m - projective curvature tensor on a Lorentzian para – Sasakian manifolds

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728.Volume 6, Issue 1 (Mar. - Apr. 2013), PP 19-23
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m - projective curvature tensor on a Lorentzian para – Sasakian
manifolds
Amit Prakash1
, Mobin Ahmad2
and Archana Srivastava3
1
Department of Mathematics, National Institute of Technology, Kurukshetra,
Haryana - 136119, India.
2
Department of Mathematics, Integral University, Kursi Road Lucknow - 226 026, India,
3
Department of Mathematics, S. R. Institute of Management and Technology, BKT, Lucknow - 227
202, India,
Abstract: In this paper we studied m-projectively flat, m-projectively conservative, 𝜑-m-projectively flat LP-
Sasakian manifold. It has also been proved that quasi m- projectively flat LP-Sasakian manifold is locally
isometric to the unit sphere 𝑆 𝑛
(1) if and only if 𝑀 𝑛
is m-projectively flat.
Keywords – Einstein manifold, m-projectively flat, m-projectively conservative, quasi m-projectively flat,
𝜑-m-projectively flat.
I. Introduction
The notion of Lorentzian para contact manifold was introduced by K. Matsumoto. The properties of
Lorentzian para contact manifolds and their different classes, viz LP-Sasakian and LSP-Sasakian manifolds
have been studied by several authors. In [13], M.Tarafdar and A. Bhattacharya proved that a LP-Sasakian
manifold with conformally flat and quasi - conformally flat curvature tensor is locally isometric with a unit
sphere 𝑆 𝑛
(1). Further, they obtained that an LP-Sasakian manifold with 𝑅 𝑋, 𝑌 ∙ 𝐶 = 0 is locally isometric
with a unit sphere 𝑆 𝑛
1 , where 𝐶 is the conformal curvature tensor of type (1, 3) and 𝑅 𝑋, 𝑌 denotes the
derivation of tensor of tensor algebra at each point of the tangent space. J.P. Singh [10] proved that an m-
projectively flat para-Sasakian manifold is an Einstein manifold. He has also shown that if in an Einstein P-
Sasakian manifold 𝑅 𝜉, 𝑋 ∙ 𝑊 = 0 holds, then it is locally isometric with a unit sphere 𝐻 𝑛
1 . Also an n-
dimensional 𝜂-Einstien P-Sasakian manifold satisfying 𝑊 𝜉, 𝑋 ∙ 𝑅 = 0 if and only if either manifold is locally
isometric to the hyperbolic space 𝐻 𝑛
−1 or the scalar curvature tensor 𝑟 of the manifold is −𝑛 𝑛 − 1 . S.K.
Chaubey [18], studied the properties of m-projective curvature tensor in LP-Sasakian, Einstein LP-Sasakian and
𝜂-Einstien LP-Sasakian manifold. LP-Sasakian manifolds have also studied by Matsumoto and Mihai [4],
Takahashi [11], De, Matsumoto and Shaikh [2], Prasad & De [9], Venkatesha and Bagewadi[14].
