PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 239–47 DOI: 10.2298/PIM1409239B

ON PARA-SASAKIAN MANIFOLDS ADMITTING SEMI-SYMMETRIC METRIC CONNECTION

Ajit Barman

Communicated by Stevan Pilipović

Abstract. We study a Para-Sasakian manifold admitting a semi-symmetric metric connection whose projective curvature tensor satisfies certain curvature conditions.

1. Introduction

In [19], Takahashi introduced the notion of locally ϕ-symmetric Sasakian manifolds as a weaker version of local symmetry of such manifolds. In respect of contact geometry, the notion of ϕ-symmetric was introduced and studied by Boeckx, Buecken and Vanhecke [4] with several examples. In [5], De studied the notion of ϕ-symmetry with several examples for Kenmotsu manifolds. In 1977, Adati and Matsumoto defined para-Sasakian and special para-Sasakian manifolds [2], which are special classes of an almost paracontact manifold introduced by Sato [17]. Para-Sasakian manifolds have been studied by Tarafdar and De [20], De and Pathak [11], Matsumoto, Ianus and Mihai [15], Matsumoto [14] and many others.

Hayden [13] introduced semi-symmetric linear connections on a Riemannian manifold. Let M be an n-dimensional Riemannian manifold of class C^{∞} endowed with the Riemannian metric g and ∇ be the Levi-Civita connection on (M^{n}, g).

A linear connection ¯∇ defined on (M^{n}, g) is said to be semi-symmetric [12] if its torsion tensor T is of the form T (X, Y) = η(Y)X−η(X)Y, where η is a 1-form and ξ is a vector field defined by η(X) = g(X, ξ), for all vector fields X ∈ χ(M^{n}), χ(M^{n}) is the set of all differentiable vector fields on M^{n}. A semi-symmetric connection ¯∇ is called a semi-symmetric metric connection [13] if it further satisfies ∇g = 0. A relation between the semi-symmetric metric connection ¯∇ and the Levi-Civita connection ∇ on (M^{n}, g) has been obtained by Yano [21] which is given by

(1.1)  ¯∇_{X} Y = ∇_{X} Y + η(Y)X − g(X, Y)ξ.

2010 Mathematics Subject Classification: 53C15, 53C25.

Key words and phrases: para-Sasakian manifold, semi-symmetric metric connection, recurrent, η-Einstein, ξ-projectively flat, locally ϕ-projectively symmetric manifold.

We also have ( ¯∇_{X} η)(Y) = (∇_{X} η)Y − η(X)η(Y) + η(ξ)g(X, Y). Further, a relation between the curvature tensor R̄ of the semi-symmetric metric connection ¯∇ and the curvature tensor R of the Levi-Civita connection ∇ is given by

(1.2) R̄(X, Y)Z = R(X, Y)Z + α(X, Z)Y − α(Y, Z)X + g(X, Z)QY − g(Y, Z)QX,

where α is a tensor field of type (0,2) and Q is a tensor field of type (1,1) which is given by

(1.3)  α(Y, Z) = g(QY, Z) = (D_{Y}η)(Z) − η(Y)η(Z) + \frac{1}{2}η(ξ)g(Y, Z).

From (1.2) and (1.3), we obtain

˜R̄(X, Y, Z, W) = R̃(X, Y, Z, W) − α(Y, Z)g(X, W) + α(X, Z)g(Y, W)
− g(Y, Z)α(X, W) + g(X, Z)α(Y, W),

where ˜ R̄(X, Y, Z, W) = g(R̄(X, Y)Z, W), R̃(X, Y, Z, W) = g(R(X, Y)Z, W).

The semi-symmetric metric connections have been studied by several authors such as Yano [21], Amur and Pujar [1], Prvanović [16], De and Biswas [10], Sharfuddin and Hussain [18], Binh [3], De [6, 7], De and De [8, 9] and many others.

