EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 1, No. 2, 2008, (21-31)

ISSN 1307-5543 – www.ejpam.com

Some results on K-contact and Trans-Sasakian Manifolds

Bagewadi Channabasappa1,∗, Basavarajappa N.S2, Prakasha D.G 1, and Venkatesha1

1 Department of Mathematics and Computer Science, Kuvempu University, Jnana Sahyadri-577
451, Shimoga, Karnataka, INDIA.
2 Department of Mathematics, SBM Jain College of Engineering, 562 112, Jakkasandra BAN-
GALORE.

Abstract. We obtain results on the vanishing of divergence of Pseudo projective curvature tensor
P̃ with respect to semi-symmetric metric connection on k-contact and trans-Sasakian manifolds.

AMS subject classifications: 53D15,53B15.

Key words: k-contact manifold, Trans-Sasakian manifold, Pseudo projective curvature tensor,
η-Einstein manifold.

1. Introduction

In 1924, Friedman and Schouten [11] introduced the notion of semi-symmetric lin-
ear connection on a differentiable manifold. Then in 1932, Hayden [14] introduced
the idea of metric connection with a torsion on a Riemannain manifold. A systematic
study of semi-symmetric metric connection on a Riemannain manifold has been given by
Yano [18] in 1970 and later studied by K.S.Amur and S.S.Pujar [1], C.S.Bagewadi [2],
U.C.De et al [10], Sharafuddin and Hussain [16] and others. The authors U.C.De [10]
and C.S.Bagewadi et al [ [3, 12]] have obtained results on the conservativeness of Pro-
jective, Pseudo projective, Conformal, Concircular, Quasi conformal curvature tensors on
k-contact, Kenmotsu and trans-sasakian manifolds.

In this paper we extend the conservativeness of Pseudo projective curvature tensor
to k-contact and trans-Sasakian manifolds admitting semi-symmetric metric connection.
After preliminaries in section 2, we study in section 3 the Pseudo projective curvature
tensor with respect to semi-symmetric metric connection on k-contact manifold. In the
section 4 we study some properties regarding Pseudo projective curvature tensor with
respect to this connection on trans-Sasakian manifold under the condition φ(gradα) =
(n− 2)gradβ and obtained some interesting results.

∗Corresponding author. Email addresses: prof bagewadi@yahoo.co.in (Bagewadi Channabasappa),
nsb sbmjce@yahoo.co.in (Basavarajappa N.S)

http://www.ejpam.com 21 c© 2007 EJPAM All rights reserved.



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 22

2. Preliminaries

Let Mn be an almost contact metric manifold [9] with an almost contact metric struc-
ture (φ, ξ, η, g), that is, φ is a (1, 1) tensor field, ξ is a vector field; η is a 1-form and g is a
compatible Riemannian metric such that

φ2 = −I + η ⊗ ξ, η(ξ) = 1, φ(ξ) = 0, η.φ = 0, (2.1)

g(φX, φY ) = g(X,Y )− η(X)η(Y ), (2.2)

g(X,φY ) = −g(φX, Y ), g(X, ξ) = η(X), (2.3)

for all X,Y ∈ TM .
If Mn is a k-contact Riemannian manifold, then besides (2.1), (2.2) and (2.9 ) the

following relations hold [15]:

∇Xξ = −φX, (2.4)

(∇Xη)(Y ) = −g(φX, Y ), (2.5)

S(X, ξ) = (n− 1)η(X), (2.6)

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

for any vector fields X, Y , where R and S denote respectively the curvature tensor of type
(1, 3) and the Ricci tensor of type (0, 2).

An almost contact metric structure (φ, ξ, η, g) in M is called trans-Sasakian structure
[14] if (M × R, J,G) belongs to the class w4 [ [8], [13]] where J is the almost complex
structure on M ×R defined by J(X,λd/dt) = (φX − λξ, η(X)d/dt) for all vector fields X
on M and smooth functions λ on M ×R and G is the product metric on M ×R. This may
be expressed by the condition [8]

(∇Xφ)Y = α(g(X,Y )ξ − η(Y )X) + β(g(φX, Y )ξ − η(Y )φX) (2.8)

for some smooth functions functions α and β on M , and we say that the trans-Sasakian
structure is of type (α, β).