In this paper, we studied the properties of LP-Sasakian manifolds equipped with m-projective
curvature tensor. Section 1 is introductory. Section 2 deals with brief account of Lorentzian para-Sasakian
manifolds. In section 3, we proved that an m-projectively flat LP-Sasakian manifold is an Einstein manifold and
an LP-Sasakian manifold satisfying 𝐶1
1
𝑊 𝑌, 𝑍 = 0 is of constant curvature is m-projectively flat. In section
4, we proved that an Einstein LP-Sasakian manifold is m-projectively conservative if and only if the scalar
curvature is constant. In section 5, we proved that an n-dimensional 𝜑-m-projectively flat LP-Sasakian manifold
is an 𝜂-Einstein manifold. In last, we proved that an n-dimensional quasi m - projectively flat LP-Sasakian
manifold 𝑀 𝑛
is locally isometric to the unit sphere 𝑆 𝑛
(1) if and only if 𝑀 𝑛
II. Preliminaries
An n- dimensional differentiable manifold 𝑀 𝑛
is a Lorentzian para-Sasakian (LP-Sasakian) manifold, if
it admits a (1, 1) - tensor field 𝜙, a vector field 𝜉, a 1-form 𝜂 and a Lorentzian metric 𝑔 which satisfy
𝜙2
𝑋 = 𝑋 + 𝜂(𝑋)𝜉 , (2.1)
𝜂 𝜉 = −1, (2.2)
𝑔 𝜙𝑋, 𝜙𝑌 = 𝑔 𝑋, 𝑌 + 𝜂 𝑋 𝜂 𝑌 , (2.3)
𝑔 𝑋, 𝜉 = 𝜂(𝑋), (2.4)
𝐷𝑋 𝜙 𝑌 = 𝑔 𝑋, 𝑌 𝜉 + 𝜂 𝑌 𝑋 + 2𝜂 𝑋 𝜂 𝑌 𝜉, (2.5)
and 𝐷𝑋 𝜉 = 𝜙𝑋, (2.6)
for arbitrary vector fields X and Y, where D denote the operator of covariant differentiation with respect to
Lorentzian metric g, (Matsumoto, (1989) and Matsumoto and Mihai, (1988)).
In an LP-Sasakian manifold 𝑀 𝑛
with structure(𝜙, 𝜉, 𝜂, 𝑔), it is easily seen that

m-projective curvature tensor on a Lorentzian para – Sasakian manifolds
𝑎 𝜙𝜉 = 0 𝑏 𝜂 𝜙𝑋 = 0 𝑐 𝑟𝑎𝑛𝑘 𝜙 = 𝑛 − 1 (2.7)
Let us put 𝐹 𝑋, 𝑌 = 𝑔 𝜙𝑋, 𝑌 , (2.8)
then the tensor field 𝐹 is symmetric (0, 2) tensor field
𝐹 𝑋, 𝑌 = 𝐹 𝑌, 𝑋 , (2.9)
𝐹 𝑋, 𝑌 = 𝐷𝑋 𝜂 𝑌 , (2.10)
and 𝐷𝑋 𝜂 𝑌 − 𝐷 𝑌 𝜂 𝑋 = 0. (2.11)
An LP- Sasakian manifold 𝑀 𝑛
is said to be Einstein manifold if its Ricci tensor S is of the form
𝑆 𝑋, 𝑌 = 𝑘𝑔(𝑋, 𝑌). (2.12)
An LP- Sasakian manifold 𝑀 𝑛
is said to be 𝑎𝑛 𝜂 -Einstein manifold if its Ricci tensor S is of the form
𝑆 𝑋, 𝑌 = 𝛼𝑔 𝑋, 𝑌 + 𝛽𝜂 𝑋 𝜂 𝑌 , (2.13)
for any vector fields X and Y, where 𝛼, 𝛽 are the functions on 𝑀 𝑛
.
Let 𝑀 𝑛
be an 𝑛-dimensional LP-Sasakian manifold with structure 𝜑, 𝜉, 𝜂, 𝑔 . Then we have
(Matsumoto and Mihai, (1998) and Mihai, Shaikh and De,(1999)).
𝑔 𝑅 𝑋, 𝑌 𝑍, 𝜉 = 𝜂 𝑅 𝑋, 𝑌 𝑍 = 𝑔 𝑌, 𝑍 𝜂 𝑋 − 𝑔 𝑋, 𝑍 𝜂 𝑌 (2.14)
𝑅 𝜉, 𝑋 𝑌 = 𝑔 𝑋, 𝑌 𝜉 − 𝜂(𝑌)𝑋 , (2.15) 𝑎
𝑅 𝜉, 𝑋 𝜉 = 𝑋 + 𝜂 𝑋 𝜉 , (2.15) 𝑏
𝑅 𝑋, 𝑌 𝜉 = 𝜂 𝑌 𝑋 − 𝜂 𝑋 𝑌 , (2.15) 𝑐
𝑆 𝑋, 𝜉 = 𝑛 − 1 𝜂 𝑋 , (2.16)
𝑆 𝜑𝑋, 𝜑𝑌 = 𝑆 𝑋, 𝑌 + 𝑛 − 1 𝜂 𝑋 𝜂 𝑌 , (2.17)
for any vector fields 𝑋, 𝑌, 𝑍 ; where 𝑅 𝑋, 𝑌 𝑍 is the Riemannian curvature tensor of type (1, 3). 𝑆 is a Ricci
tensor of type (0, 2), 𝑄 is Ricci tensor of type (1, 1) and 𝑟 is the scalar curvature.