The projective curvature tensor is an important tensor from the differential geometric point of view. Let M be a n-dimensional Riemannian manifold. If there exists a one-to-one correspondence between each coordinate neighbourhood of M and a domain in Euclidian space such that any geodesic of the Riemannian manifold corresponds to a straight line in the Euclidean space, then M is said to be locally projectively flat. For n ⩾ 1, M is locally projectively flat if and only if the projective curvature tensor P vanishes. Here the projective curvature tensor P with respect to the semi-symmetric metric connection is defined by

(1.4)  P̄(X, Y)Z = R̄(X, Y)Z − \frac{1}{2n} [S̄(Y, Z)X − S̄(X, Z)Y],

From (1.4), it follows that

˜P̄ (X, Y, Z, W)= ˜ R̄(X, Y, Z, W) − \frac{1}{2n} [S̄(Y, Z)g(X, W) − S̄(X, Z)g(Y, W)],
˜P̄ (X, Y, Z, W)= g(P̄(X, Y)Z, W),

for X, Y, Z, W ∈ χ(M), where S̄ is the Ricci tensor with respect to the semisymmetric metric connection. In fact M is projectively flat if and only if it is of constant curvature [22]. Thus the projective curvature tensor is the measure of the failure of a Riemannian manifold to be of constant curvature.

ON PARA-SASAKIAN MANIFOLDS ADMITTING SEMI-SYMMETRIC METRIC ... 241

The paper is organized as follows: Section 2 is equipped with some prerequisites about P-Sasakian manifolds. In section 3, we establish the relation of the curvature tensor between the Levi-Civita connection and the semi-symmetric metric connection of a P-Sasakian manifold. A P-Sasakian manifold whose curvature tensor of manifold is covariant constant with respect to the semi-symmetric metric connection and manifold if recurrent with respect to the Levi-Civita connection is studied in Section 4. Section 5 is devoted to study ξ-projectively flat P-Sasakian manifolds with respect to the semi-symmetric metric connection. Finally, we investigate locally ϕ-projectively symmetric P-Sasakian manifolds with respect to the semi-symmetric metric connection.

2. P-Sasakian manifolds

An n-dimensional differentiable manifold M is said to admit an almost paracontact Riemannian structure (ϕ, ξ, η, g), where ϕ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form and g is the Riemannian metric on M such that

ϕξ = 0, η(ϕX) = 0, η(ξ) = 1, g(X, ξ) = η(X), (2.1)

ϕ^{2}(X) = X − η(X)ξ, (2.2)

g(ϕX, ϕY) = g(X, Y) − η(X)η(Y), (2.3)

(∇_{X} η)Y = g(X, ϕY) = (∇_{Y}η)X, (2.4)

for any vector fields X, Y on M. In addition, if (ϕ, ξ, η, g), satisfy the equations

dη = 0, ∇_{X} ξ = ϕX, (2.5)

(∇_{X} ϕ)Y = −g(X, Y)ξ − η(Y)X + 2η(X)η(Y)ξ, (2.6)

then M is called a para-Sasakian manifold or briefly a P-Sasakian manifold. It is known [2, 17] that in a P-Sasakian manifold the following relations hold:

η(R(X, Y)Z) = g(X, Z)η(Y) − g(Y, Z)η(X), (2.7)

R(ξ, X)Y = η(Y)X − g(X, Y)ξ, (2.8)

R(ξ, X)ξ = X − η(X)ξ, (2.9)

R(X, Y)ξ = η(X)Y − η(Y)X, (2.10)

S(X, ξ) = −(n − 1)η(X), (2.11)

S(ϕX, ϕY) = S(X, Y) + (n − 1)η(X)η(Y), (2.12)

where R and S are the curvature tensor and the Ricci tensor of the Levi-Civita connection respectively.

