The object of the present paper is to study (LCS) _n-manifolds with vanishing quasi-conformal curvature tensor. (LCS) _n-manifolds satisfying Ricci-symmetric condition are also characterized.

1. Introduction

Recently, in [1], Shaikh introduced and studied Lorentzian concircular structure manifolds (briefly (LCS)-manifold) which generalizes the notion of LP-Sasakian manifolds, introduced by Matsumoto [2].

Generalizing the notion of LP-Sasakian manifold in 2003 [1], Shaikh introduced the notion of (LCS) _n-manifolds along with their existence and applications to the general theory of relativity and cosmology. Also, Shaikh and his coauthors studied various types of (LCS) _n-manifolds by imposing the curvature restrictions (see [3–6]). In [7, 8], the authors also studied (LCS) _2n+1-manifolds.

The submanifold of an (LCS) _n-manifold is studied by Atceken and Hui [9, 10] and Shukla et al. [11]. In [12], Yano and Sawaki introduced the quasi-conformal curvature tensor, and later it was studied by many authors with curvature restrictions on various structures [13].

After then, the same author studied weakly symmetric (LCS) _n-manifolds by several examples and obtain various results in such manifolds. In [7], authors shown that a pseudo projectively flat and pseudo projectively recurrent (LCS) _n manifolds are η-Einstein manifold.

On the other hand, in [5], authors proved the existence of ϕ-recurrent (LCS) ₃ manifold which is neither locally symmetric nor locally ϕ-symmetric by nontrivial examples. Furthermore, they also give the necessary and sufficient conditions for a (LCS) _n-manifold to be locally ϕ-recurrent.

In this study, we have investigated the quasi-conformal flat (LCS) _n-manifolds satisfying properties such as Ricci-symmetric, locally symmetric, and η-Einstein. Finally, we give an example for η-Einstein manifolds.

2. Preliminaries

An n-dimensional Lorentzian manifold M is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric tensor g, that is, M admits a smooth symmetric tensor field g of the type (2,0) such that, for each p ∈ M,

()

is a nondegenerate inner product of signature (−, +, +, …, +). In such a manifold, a nonzero vector X_p ∈ T_M(p) is said to be timelike (resp., nonspacelike, null, and spacelike) if it satisfies the condition g_p(X_p, X_p) < 0 (resp., ≤0, =0, >0). These cases are called casual character of the vectors.

Definition 1. In a Lorentzian manifold (M, g), a vector field P defined by

()

for any X ∈ Γ(TM) is said to be a concircular vector field if

()

for Y ∈ Γ(TM), where α is a nonzero scalar function, A is a 1-form, w is also closed 1-form, and ∇ denotes the Levi-Civita connection on M [7].

Let M be a Lorentzian manifold admitting a unit timelike concircular vector field ξ, called the characteristic vector field of the manifold. Then we have

()

Since ξ is a unit concircular unit vector field, there exists a nonzero 1-form η such that

()

The equation of the following form holds:

()

for all X, Y ∈ Γ(TM), where α is a nonzero scalar function satisfying

()

ρ being a certain scalar function given by ρ = −ξ(α).

Let us put

()

then from (6) and (8), we can derive

()

which tell us that ϕ is a symmetric (1,1)-tensor. Thus the Lorentzian manifold M together with the unit timelike concircular vector field ξ, its associated 1-form η, and (1,1)-type tensor field ϕ is said to be a Lorentzian concircular structure manifold.

A differentiable manifold M of dimension n is called (LCS)-manifold if it admits a (1,1)-type tensor field ϕ, a covariant vector field η, and a Lorentzian metric g which satisfy

()

for all X ∈ Γ(TM). Particularly, if we take α = 1, then we can obtain the LP-Sasakian structure of Matsumoto [2].

Also, in an (LCS) _n-manifold M, the following relations are satisfied (see [3–6]):

()

for all vector fields X, Y, Z on M, where R and S denote the Riemannian curvature tensor and Ricci curvature, respectively, Q is also the Ricci operator given by S(X, Y) = g(QX, Y).

Now let (M, g) be an n-dimensional Riemannian manifold; then the concircular curvature tensor

, the Weyl conformal curvature tensor C, and the pseudo projective curvature tensor

are, respectively, defined by

()

where a and b are constants such that a, b ≠ 0, and τ is also the scalar curvature of M [7].

For an n-dimensional (LCS) _n-manifold the quasi-conformal curvature tensor

is given by

()

for all X, Y, Z ∈ Γ(TM) [14].

The notion of quasi-conformal curvature tensor was defined by Yano and Swaki [12]. If a = 1 and b = −1/(n − 1), then quasi-conformal curvature tensor reduces to conformal curvature tensor.

3. Quasi-Conformally Flat (LCS)_n-Manifolds and Some of Their Properties

For an n-dimensional quasi-conformally flat (LCS) _n-manifold, we know for Z = ξ from (23),

()

Here, taking into account of (16), we have

()

Let Y = ξ be in (25); then also by using (18) we obtain

()

Taking the inner product on both sides of the last equation by Y, we obtain

()

that is,

()

Now we are in a proposition to state the following.