Let M be a n-dimensional trans-Sasakian manifold. From (2.8)it is easy to see that

∇Xξ = −αφX + β(X − η(X)ξ), (2.9)

(∇Xη)Y = −αg(φX, Y ) + βg(φX, φY ). (2.10)

In a n-dimensional trans-Sasakian manifold, we have

R(ξ,X)ξ = (α2 − β2 − ξβ)(η(X)ξ −X), (2.11)

2αβ + ξα = 0, (2.12)

S(X, ξ) = ((n− 1)(α2 − β2)− ξβ)η(X)− (n− 2)Xβ − (φX)α. (2.13)

If in a n-dimensional trans Sasakian manifold of type(α, β), we have [4]

φ(gradα) = (n− 2)gradβ, (2.14)



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 23

then (2.11) and (2.13) reduces to

R(ξ,X)ξ = (α2 − β2)(η(X)ξ −X), (2.15)

S(X, ξ) = (n− 1)(α2 − β2)η(X). (2.16)

In this paper we study trans Sasakian manifold under the condition (2.14).
Let (Mn, g) be an n-dimensional Riemannian manifold of class C∞ with metric tensor

g and let ∇ be the Levi-Civita connection on Mn. A linear connection ∇̃ on (Mn, g) is said
to be semi symmetric [16] if the torsion tensor T of the connection ∇̃ satisfies

T (X,Y ) = π(Y )X − π(X)Y, (2.17)

where π is a 1-form on Mn with ρ as associated vector field, i.e., π(X) = g(X, ρ) for any
differentiable vector field X on Mn.

A semi-symmetric connection ∇̃ is called semi-symmetric metric connection [5] if it
further satisfies ∇̃g = 0.

In an almost contact manifold semi-symmetric metric connection is defined by identi-
fying the 1-form π of (2.17) with the contact-form η, i.e., by setting [16]

T (X,Y ) = η(Y )X − η(X)Y (2.18)

with ξ as associated vector field. i.e., g(X, ξ) = η(X).
The relation between the semi-symmetric metric connection ∇̃ and the Levi-Civita

connection ∇ of (Mn, g) has been obtained by K.Yano [18], which is given by

∇̃XY = ∇XY + η(Y )X − g(X,Y )ξ, (2.19)

where η(Y ) = g(Y, ξ).
Further, a relation between the curvature tensor R and R̃ of type (1, 3) of the connec-

tions ∇ and ∇̃ respectively is given by [18].

R̃(X,Y )Z = R(X,Y )Z −K(Y,Z)X +K(X,Z)Y − g(Y, Z)FX + g(X,Z)FY. (2.20)

where K is a tensor field of type (0, 2) defined by

K(Y, Z) = g(FY,Z) = (∇Y η)(Z)− η(Y )η(Z) +
1
2
η(ξ)g(Y,Z), (2.21)

for any vector fields X and Y .
From (2.20), it follows that

S̃(Y,Z) = S(Y, Z)− (n− 2)K(Y,Z)− a.g(Y, Z) (2.22)

where S̃ denotes the Ricci tensor with respect to ∇̃ and a = Tr.K. Differentiating (2.22)
covariantly with respect to X, we obtain [6]

(∇̃X S̃)(Y,Z) = (∇XS)(Y,Z)− (n− 2)(∇XK)(Y, Z)− η(Y )S(X,Z)− η(Z)S(X,Y )
+(n− 2)η(Y )K(X,Z) + (n− 2)η(Z)K(Y,X) + g(X,Y )S(ξ, Z)
+g(X,Z)S(Y, ξ)− (n− 2)g(X,Z)K(Y, ξ)− (n− 2)g(X,Y )K(Z, ξ)(2.23)



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 24

Now let ei be an orthogonal basis of the tangent space at each point of the manifold
Mn for i = 1, 2, ...., n. Putting Y = Z = ei in (2.23 ) and then taking summation over the
index i, we get