3. Curvature tensor of a P-Sasakian manifold with respect to the semi-symmetric metric connection

Theorem 3.1. For a P-Sasakian manifold M with respect to the semi-symmetric metric connection ¯∇

(i) The curvature tensor R̄ is given by (3.3), (ii) The Ricci tensor S̄ is given by (3.5), (iii) The scalar curvature r̄ is given by (3.6), (iv) R̄(X, Y)Z = −R̄(Y, X)Z,

(v) η(R̄(X, Y)Z) = η(Y)g(X, Z)−η(X)g(Y, Z)+η(Y)g(X, ϕZ)−η(X)g(Y, ϕZ), (vi) The Ricci tensor S̄ is symmetric, (vii) S̄(Y, ξ) = −(n − 1 + γ)η(Y), (viii) ( ¯∇_{W} ϕ)(X) = −g(X, W)ξ−η(X)W +2η(X)η(W)ξ−g(X, ϕW)ξ−η(X)ϕW, (ix) ( ¯∇_{W} η)(X) = g(X, ϕW) − η(X)η(W) + g(X, W), (x) ¯∇_{W} ξ = ϕW + W − η(W)ξ.

Proof. Using (2.4) and (2.1) in (1.3), we get

(3.1)  α(X, Y) = g(QX, Y) = g(X, ϕY) − η(X)η(Y) + \frac{1}{2}g(X, Y).

From (3.1) implies that

(3.2)  QX = ϕX − η(X)ξ + \frac{1}{2}X.

Again using (3.1) and (3.2) in (1.2), we have

(3.3)  R̄(X, Y)Z = R(X, Y)Z + g(X, ϕZ)Y − η(X)η(Z)Y − g(Y, ϕZ)X
+ η(Y)η(Z)X + g(X, Z)Y − g(Y, Z)X + g(X, Z)ϕY
− g(Y, Z)ϕX − g(X, Z)η(Y)ξ + g(Y, Z)η(X)ξ.

From (3.3), we obtain that the curvature tensor R̄ satisfies R̄(X, Y)Z = −R̄(Y, X)Z. Using (2.7) and (2.1) in (3.3), implies that

η(R̄(X, Y)Z) = η(Y)g(X, Z) − η(X)g(Y, Z) + η(Y)g(X, ϕZ) − η(X)g(Y, ϕZ).

Taking the inner product of (3.3) with W, it follows that

˜R̄(X, Y, Z, W) = R̃(X, Y, Z, W) + g(X, ϕZ)g(Y, W) − η(X)η(Z)g(Y, W)

− g(Y, ϕZ)g(X, W) + η(Y)η(Z)g(X, W) + g(X, Z)g(Y, W)

− g(Y, Z)g(X, W) + g(X, Z)g(ϕY, W) − g(Y, Z)g(ϕX, W) (3.4)

− g(X, Z)η(Y)η(W) + g(Y, Z)η(X)η(W) .

Let {e_{1}, …, e_{n}} be a local orthonormal basis of vector fields in M. Then by putting X = W = e_{i} in (3.4), summing over i, 1 ⩽ i ⩽ n, and using (2.1), we obtain

(3.5) S̄(Y, Z) = S(Y, Z) − (n − 2)g(Y, ϕZ) + (n − 2)η(Y)η(Z) − (n − 2 + γ)g(Y, Z),

where trace of ϕ = γ. Again by putting Y = Z = e_{i} in (3.5), summing over i, 1 ⩽ i ⩽ n and using (2.1), we get

(3.6)  r̄ = r − 2(n − 1)γ − (n − 1)(n − 2)

where r̄ and r are the scalar curvatures with respect to the semi-symmetric metric connection and the Levi-Civita connection respectively. Again putting Z = ξ in (3.5) and using (2.1) and (2.11), we get S̄(Y, ξ) = −(n − 1 + γ)η(Y). Using (1.1), (2.1) and (2.6), implies that

(3.7) ( ¯∇_{W} ϕ)(X) = −g(X, W)ξ − η(X)W + 2η(X)η(W)ξ − g(X, ϕW)ξ − η(X)ϕW.

Using (1.1), (2.1) and (2.4), it follows that

(3.8)  ( ¯∇_{W} η)(X) = g(X, ϕW) − η(X)η(W) + g(X, W).

Again using (1.1), (2.1) and (2.5), we get

{□ \atop (3.9)} ¯∇_{W} ξ = ϕW + W − η(W)ξ.