𝑔 𝑄𝑋, 𝑌 = 𝑆(𝑋, 𝑌) for all 𝑋, 𝑌.
m-projective curvature tensor 𝑊 on an Riemannian manifold (𝑀 𝑛
, 𝑔) (𝑛 > 3) of type (1, 3) is defined as
follows (G.P.Pokhariyal and R.S. Mishra (1971)).
𝑊 𝑋, 𝑌 𝑍 = 𝑅 𝑋, 𝑌 𝑍 −
1
2(𝑛−1)
𝑆 𝑌, 𝑍 𝑋 − 𝑆 𝑋, 𝑍 𝑌 + 𝑔(𝑌, 𝑍)𝑄𝑋 − 𝑔(𝑋, 𝑍)𝑄𝑌 , (2.18)
so that ′𝑊 𝑋, 𝑌, 𝑍, 𝑈 ≝ 𝑔 𝑊 𝑋, 𝑌 𝑍, 𝑈 =′
𝑊 𝑍, 𝑈, 𝑋, 𝑌 .
On an 𝑛- dimensional LP-Sasakian manifold, the Concircular curvature tensor 𝐶 is defined as
𝐶 𝑋, 𝑌 𝑍 = 𝑅 𝑋, 𝑌 𝑍 −
𝑟
𝑛 𝑛−1
𝑔 𝑌, 𝑍 𝑋 − 𝑔 𝑋, 𝑍 𝑌 . (2.19)
Now, in view of 𝑆 𝑋, 𝑌 =
𝑟
𝑛
𝑔 𝑋, 𝑌 , (2.18) becomes
𝑊 𝑋, 𝑌 𝑍 = 𝐶 𝑋, 𝑌 𝑍.
Thus, in an Einstein LP-Sasakian manifold, m-projective curvature tensor 𝑊 and the concircular curvature
tensor 𝐶 concide.
III. m-projectively flat LP-Sasakian manifold
In this section we assume that 𝑊 𝑋, 𝑌 𝑍 = 0.
Then from (2.18), we get
𝑅 𝑋, 𝑌 𝑍 =
1
2(𝑛−1)
𝑆 𝑌, 𝑍 𝑋 − 𝑆 𝑋, 𝑍 𝑌 + 𝑔(𝑌, 𝑍)𝑄𝑋 − 𝑔(𝑋, 𝑍)𝑄𝑌 . (3.1)
Contracting (3.1) with respect to 𝑋, we get
𝑆 𝑌, 𝑍 =
𝑟
𝑛
𝑔(𝑌, 𝑍). (3.2)
Hence we can state the following theorem.
Theorem 3.1: Let 𝑀 𝑛
be an 𝑛-dimensional m-projectively flat LP-Sasakian manifold, then 𝑀 𝑛
be an Einstien
manifold.
Contracting (2.18) with respect to 𝑋, we get
𝐶1
1
𝑊 𝑌, 𝑍 = 𝑆 𝑌, 𝑍 −
1
2(𝑛−1)
𝑛𝑆 𝑌, 𝑍 − 𝑆 𝑌, 𝑍 + 𝑟𝑔(𝑌, 𝑍) − 𝑔(𝑌, 𝑍) , (3.3)
𝑂𝑟, 𝐶1
1
𝑊 𝑌, 𝑍 =
𝑛
2(𝑛−1)
𝑆 𝑌, 𝑍 −
𝑟
𝑛
𝑔 𝑌, 𝑍 , (3.4)
where 𝐶1
1
𝑊 𝑌, 𝑍 is the contraction of 𝑊 𝑋, 𝑌 𝑍 with respect to 𝑋.