Theorem 2. If an n-dimensional (LCS) _n-manifold M is quasi-conformally flat, then M is an η-Einstein manifold.

Now, let {e₁, e₂, …, e_n−1, ξ} be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = Y = e_i, ξ in (28), and taking summation for 1 ≤ i ≤ n − 1, we have

()

In view of (28) and (29), we obtain

()

which is equivalent to

()

for any X ∈ Γ(TM).

By using (29) and (31) in (23) for a quasi-conformally flat (LCS) _n-manifold M, we get

()

for all X, Y, Z ∈ Γ(TM). If we consider Schur′s Theorem, we can give the following the theorem.

Theorem 3. A quasi-conformally flat (LCS) _n-manifold M (n > 1) is a manifold of constant curvature (α² − ρ) provided that a + b(n − 2) ≠ 0.

Now let us consider an (LCS) _n-manifold M which is conformally flat. Thus we have from (21) that

()

for all vector fields X, Y, Z tangent to M. Setting Z = ξ in (33) and using (16), (18) we have

()

If we put Y = ξ in (34) and also using (18), we obtain

()

Corollary 4. A conformally flat (LCS) _n-manifold is an η-Einstein manifold.

Generalizing the notion of a manifold of constant curvature, Chen and Yano [15] introduced the notion of a manifold of quasi-constant curvature which can be defined as follows:

Definition 5. A Riemannian manifold is said to be a manifold of quasi-constant curvature if it is conformally flat and its curvature tensor of type (0,4) is of the form

()

for all X, Y, Z, W ∈ Γ(TM), where a, b are scalars of which b ≠ 0 and A is a nonzero 1-form (for more details, we refer to [13, 16]).

Thus we have the following theorem for (LCS) _n-conformally flat manifolds.

Theorem 6. A conformally flat (LCS) _n-manifold is a manifold of quasi-constant curvature.

Proof. From (33) and (35), we obtain

()

This implies (36) for

()

This proves our assertion.

Next, differentiating the (19) covariantly with respect to W, we get

()

for any X, Y ∈ Γ(TM). Making use of the definition of ∇S and (8), we have

()

Thus we have

()

Here taking account of (17), we arrive at

()

Again, by using (13), (18), and (19), we reach

()

Thus we have the following theorem.

Theorem 7. If an (LCS) _n-manifold M is Ricci-symmetric; then α² − ρ is constant.

Proof. If n > 1-dimensional (LCS) _n-manifold M is Ricci-symmetric, then from (43) we conclude that

()

It follows that

()

from which

()

which is equivalent to

()

that is,

()

which proves our assertion.

Since ∇R = 0 implies that ∇S = 0, we can give the following corollary.

Corollary 8. If an n-dimensional (LCS) _n-manifold M is locally symmetric, then α² − ρ is constant.

Now, taking the covariant derivation of the both sides of (18) with respect to Y, we have

()

From the definition of the covariant derivation of Ricci-tensor, we have

()

If an (LCS) _n-manifold M Ricci symmetric, then Theorem 7 and (43) imply that

()

This leads us to state the following.

Theorem 9. If an (LCS) _n-manifold M is Ricci symmetric, then it is an Einstein manifold.

Corollary 10. If an (LCS) _n-manifold M is locally symmetric, then it is an Einstein manifold.

In this section, an example is used to demonstrate that the method presented in this paper is effective. But this example is a special case of Example 6.1 of [6].

Example 11. Now, we consider the 3-dimensional manifold

()

where (x, y, z) denote the standard coordinates in ℝ³. The vector fields

()

are linearly independent of each point of M. Let g be the Lorentzian metric tensor defined by

()

for i, j = 1,2, 3. Let η be the 1-form defined by η(Z) = g(Z, e₃) for any Z ∈ Γ(TM). Let ϕ be the (1,1)-tensor field defined by

()

Then using the linearity of ϕ and g, we have η(e₃) = −1,

()

for all Z, W ∈ Γ(TM). Thus for ξ = e₃, (ϕ, ξ, η, g) defines a Lorentzian paracontact structure on M.

Now, let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g, and let R be the Riemannian curvature tensor of g. Then we have

()

Making use of the Koszul formulae for the Lorentzian metric tensor g, we can easily calculate the covariant derivations as follows:

()

From the previously mentioned, it can be easily seen that (ϕ, ξ, η, g) is an (LCS) ₃-structure on M, that is, M is an (LCS) ₃-manifold with α = −1 and ρ = 0. Using the previous relations, we can easily calculate the components of the Riemannian curvature tensor as follows:

()

By using the properties of R and definition of the Ricci tensor, we obtain

()

Thus the scalar curvature τ of M is given by

()

On the other hand, for any Z, W ∈ Γ(TM), Z and W can be written as

and

, where f_i and g_i are smooth functions on M. By direct calculations, we have

()

Since η(Z) = −f₃ and η(W) = −g₃ and g(Z, W) = f₁g₁ + f₂g₂ − f₃g₃, we have

()

This tell us that M is an η-Einstein manifold.