4. A P-Sasakian manifold (M^{n}, g) whose curvature tensor of manifold is covariant constant with respect to the semi-symmetric metric connection and M is recurrent with respect to the Levi-Civita

connection

Theorem 4.1. If an n-dimensional P-Sasakian manifold whose curvature tensor of manifold is covariant constant with respect to the semi-symmetric metric connection and the manifold is recurrent with respect to the Levi-Civita connection and the associated 1-form A is equal to the associated 1-form η, then the manifold is an η-Einstein manifold.

Definition 4.1. A P-Sasakian manifold M with respect to the Levi-Civita connection is called recurrent if its curvature tensor R satisfies the condition

(4.1)  (∇_{W}R)(X, Y)Z = A(W)R(X, Y)Z,

where A is the 1-form.

Definition 4.2. A P-Sasakian manifold M is said to be an η-Einstein manifold if its Ricci tensor S of the Levi-Civita connection is of the form

S(Z, W) = ag(Z, W) + bη(Z)η(W),

where a and b are smooth functions on the manifold.

Proof. Using (1.1), (2.7), (2.8) and (2.10), we obtain

( ¯∇_{W} R)(X, Y)Z = ¯∇_{W} R(X, Y)Z − R( ¯∇_{W} X, Y)Z − R(X, ¯∇_{W} Y)Z (4.2)

− R(X, Y) ¯∇_{W} Z = (∇_{W}R)(X, Y)Z − R̃(X, Y, Z, W)ξ
− η(X)R(W, Y)Z − η(Y)R(X, W)Z − η(Z)R(X, Y)W
+ η(Y)g(X, Z)W − η(X)g(Y, Z)W + η(Z)g(X, W)Y
− g(X, W)g(Y, Z)ξ − η(Z)g(Y, W)X + g(X, Z)g(Y, W)ξ
+ η(X)g(Z, W)Y − η(Y)g(Z, W)X .

Suppose ( ¯∇_{W} R)(X, Y)Z = 0, then from (4.2), we get

(∇_{W}R)(X, Y)Z − R̃(X, Y, Z, W)ξ − η(X)R(W, Y)Z − η(Y)R(X, W)Z (4.3)

− η(Z)R(X, Y)W + η(Y)g(X, Z)W − η(X)g(Y, Z)W
+ η(Z)g(X, W)Y − g(X, W)g(Y, Z)ξ − η(Z)g(Y, W)X
+ g(X, Z)g(Y, W)ξ + η(X)g(Z, W)Y − η(Y)g(Z, W)X = 0.

Using (4.1) in (4.3), we have

A(W)R(X, Y)Z − R̃(X, Y, Z, W)ξ − η(X)R(W, Y)Z − η(Y)R(X, W)Z (4.4)

− η(Z)R(X, Y)W + η(Y)g(X, Z)W − η(X)g(Y, Z)W
+ η(Z)g(X, W)Y − g(X, W)g(Y, Z)ξ − η(Z)g(Y, W)X
+ g(X, Z)g(Y, W)ξ + η(X)g(Z, W)Y − η(Y)g(Z, W)X = 0.

Now contracting X in (4.4) and using (2.1) and (2.7), it follows that

(4.5)  A(W)S(Y, Z)− η(Y)S(Z, W) − η(Z)S(Y, W)
− (n − 1)η(Z)g(Y, W) − (n − 1)η(Y)g(Z, W) = 0.

Putting Y = ξ in (4.5) and using (2.1) and (2.11), we obtain

(4.6)  S(Z, W) = (1 − n)g(Z, W) + (1 − n)A(W)η(Z).

Suppose the associated 1-form A is equal to the associated 1-form η, then from (4.6), we get S(Z, W) = (1 − n)g(Z, W) + (1 − n)η(W)η(Z). Therefore, S(Z, W) = ag(Z, W) + bη(Z)η(W), where a = (1 − n) and b = (1 − n). □

5. ξ-projectively flat P-Sasakian manifolds with respect to the semi-symmetric metric connection

Theorem 5.1. An n-dimensional P-Sasakian manifold is ξ-projectively flat with respect to the semi-symmetric metric connection if and only if the manifold is also ξ-projectively flat with respect to the Levi-Civita connection provided the vector fields X and Y are horizontal vector fields.