If 𝐶1
1
𝑊 𝑌, 𝑍 = 0, then from (3.4), we get
𝑆 𝑌, 𝑍 =
𝑟
𝑛
𝑔 𝑌, 𝑍 . (3.5)
Using (3.5) in (3.1), we get
′𝑅 𝑋, 𝑌, 𝑍, 𝑊 =
𝑟
𝑛 𝑛−1
𝑔 𝑌, 𝑍 𝑔 𝑋, 𝑊 − 𝑔 𝑋, 𝑍 𝑔 𝑌, 𝑊 . (3.6)
Theorem 3.2. An m-projectively flat LP-Sasakian manifold satisfying 𝐶1
1
𝑊 𝑌, 𝑍 = 0 is a manifold of
constant curvature.

Using (3.5) and (3.6) in (2.18), we get
𝑊 𝑋, 𝑌 𝑍 = 0,
i.e. the manifold 𝑀 𝑛
Theorem 3.3. An LP-Sasakian manifold (𝑀 𝑛
, 𝑔) (𝑛 > 3) satisfying 𝐶1
1
𝑊 𝑌, 𝑍 = 0, is of constant curvature
IV. Einstein LP-Sasakian manifold satisfying 𝒅𝒊𝒗 𝑾 𝑿, 𝒀 𝒁 = 𝟎
Definition 4.1. A manifold (𝑀 𝑛
, 𝑔) (𝑛 > 3) is called m-projectively conservative if (Hicks N.J.(1969)),
𝑑𝑖𝑣 𝑊 = 0, (4.1)
where 𝑑𝑖𝑣 denotes divergence.
Now differentiating (2.18) covariently, we get
𝐷 𝑈 𝑊 𝑋, 𝑌 𝑍 = 𝐷 𝑈 𝑅 𝑋, 𝑌 𝑍 −
1
2(𝑛 − 1)
[ 𝐷 𝑈 𝑆 𝑌, 𝑍 𝑋 − 𝐷 𝑈 𝑆 𝑋, 𝑍 𝑌 + 𝑔(𝑌, 𝑍) 𝐷 𝑊 𝑄 𝑋
−𝑔(𝑋, 𝑍) 𝐷 𝑊 𝑄 𝑌]. (4.2)
Which gives on contraction
𝑑𝑖𝑣 𝑊 𝑋, 𝑌 𝑍 = 𝑑𝑖𝑣 𝑅 𝑋, 𝑌 𝑍 −
1
2 𝑛−1
𝐷𝑋 𝑆 𝑌, 𝑍 − 𝐷 𝑌 𝑆 𝑋, 𝑍 + 𝑔 𝑌, 𝑍 𝑑𝑖𝑣 𝑄 𝑋 −
𝑔𝑋, 𝑍𝑑𝑖𝑣𝑄𝑌. (4.3)
But 𝑑𝑖𝑣 𝑄 =
1
2
𝑑𝑟, using in (4.3), we get
𝑑𝑖𝑣 𝑊 𝑋, 𝑌 𝑍 = 𝑑𝑖𝑣 𝑅 𝑋, 𝑌 𝑍 −
1
2 𝑛−1
𝐷𝑋 𝑆 𝑌, 𝑍 − 𝐷 𝑌 𝑆 𝑋, 𝑍 +
1
2
𝑔 𝑌, 𝑍 𝑑𝑟 𝑋 −
12𝑔𝑋, 𝑍𝑑𝑟𝑌. (4.4)
But from (Eisenhart L.P.(1926)), we have
𝑑𝑖𝑣 𝑅 𝑋, 𝑌 𝑍 = 𝐷𝑋 𝑆 𝑌, 𝑍 − 𝐷 𝑌 𝑆 𝑋, 𝑍 . (4.5)
𝑑𝑖𝑣 𝑊 𝑋, 𝑌 𝑍 =
(2𝑛−3)
2(𝑛−1)
𝐷𝑋 𝑆 𝑌, 𝑍 − 𝐷 𝑌 𝑆 𝑋, 𝑍 −
1
4 𝑛−1
𝑔 𝑌, 𝑍 𝑑𝑟 𝑋 − 𝑔 𝑋, 𝑍 𝑑𝑟 𝑌 . (4.6)
If LP-Sasakian manifold is an Einstein manifold, then from (1.12) and (4.5), we get
𝑑𝑖𝑣 𝑅 𝑋, 𝑌 𝑍 = 0. (4.7)
From (4.6) and (4.7), we get
𝑑𝑖𝑣 𝑊 𝑋, 𝑌 𝑍 = −
1
4(𝑛−1)
[ 𝑔 𝑌, 𝑍 𝑑𝑟 𝑋 − 𝑔 𝑋, 𝑍 𝑑𝑟 𝑌 ]. (4.8)
From (4.1) and (4.8), we get
𝑔 𝑌, 𝑍 𝑑𝑟 𝑋 − 𝑔 𝑋, 𝑍 𝑑𝑟 𝑌 = 0,
which shows that 𝑟 is constant. Again if 𝑟 is constant then from (4.8), we get
𝑑𝑖𝑣 𝑊 𝑋, 𝑌 𝑍 = 0.