∇̃X r̃ = ∇Xr − (n− 2)(∇Xa) (2.24)

Further, since ξ is a killing vector in k-contact manifold. S, α, r, and a are invariant under
it, i.e.,

LξS = 0, Lξr = 0 (2.25)

LξK = 0, Lξa = 0 (2.26)

We recall some definitions which are used in later section, A Riemannian manifold is said
to be η-Einstein manifold if the Ricci tensor S is of the form

S(X,Y ) = λg(X,Y ) + µη(X)η(Y )

where λ, µ are the associated functions on the manifold. A Riemannian manifold is said
to be cyclic-Ricci tensor, if the Ricci tensor S satisfies the condition

(∇XS)(Y, Z) + (∇Y S)(Z,X) + (∇ZS)(X,Y ) = 0

3. k-contact Manifold Admitting a Semi-symmetric Metric Connection With

Div.P̃ = 0

The pseudo projective curvature tensor on a Riemannian manifold is given by ( [7],
[17])

P̃ (X,Y )Z = aR(X,Y )Z + b(S(Y,Z)X − S(X,Z)Y ]
r

n

[
a

n− 1
+ b

]
[g(Y,Z)X − g(X,Z)Y ]. (3.1)

In this section we prove the following: If a k-contact manifold Mn (n > 2) admits a semi-
symmetric metric connection and if the Pseudo projective curvature tensor with respect
to this connection is conservative, then the manifold is η-Einstein; the scalar curvature of
such a manifold is given by (3.14).

Proof. : Let us suppose that in a k-contact ManifoldMn with respect to semi-symmetric
metric connection Div.C = 0 where Div denotes the divergence.
Differentiate (3.1) covariantly and then contracting we get Div.P̃ . By virtue of conserva-



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 25

tiveness of P̃ i.e.,div.P̃ = 0, we obtain

(a+ b)[(∇XS)(Y,Z)− (∇Y S)(X,Z)]− [a+ b(n− 2)][(∇XK)(Y,Z)− (∇YK)(X,Z)]
= (a− b)[S(Y,Z)η(X)− S(X,Z)η(Y )]− a(n− 1)η(R(X,Y )Z) + a.S(X,Y )η(Z)

+a(n−A− 1)[g(Y, Z)η(X)− g(X,Z)η(Y )] + b[g(Y, Z)S(X, ξ)− g(X,Z)S(Y, ξ)](3.2)

+(a+ b(n− 2))[K(X,Y )η(Z)−K(X,Z)η(Y ) +K(Y,X)η(Z)−K(Y,Z)η(X)]

−b(n− 2)[g(Y, Z)K(X, ξ) + g(X,Z)K(Y, ξ)] +
1
n

[
a+ (n− 1)

(n− 1)

]
[g(Y,Z)∇Xr

−g(X,Z)∇Y r] +
[
a+

1
n

a+ (n− 1)
(n− 1)

]
[g(Y,Z)∇XA− g(X,Z)∇YA].

By virtue of (2.1) and (2.4) we obtain from (2.21) that

K(X,Y ) = g(X,φY )− η(X)η(Y ) +
1
2
g(X,Y ). (3.3)

K(X, ξ) = −1
2
η(X) (3.4)

LX = −φX − η(X)ξ +
1
2
X. (3.5)

Now putting X = ξ in (3.2), then using (2.1), (2.6),(2.7),(3.3) and (3.4), we get

(a+ b)[(∇ξS)(Y,Z)− (∇Y S)(ξ, Z)]− [a+ n(n− 2)][(∇ξK)(Y,Z)− (∇YK)(ξ, Z)]

= [a+ n(n− 2)])g(φY,Z) + (a− b)S(Y,Z) +
[
a

(
A+

1
2

)
+ b(2n− 3)

]
g(Y,Z)

−
[
a

(
A+

1
2

)
+ b(n− 2)

]
η(Y )η(Z) +

1
n

[
a+ (n− 1)

(n− 1)

]
[g(Y, Z)∇ξr − η(Z)∇Y r](3.6)

+
[
a+

1
n

a+ (n− 1)
(n− 1)

]
[g(Y,Z)∇ξA− η(Z)∇YA].