Acknowledgment

The authors would like to thank the reviewers for the extremely carefully reading and for many important comments, which improved the paper considerably.

References

1 Shaikh A. A., On Lorentzian almost paracontact manifolds with a structure of the concircular type, Kyungpook Mathematical Journal. (2003) 43, no. 2, 305–314, MR1983436, ZBL1054.53056.
Google Scholar
2 Matsumoto K., On Lorentzian paracontact manifolds, Bulletin of Yamagata University. (1989) 12, no. 2, 151–156, MR994289, ZBL0675.53035.
Google Scholar
3 Shaikh A. A., Some results on (LCS)_n-manifolds, Journal of the Korean Mathematical Society. (2009) 46, no. 3, 449–461, https://doi.org/10.4134/JKMS.2009.46.3.449, MR2515129.
10.4134/JKMS.2009.46.3.449
Web of Science® Google Scholar
4 Shaikh A. A. and Hui S. K., On generalized ρ-recurrent (LCS) _n-manifolds, 1309, Proceedings of the ICMS International Conference on Mathematical Science, 2010, 419–429, American Institute of Physics Conference Proceedings, https://doi.org/10.1063/1.3525143.
10.1063/1.3525143
Google Scholar
5 Shaikh A. A., Basu T., and Eyasmin S., On the existence of ϕ-recurrent (LCS) _n-manifolds, Extracta Mathematicae. (2008) 23, no. 1, 71–83, MR2449997.
Google Scholar
6 Shaikh A. A. and Binh T. Q., On weakly symmetric (LCS) _n-manifolds, Journal of Advanced Mathematical Studies. (2009) 2, no. 2, 103–118, MR2583641, ZBL1224.53039.
Google Scholar
7 Sreenivasa G. T., Venkatesha, and Bagewadi C. S., Some results on (LCS) _2n+1-manifolds, Bulletin of Mathematical Analysis and Applications. (2009) 1, no. 3, 64–70, MR2578117.
Google Scholar
8 Yadav S. K., Dwivedi P. K., and Suthar D., On (LCS) _2n+1-manifolds satisfying certain conditions on the concircular curvature tensor, Thai Journal of Mathematics. (2011) 9, no. 3, 597–603, MR2862053.
Google Scholar
9 Atceken M., On geometry of submanifolds of (LCS) _n-manifolds, International Journal of Mathematics and Mathematical Sciences. (2012) 2012, 11, 304647, https://doi.org/10.1155/2012/304647, MR2888754.
10.1155/2012/304647
Google Scholar
10 Hui S. K. and Atceken M., Contact warped product semi-slant submanifolds of (LCS) _n-manifolds, Acta Universitatis Sapientiae. Mathematica. (2011) 3, no. 2, 212–224, MR2915836.
Google Scholar
11 Shukla S. S., Shukla M. K., and Prasad R., Slant submanifold of (LCS) _n-manifolds, to appear in Kyungpook Mathematical Journal.
Google Scholar
12 Yano K. and Sawaki S., Riemannian manifolds admitting a conformal transformation group, Journal of Differential Geometry. (1968) 2, 161–184, MR0233314, ZBL0167.19802.
10.4310/jdg/1214428253
Google Scholar
13 Shaikh A. A. and Jana S. K., On weakly quasi-conformally symmetric manifolds, SUT Journal of Mathematics. (2007) 43, no. 1, 61–83, MR2417157, ZBL1164.53011.
10.55937/sut/1189443276
Google Scholar
14 Kumar R. and Prasad B., On (LCS) _n-manifolds, to appear in Thai Journal of Mathematics.
Google Scholar
15 Chen B.-Y. and Yano K., Hypersurfaces of a conformally flat space, Tensor. (1972) 26, 318–322, MR0331283, ZBL0257.53027.
Google Scholar
16 Shaikh A. A. and Jana S. K., On weakly symmetric Riemannian manifolds, Publicationes Mathematicae Debrecen. (2007) 71, no. 1-2, 27–41, MR2340032, ZBL1136.53019.
10.5486/PMD.2007.3557
Web of Science® Google Scholar

All articles

Some Curvature Properties of (LCS) _n-Manifolds

Abstract

1. Introduction

2. Preliminaries

3. Quasi-Conformally Flat (LCS)_n-Manifolds and Some of Their Properties

Acknowledgment

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Some Curvature Properties of (LCS) n-Manifolds

Abstract

1. Introduction

2. Preliminaries

3. Quasi-Conformally Flat (LCS)n-Manifolds and Some of Their Properties

Acknowledgment

References

References

Related

Information

Some Curvature Properties of (LCS) _n-Manifolds

3. Quasi-Conformally Flat (LCS)_n-Manifolds and Some of Their Properties