Theorem 4.1. An Einstein LP-Sasakian manifold (𝑀 𝑛
, 𝑔) (𝑛 > 3) is m-projectively conservative if and only if
the scalar curvature is constant.
V. 𝝋- m-projectively flat LP-Sasakian manifold
Definition 5.1. A differentiable manifold 𝑀 𝑛
, 𝑔 , 𝑛 > 3, satisfying the condition
𝜑2
𝑊 𝜑𝑋, 𝜑𝑌 𝜑𝑍 = 0, (5.1)
is called 𝜑-m - projectively flat LP-Sasakian manifold.(Cabrerizo, Fernandez, Fernandez and Zhen (1999)).
Suppose that 𝑀 𝑛
, 𝑔 , 𝑛 > 3 is a 𝜑 - m - projectively flat LP-Sasakian manifold. It is easy to see that
𝜑2
𝑊 𝜑𝑋, 𝜑𝑌 𝜑𝑍 = 0, holds if and only if
𝑔 𝑊 𝜑𝑋, 𝜑𝑌 𝜑𝑍, 𝜑𝑊 = 0,
for any vector fields 𝑋, 𝑌, 𝑍, 𝑊.
By the use of (2.18), 𝜑 – m - projectively flat means
′𝑅 𝜑𝑋, 𝜑𝑌, 𝜑𝑍, 𝜑𝑊 =
1
2(𝑛−1)
𝑆 𝜑𝑌, 𝜑𝑍 𝑔 𝜑𝑋, 𝜑𝑊 − 𝑆 𝜑𝑋, 𝜑𝑍 𝑔 𝜑𝑌, 𝜑𝑊 +
𝑔 𝜑𝑌, 𝜑𝑍 𝑆 𝜑𝑋, 𝜑𝑊 − 𝑔 𝜑𝑋, 𝜑𝑍 𝑆 𝜑𝑌, 𝜑𝑊
, (5.2)
where ′𝑅 𝑋, 𝑌, 𝑍, 𝑊 = 𝑔 𝑅 𝑋, 𝑌 𝑍, 𝑊 .