From (2.25) and (2.26), we obtain

(∇ξS)(Y,Z) = −S(∇Y ξ, Z)− S(Y,∇Zξ), (∇ξr) = 0, (3.7)

(∇ξK)(Y,Z) = −K(∇Y ξ, Z)−K(Y,∇Zξ), (∇ξa) = 0, (3.8)

respectively.
By using(3.7)and(3.8) in (3.6), we have

(a+ b)[−S(∇Y ξ, Z)− S(Y,∇Zξ)− (∇Y S)(ξ, Z)]
= [a+ n(n− 2)][−K(∇Y ξ, Z)−K(Y,∇Zξ)− (∇YK)(ξ, Z)]

+[a+ n(n− 2)])g(φY,Z) + (a− b)S(Y, Z) +
[
a

(
A+

1
2

)
+ b(2n− 3)

]
g(Y,Z)

−
[
a

(
A+

1
2

)
+ b(n− 2)

]
η(Y )η(Z) +

1
n

[
a+ (n− 1)

(n− 1)

]
[g(Y,Z)∇ξr − η(Z)∇Y r](3.9)

+
[
a+

1
n

a+ (n− 1)
(n− 1)

]
[g(Y,Z)∇ξA− η(Z)∇YA].



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 26

Using (2.4) ,(2.6) and (3.4) in (3.9), we get

−(a+ b)S(φY,Z)− (a− b)S(Y,Z)

=
[
a

(
A+

3
2

)
+ b(3n− 5)

]
g(Y,Z)−

[
a

(
A+

3
2

)
− 2b(n− 2)

]
η(Y )η(Z)

+[a(n+ 1) + b(3n− 5)]g(φY,Z)− 1
n

[
a+ (n− 1)

(n− 1)

]
η(Z)∇Y r

−
[
a+

1
n

a+ (n− 1)
(n− 1)

]
η(Z)∇YA. (3.10)

Next, by replacing Z by φZ in above and then using (2.1), we obtain

−(a+ b)S(φY, φZ)− (a− b)S(Y, φZ)[
a

(
A+

3
2

)
+ b(3n− 5)

]
g(Y, φZ) + [a(n+ 1) + b(3n− 5)] g(φY, φZ) (3.11)

Interchanging Y and Z in (3.11), we have

−(a+ b)S(φY, φZ)− (a− b)S(φY,Z)[
a

(
A+

3
2

)
+ b(3n− 5)

]
g(φY,Z) + [a(n+ 1) + b(3n− 5)] g(φY, φZ) (3.12)

By adding (3.11) with (3.12), and then by using the skew-symmetric property of φ, one
can get

S(Y,Z) = P1.g(Y, Z) +Q1.η(Y )η(Z) (3.13)

whereP1 =
[
− a

a+ b
(n− 1)− b

a+ b
(3n− 5)

]
and

Q1 =
[

2a+ b

a+ b
(n− 1) +

b

a+ b
(3n− 5)

]
.

There fore the manifold is η-Einstein.
Let ei be an orthogonal basis of the tangent space at each point of the manifold Mn for
i = 1, 2, ...., n. Putting Y = Z = ei in (3.13) and then taking summation over the index i,
we get

r = −(n− 1)(n− 2)
(a+ 3b)
(a+ b)

. (3.14)

This proves the theorem.

Suppose in k-contact manifold admitting a semi-symmetric metric connection, the
Pseudo projective curvature tensor with respect to this connection is conservative. Then
the manifold has a cyclic-Ricci tensor with respect to Levi-Civita connection; and moreover
the scalar curvature of the manifold is constant if and only if the vector field ξ is harmonic
provided (a+ b) 6= 0.