Proof. Using (3.3) in (1.4), we have

P̄(X, Y)Z = R(X, Y)Z + g(X, ϕZ)Y − η(X)η(Z)Y − g(Y, ϕZ)X + η(Y)η(Z)X

+ g(X, Z)Y − g(Y, Z)X + g(X, Z)ϕY − g(Y, Z)ϕX − g(X, Z)η(Y)ξ
(5.1)  + g(Y, Z)η(X)ξ − \frac{1}{n−1} [S̄(Y, Z)X − S̄(X, Z)Y] .

Using (3.5) in (5.1), it follows that

P̄(X, Y)Z = P (X, Y)Z + \frac{1}{n−1} [g(X, ϕZ)Y − g(Y, ϕZ)X − η(X)η(Z)Y
+ η(Y)η(Z)X + (1 − γ)g(X, Z)Y − (1 − γ)g(Y, Z)X]

+ g(X, Z)ϕY − g(Y, Z)ϕX − g(X, Z)η(Y)ξ + g(Y, Z)η(X)ξ (5.2)

where

(5.3)  P (X, Y)Z = R(X, Y)Z − \frac{1}{n−1} [S(Y, Z)X − S(X, Z)Y],

is the projective curvature tensor with respect to the Levi-Civita connection. Putting Z = ξ in (5.2) and using (2.1), we obtain

(5.4) P̄(X, Y)ξ = P (X, Y)ξ + \frac{1}{n−1} [γη(Y)X − γη(X)Y] + η(X)ϕY − η(Y)ϕX.

Suppose X and Y are orthogonal to ξ; then from (5.4), we get

P̄(X, Y)ξ = P (X, Y)ξ,

concluding the proof. □

6. Locally ϕ-projectively symmetric P-Sasakian manifolds with respect to the semi-symmetric metric connection

Theorem 6.1. An n-dimensional P-Sasakian manifold is locally ϕ-projectively symmetric with respect to the semi-symmetric metric connection if and only if the manifold is also locally ϕ-projectively symmetric with respect to the Levi-Civita connection.

Definition 6.1. A P-Sasakian manifold M with respect to the semi-symmetric metric connection is said to be locally ϕ-projectively symmetric if

ϕ^{2}(( ¯∇_{W} P̄)(X, Y)Z) = 0,

for all vector fields X, Y, Z, W are orthogonal to ξ.

Proof. Using (1.1), we get

( ¯∇_{W} P)(X, Y)Z = ¯∇_{W} P (X, Y)Z−P ( ¯∇_{W} X, Y)Z−P (X, ¯∇_{W} Y)Z−P (X, Y) ¯∇_{W} Z

= (∇_{W}P)(X, Y)Z + η(P (X, Y)Z)W − η(X)P (W, Y)Z − η(Y)P (X, W)Z

− η(Z)P (X, Y)W − P̃(X, Y, Z, W)ξ + g(X, W)P (ξ, Y)Z

(6.1)  + g(Y, W)P (X, ξ)Z + g(Z, W)P (X, Y)ξ.

Putting X = ξ in (5.3) and using (2.8) and (2.11), we have

(6.2)  P (ξ, Y)Z = −g(Y, Z)ξ − \frac{1}{n−1} S(Y, Z)ξ.

Putting Y = ξ in (5.3) and using (2.8) and (2.11), it follows that

(6.3)  P (X, ξ)Z = g(X, Z)ξ + \frac{1}{n−1} S(X, Z)ξ.

Again putting Z = ξ in (5.3) and using (2.10) and (2.11),

(6.4)  P (X, Y)ξ = 0.

Using (2.7), (5.3), (6.2), (6.3), (6.4) in (6.1), we obtain

( ¯∇_{W} P)(X, Y)Z = (∇_{W}P)(X, Y)Z − η(X)P (W, Y)Z − η(Y)P (X, W)Z
− η(Z)P (X, Y)W + η(Y)g(X, Z)W − η(X)g(Y, Z)W
− \frac{1}{n−1} [η(X)S(Y, Z)W − η(Y)S(X, Z)W ] − P̃(X, Y, Z, W)ξ
− g(X, W)[g(Y, Z)ξ + \frac{1}{n−1} S(Y, Z)ξ]
(6.5)  + g(Y, W)[g(X, Z)ξ + \frac{1}{n−1} S(X, Z)ξ] .