Let {𝑒1, 𝑒2, … , 𝑒 𝑛−1, 𝜉} be a local orthonormal basis of vector fields in 𝑀 𝑛
by using the fact that
{𝜑𝑒1, 𝜑𝑒2, … , 𝜑𝑒 𝑛−1, 𝜉} is also a local orthonormal basis, if we put 𝑋 = 𝑊 = 𝑒𝑖 in (5.2) and sum up with respect
to 𝑖, then we have
′𝑅(𝜑𝑒𝑖
𝑛−1
𝑖=1
, 𝜑𝑌, 𝜑𝑍, 𝜑𝑒𝑖) =
1
2 𝑛 − 1
[𝑆 𝜑𝑌, 𝜑𝑍 𝑔 𝜑𝑒𝑖, 𝜑𝑒𝑖
𝑛−1
𝑖=1
− 𝑆 𝜑𝑒𝑖, 𝜑𝑍 𝑔 𝜑𝑌, 𝜑𝑒𝑖
+𝑔 𝜑𝑌, 𝜑𝑍 𝑆 𝜑𝑒𝑖, 𝜑𝑒𝑖 − 𝑔 𝜑𝑒𝑖, 𝜑𝑍 𝑆 𝜑𝑌, 𝜑𝑒𝑖 ]. (5.3)
On an LP-Sasakian manifold, we have (𝑂 𝑧𝑔𝑢r (2003))
′𝑅(𝜑𝑒𝑖
𝑛−1
𝑖=1
, 𝜑𝑌, 𝜑𝑍, 𝜑𝑒𝑖) = 𝑆 𝜑𝑌, 𝜑𝑍 + 𝑔 𝜑𝑌, 𝜑𝑍 , (5.4)
𝑆(
𝑛−1
𝑖=1
𝜑𝑒𝑖, 𝜑𝑒𝑖) = 𝑟 + 𝑛 − 1 , (5.5)
𝑔 𝜑𝑒𝑖, 𝜑𝑍 𝑆(𝜑𝑌,
𝑛−1
𝑖=1
𝜑𝑒𝑖) = 𝑆 𝜑𝑌, 𝜑𝑍 , (5.6)
𝑔(
𝑛−1
𝑖=1
𝜑𝑒𝑖, 𝜑𝑒𝑖) = 𝑛 + 1 , (5.7)
𝑔 𝜑𝑒𝑖, 𝜑𝑍 𝑔(𝜑𝑌,
𝑛−1
𝑖=1
𝜑𝑒𝑖) = 𝑔 𝜑𝑌, 𝜑𝑍 , (5.8)
so, by the virtue of (5.4)−(5.8), the equation(5.3) takes the form
𝑆 𝜑𝑌, 𝜑𝑍 =
𝑟
𝑛−1
− 1 𝑔 𝜑𝑌, 𝜑 𝑍 . (5.9)
By making the use of (2.3) and (2.17) in (5.9), we get
𝑆 𝑌, 𝑍 =
𝑟
𝑛−1
− 1 𝑔 𝑌, 𝑍 +
𝑟
𝑛−1
− 𝑛 𝜂 𝑌 𝜂 𝑍 .
Theorem 5.1. Let 𝑀 𝑛
be an 𝑛-dimenstional 𝑛 > 3, 𝜑 – m - projectively flat LP-Sasakian manifold, then 𝑀 𝑛
is
an 𝜂 −Einstein manifold with constants 𝛼 =
𝑟
𝑛−1
− 1 and 𝛽 =
𝑟
𝑛−1
− 𝑛 .
VI. quasi m-projectively flat LP-Sasakian manifold
Definition 6.1. An LP-Sasakian manifold 𝑀 𝑛
is said to be quasi m-projectively flat, if
𝑔 𝑊 𝑋, 𝑌 𝑍, 𝜑𝑈 = 0, (6.1)
for any vector fields 𝑋, 𝑌, 𝑍, 𝑈.
From (2.18), we get
𝑔 𝑊 𝑋, 𝑌 𝑍, 𝜑𝑈 = 𝑔(𝑅 𝑋, 𝑌 𝑍, 𝜑𝑈) −
1
2 𝑛−1
[𝑆 𝑌, 𝑍 𝑔(𝑋, 𝜑𝑈) − 𝑆 𝑋, 𝑍 𝑔(𝑌, 𝜑𝑈)
+ 𝑔 𝑌, 𝑍 𝑆 𝑋, 𝜑𝑈 − 𝑔 𝑋, 𝑍 𝑆 𝑌, 𝜑𝑈 ]. (6.2)
Let {𝑒1, 𝑒2, … , 𝑒 𝑛−1, 𝜉} be a local orthonormal basis of vector fields in 𝑀 𝑛
by using the fact that
{𝜑𝑒1, 𝜑𝑒2, … , 𝜑𝑒 𝑛−1, 𝜉} is also a local orthonormal basis, if we put 𝑋 = 𝜑𝑒𝑖 , 𝑈 = 𝑒𝑖 in (5.2) and sum up with