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 27

Proof. Differentiating (3.13) covariantly with respect to X, we have

(∇XS)(Y,Z) =
[

2a+ b

a+ b
(n− 1) +

b

a+ b
(3n− 5)

]
[g(φY,X)η(Z) + g(φX,Z)η(Y )] (3.15)

Similarly

(∇Y S)(Z,X) =
[

2a+ b

a+ b
(n− 1) +

b

a+ b
(3n− 5)

]
[g(φY,Z)η(X) + g(φX, Y )η(Z)](3.16)

(∇ZS)(X,Y ) =
[

2a+ b

a+ b
(n− 1) +

b

a+ b
(3n− 5)

]
[g(φX,Z)η(Y ) + g(φZ, Y )η(X)].(3.17)

Adding the equations (3.15), (3.16) and (3.17), then using skew-symmetry of φ, we obtain

(∇XS)(Y, Z) + (∇Y S)(Z,X) + (∇ZS)(X,Y ) = 0 (3.18)

Thus the manifold has a cyclic-Ricci tensor.
Taking an orthonormal frame field and contracting (3.15) over X and Z, we obtain

dr(Y ) =
[

2a+ b

a+ b
(n− 1) +

b

a+ b
(3n− 5)

]
ψη(Y ) (3.19)

where ψ = Tr.φ. From (3.19), it follows that

dr(Y ) = 0 if and only ψ = 0 provided (a+ b) 6= 0. (3.20)

4. Trans-Sasakian Manifold Admitting a Semi-symmetric Metric Connection

With Div.P̃ = 0

Here we recall some results which will be used in further. [5]: In a trans-Sasakian
manifold under the condition (2.14), we have

[(∇ξS)(Y,Z)− (∇Y S)(ξ, Z)] = βS(Y, Z)− (n− 1)(α2 − β2)βg(Y, Z) (4.1)

−(n− 1)(α2 − β2)αg(Y, φZ) + αS(Y, φZ).

[5]: For trans-Sasakian manifold under the condition (2.14), the following results are



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 28

true

(i) K(Y,Z) = αg(Y, φZ) +
(
β +

1
2

)
g(Y, Z)− (β + 1)η(Y )η(Z)

(ii) K(Y, ξ) = K(ξ, Y ) = −1
2
η(Y )

(iii) K(∇Y ξ, Z) = −α2[g(Y,Z)− η(Y )η(Z)]− 2αβg(φY,Z)

−α
2
g(φY,Z) + β

(
β +

1
2

)
[g(Y,Z)− η(Y )η(Z)] (4.2)

(iv) K(Y,∇Zξ) = α2[g(Y,Z)− η(Y )η(Z)] +
α

2
g(φY,Z)

+β
(
β +

1
2

)
[g(Y,Z)− η(Y )η(Z)].

[5]: In a trans-Sasakian manifold under the condition (2.14), we have

[(∇ξK)(Y,Z)− (∇YK)(ξ, Z)] = αg(Y, φZ)− 2αβg(φY,Z)
−[(α2 − β2)− (2β + 1)][g(Y, Z)− η(Y )η(Z)].(4.3)

In this section we prove the following: Let in a trans-Sasakian manifoldMn (n > 2) under
the condition (2.14) admits a semi-symmetric metric connection the Pseudo projective
curvature tensor with respect to this connection is conservative. Then the manifold Mn is
η-Einstein with respect to Levi-Civita connection; the scalar curvature of such a manifold
is given by (4).

Proof.
Let us suppose that in a trans-Sasakian Manifold Mn under the condition (2.14)with

respect to semi-symmetric metric connection Div.P̃ = 0.
Putting X = ξ in (3.2) then using (2.1),(2.3), (2.16) and (4.2(ii)) we get

(a+ b)[(∇ξS)(Y, Z)− (∇Y S)(ξ, Z)]− (a+ b(n− 2))[(∇ξK)(Y,Z)− (∇YK)(ξ, Z)]
= (a− b)S(Y,Z) + (a+ b(n− 2))αg(Y, φZ)− a(n− 1)η(R(ξ, Y )Z)

+
[
(a+ b(n− 2))

(
β +

1
2

)
+ a(n−A− 1) + b(n− 1)(α2 − β2) +

b

2
(n− 2)

]
g(Y,Z)

−
[
a(n−A− 1) + (a+ b(n− 2))

(
β +

1
2

)
+
b

2
(n− 2)

]
η(Y )η(Z) (4.4)

+
[

1
n

a+ (n− 1)
(n− 1)

(n− 2)− a
]

[η(Z)∇YA− g(Y,Z)∇ξA]

− 1
n

[
a+ (n− 1)

(n− 1)

]
[η(Z)∇Y r − g(Y, Z)∇ξr].