Taking covariant differentiation of (5.2) with respect to W and using (3.7), (3.8), (3.9) and (6.5), we get ( ¯∇_{W} P̄)(X, Y)Z = (∇_{W}P)(X, Y)Z−η(X)P (W, Y)Z−η(Y)P (X, W)Z

− η(Z)P (X, Y)W − P̃(X, Y, Z, W)ξ

+ \frac{1}{n−1} [η(Y)S(X, Z)W − η(X)S(Y, Z)W − g(X, W)S(Y, Z)ξ + g(Y, W)S(X, Z)ξ

− η(Z)g(X, ϕW)Y + η(Z)g(Y, ϕW)X + 2η(X)η(Z)η(W)Y

− 2η(Y)η(Z)η(W)X + (n − 2)η(Z)g(X, W)Y − (n − 2)η(Z)g(Y, W)X (6.6)

− η(X)g(Z, ϕW)Y + η(Y)g(Z, ϕW)X − η(X)g(Z, W)Y + η(Y)g(Z, W)X ]

− η(Z)g(X, W)Y + η(Z)g(Y, W)X − g(X, Z)g(Y, W)ξ + g(X, W)g(Y, Z)ξ

− η(Y)g(X, Z)W + η(X)g(Y, Z)W + 4η(Y)η(W)g(X, Z)ξ − 4η(X)η(W)g(Y, Z)ξ

−2g(X, Z)g(Y, ϕW)ξ+2g(X, ϕW)g(Y, Z)ξ−2η(Y)g(X, Z)ϕW +2η(X)g(Y, Z)ϕW .

Now applying ϕ^{2} on both sides of (6.6) and using (2.1) and (2.2), it follows that

ϕ^{2}(( ¯∇_{W} P̄)(X, Y)Z) = ϕ^{2}((∇_{W}P)(X, Y)Z)−η(X)P (W, Y)Z+η(X)η(P (W, Y)Z)ξ

−η(Y)P (X, W)Z + η(Y)η(P (X, W)Z)ξ − η(Z)P (X, Y)W + η(Z)η(P (X, Y)W)ξ

+\frac{1}{n−1} [η(Y)S(X, Z)W −η(Y)η(W)S(X, Z)ξ−η(X)S(Y, Z)W +η(X)η(W)S(Y, Z)ξ

−η(Z)g(X, ϕW)Y +η(Z)η(Y)g(X, ϕW)ξ+η(Z)g(Y, ϕW)X −η(Z)η(X)g(Y, ϕW)ξ

+2η(X)η(Z)η(W)Y − 2η(Y)η(Z)η(W)X + (n − 2)η(Z)g(X, W)Y

+η(Z)η(Y)g(X, W)ξ − (n − 2)η(Z)g(Y, W)X − η(Z)η(X)g(Y, W)ξ (6.7)

−η(X)g(Z, ϕW)Y + η(Y)g(Z, ϕW)X − η(X)g(Z, W)Y + η(Y)g(Z, W)X ]
−η(Z)g(X, W)Y + η(Z)g(Y, W)X − η(Y)g(X, Z)W + η(Y)η(W)g(X, Z)ξ
+η(X)g(Y, Z)W − η(X)η(W)g(Y, Z)ξ − 2η(Y)g(X, Z)ϕW + 2η(X)g(Y, Z)ϕW .

Taking X, Y, Z and W are orthogonal to ξ, then from (6.7), we have

ϕ^{2}(( ¯∇_{W} P̄)(X, Y)Z) = ϕ^{2}((∇_{W}P)(X, Y)Z).

This completes the proof. □

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Department of Mathematics (Received 16 12 2012) Kabi-Nazrul Mahavidyalaya Sonamura-799181, Dist-Sepahijala Tripura, India

    ajitbarmanaw@yahoo.in