respect to 𝑖, then we have
𝑔 𝑊 𝜑𝑒𝑖, 𝑌 𝑍, 𝜑𝑒𝑖
𝑛−1
𝑖=1
= 𝑔(𝑅 𝜑𝑒𝑖, 𝑌 𝑍, 𝜑𝑒𝑖)
𝑛−1
𝑖=1
−
1
2 𝑛 − 1
[
𝑛−1
𝑖=1
𝑆 𝑌, 𝑍 𝑔 𝜑𝑒𝑖, 𝜑𝑒𝑖 −
𝑆 𝜑𝑒𝑖, 𝑍 𝑔 𝑌, 𝜑𝑒𝑖 + 𝑔 𝑌, 𝑍 𝑆 𝜑𝑒𝑖 , 𝜑𝑒𝑖 − 𝑔 𝜑𝑒𝑖, 𝑍 𝑆 𝑌, 𝜑𝑒𝑖 ]. (6.3)
On an LP-Sasakian manifold by straight forward calculation, we get
′𝑅(𝑒𝑖
𝑛−1
𝑖=1
, 𝑌, 𝑍, 𝑒𝑖) = ′𝑅(𝜑𝑒𝑖
𝑛−1
𝑖=1
, 𝑌, 𝑍, 𝜑𝑒𝑖) = 𝑆 𝑌, 𝑍 + 𝑔 𝜑𝑌, 𝜑𝑍 , (6.4)
𝑆(
𝑛−1
𝑖=1
𝜑𝑒𝑖, 𝑍)𝑔 𝑌, 𝜑𝑒𝑖 = 𝑆 𝑌, 𝑍 − 𝑛 − 1 𝜂 𝑌 𝜂 𝑍 . (6.5)
Using (5.4), (5.7), (6.4), (6.5) in (6.3), we get
𝑔(𝑊(𝜑𝑒𝑖
𝑛−1
𝑖=1
, 𝑌)𝑍, 𝜑𝑒𝑖) = 𝑆 𝑌, 𝑍 + 𝑔 𝜑𝑌, 𝜑𝑍

−
1
2(𝑛−1)
𝑛 − 1 𝑆 𝑌, 𝑍 + 𝑟 + 𝑛 − 1 𝑔 𝑌, 𝑍 + 2 𝑛 − 1 𝜂 𝑌 𝜂 𝑍 . (6.6)
𝑔(𝑊(𝜑𝑒𝑖
𝑛−1
𝑖=1
, 𝑌)𝑍, 𝜑𝑒𝑖) =
1
2
𝑆 𝑌, 𝑍 −
𝑟
𝑛
− 1 𝑔(𝑌, 𝑍) . (6.7)
If 𝑀 𝑛
is quasi m-projectively flat , then (6.7) reduces to
𝑆 𝑌, 𝑍 =
𝑟
𝑛
− 1 𝑔 𝑌, 𝑍 . (6.8)
Putting 𝑍 = 𝜉 in (6.8) and then using (2.6) and (2.16), we get
𝑟 = 𝑛 𝑛 − 1 . (6.9)
𝑆 𝑌, 𝑍 = 𝑛 − 1 𝑔 𝑌, 𝑍 . (6.10)
i.e. 𝑀 𝑛
is an Einstein manifold.
Now using (6.10) in (2.18), we get
𝑊 𝑋, 𝑌 𝑍 = 𝑅 𝑋, 𝑌 𝑍 − 𝑔 𝑌, 𝑍 𝑋 − 𝑔 𝑋, 𝑍 𝑌 . (6.11)
If LP-Sasakian manifold is m-projectively flat , then from (6.11), we get
𝑅 𝑋, 𝑌 𝑍 = [𝑔 𝑌, 𝑍 𝑋 − 𝑔 𝑋, 𝑍 𝑌 ]. (6.12)
Theorem 6.2. A quasi m-projectively flat LP-Sasakian manifold 𝑀 𝑛
is locally isomeric to the unit sphere
𝑆 𝑛
1 if and only if 𝑀 𝑛
References
[1] D. E. Blair, Contact manifolds on Riemannian geometry, Lecture Notes in Mathematics, Vol.509, Spinger-Verlag, Berlin, 1976.
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Messina, Series ΙΙ, Supplemento al 3 (1999), 149-158.
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