a LP-Sasakian manifold (n > 1) is ξ-concircularly flat with respect to the semi-symmetric non-metric connection if and only if the scalar curvature with respect to the semi-symmetric non-metric connection vanishes.

Theorem 6.1

If an LP-Sasakian manifold with respect to the semi-symmetric non-metric connection satisfies · S̄ = 0, then the scalar curvature with respect to the Levi-Civita connection is (n − 1)(α + 1), where α = g(φe_i, e_i).

In this section we construct an example on LP-Sasakian manifold with respect to the semi-symmetric non-metric connection ∇̄ which verify the result of section 5.

We consider the 5-dimensional manifold M = {(x, y, z, u, v) ∈ R⁵}, where (x, y, z, u, v) are the standard coordinate in R⁵.

We choose the vector fields

which are linearly independent at each point of M.

Let g be the Lorentzian metric defined by

and

Let η be the 1-form defined by

for any Z ∈ χ(M).

Let φ be the (1, 1)-tensor field defined by

Using the linearity of φ and g, we have

and

for any U, Z ∈ χ(M). Hence e₃ = ξ and M(φ, ξ, η, g) is a Lorentzian almost paracontact manifold.

Then we have

The Riemannian connection ∇ of the metric tensor g is given by Koszul’s formula which is given by

Taking e₃ = ξ and using Koszul’s formula we get the following

From the above calculations, the manifold under consideration satisfies η(ξ) = −1 and ∇_Xξ = φX. Therefore, the manifold is an LP-Sasakian manifold.

Using (3.1) in above equations, we obtain

By using the above results, we can easily obtain the components of the curvature tensors as follows:

and

and other curvature tensor R(e_i, e_j)e_k = R̄(e_i, e_j)e_k = 0; ∀i, j, k = 1, 2, 3, 4, 5. From these curvature tensors, we can be calculated the Ricci tensors as follows:

and

Therefore, the scalar curvature tensors r = −4 and r̄ = 0 with respect to the Levi-Civita connection and the semi-symmetric non-metric connection respectively.

Let X and Y are any two vector fields given by

and

Using the above relations of curvature tensors and scalar curvature tensor with respect to the semi-symmetric non-metric connection respectively, we get

Hence the manifold under consideration satisfies the Theorem 5.2 of Section 5.

The author is thankful to the referee for his/her valuable comments towards the improvement of my paper.