Bagewadi C.S. et al. / Eur. J. Pure Appl. Math, 1 (2008), (21-31) 29

Using (2.15) ,(4.1), (4.2(i)) and (4.3) in above, we get

(a+ b)(β − 1)S(Y,Z) + (a+ b)αS(Y, φZ)
= −[2α(β + 1)(a+ b(n− 2)) + (a+ b)(n− 1)(α2 − β2)α]g(φY,Z)

+P̀ .g(Y,Z) + Q̀.η(Y )η(Z)

+
[

1
n

a+ (n− 1)
(n− 1)

(n− 2)− a
]

[η(Z)∇YA− g(Y,Z)∇ξA] (4.5)

− 1
n

[
a+ (n− 1)

(n− 1)

]
[η(Z)∇Y r − g(Y, Z)∇ξr].

where P̀ = [a(β − 1) + b(β + 1)](n− 1)(α2 − β2) + a(n−A− 1)

+
b

2
(n− 2) + [a+ b(n− 2)]

[
2
(
β +

1
4

)
− (α2 − β2)

]
and

Q̀ = [(a− b)(α2 − β2)− b](n− 2)− a
(
n−A− 1

2

)
.

Next, by replacing Z by φZ in (4.5) and then using (2.1), we obtain

−(a+ b)αS(Y,Z)− (a+ b)(β − 1)S(φY,Z)
= −[2α(β + 1)[a+ b(n− 2) + (a+ b)(n− 1)(α2 − β2)α]g(Y,Z)
−P̀ .g(Y, φZ) + [2α(β + 1)(a+ b(n− 2))]η(Y )η(Z) (4.6)

+
[

1
n

a+ (n− 1)
(n− 1)

(n− 2)− a
]
g(Y, φZ)∇ξA−

1
n

[
a+ (n− 1)

(n− 1)

]
g(Y, φZ)∇ξr.

Interchanging Y and Z in above, we have

−(a+ b)αS(Y,Z)− (a+ b)(β − 1)S(Y, φZ)
= −[2α(β + 1)[a+ b(n− 2) + (a+ b)(n− 1)(α2 − β2)α]g(Y,Z)
−P̀ .g(φY,Z) + [2α(β + 1)(a+ b(n− 2))]η(Y )η(Z) (4.7)

+
[

1
n

a+ (n− 1)
(n− 1)

(n− 2)− a
]
g(φY,Z)∇ξA−

1
n

[
a+ (n− 1)

(n− 1)

]
g(φY,Z)∇ξr.

By adding (4.6) and (4.7), then by using skew-symmetric property of φ, one can obtain

S(Y,Z) = P2.g(Y, Z) +Q2.η(Y )η(Z). (4.8)

where P2 =
[
2

(β + 1)
(a+ b)

[a+ b(n− 2)] + (n− 1)(α2 − β2)
]

and

Q2 = −2
(β + 1)
(a+ b)

[a+ b(n− 2)].



REFERENCES 30

Therefore the manifold is η-Einstein.
Let ei be an orthogonal basis of the tangent space at each point of the manifold Mn for
i = 1, 2, ...., n. Putting Y = Z = ei in (4.8) and then taking summation over the index i,
we get

r = (n− 1)
[
2

(β + 1)
(a+ b)

[a+ b(n− 2)] + n(α2 − β2)
]
.

This proves the theorem.

5. Acknowledgment

The authors are grateful to the referrers and Eyüp Çetin for their valuable suggestions
in the improvement of the paper .

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