Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 23–32.

CERTAIN CURVATURE CONDITIONS ON

(LCS)_N-MANIFOLD ACKNOWLEDGING

CONFORMAL η-EINSTEIN SOLITON

Srabani Debnath

Abstract. The present paper focuses on the study of conformal η-Einstein soliton

on (LCS)_n-manifold satisfying certain curvature conditions.The torse-forming vector ﬁeld

has also been considered for conformal η-Einstein soliton on (LCS)_n-manifold. Finally,

an example has been set up to verify the results obtained.

Keywords: (LCS)_n- manifold, Einstein soliton, Conformal η-Einstein soliton, W₂-

curvature tensor . 2010 AMS Subject Classiﬁcation: 53C15, 53C25, 53C44.

1. Introduction

In 2012, Bhattacharyya and Debnath [2] introduced a type of generalized Ricci ﬂow

whose special case is the Einstein ﬂow. Later, in 2016, the concept of Einstein soliton

was established in diﬀerential geometry by G.Catino and L.Mazzieri [4] which represents

a self-similar solution to the Einstein ﬂow

∂g

r

= −2(S − g)

2

∂t

where g represents the Riemannian metric, S stands for the Ricci tensor and r stands for

the scalar curvature.

In 2018 [3], the concept of η-Einstein soliton on a Riemannian manifold (M, g) was de-

veloped which is represented by the following equation:

£_ξg + 2S + (2λ − r)g + 2µη ⊗ η = 0.

(1)

Here £_ξdenotes the Lie derivative along the direction of the vector ﬁeld ξ and λ, µ are

considered to be real constants. When µ becomes 0, the soliton reduces to the Einstein

soliton. If the manifold admits a constant scalar curvature, the η-Einstein soliton converts

to η-Ricci soliton [5] and furthermore, if we consider µ to be zero, it becomes a Ricci soliton

( [7], [8], [22]).

The conformal η-Ricci soliton was initiated by M.D.Siddiqi [20] in 2018 and it is deﬁned

by,

2

£_ξg + 2S + (2λ − (p + ))g + 2µη ⊗ η = 0.

(2)

n

Here p is a scalar non-dynamical ﬁeld and n represents the dimension of the manifold.

Later on, the concept of conformal Einstein soliton was introduced by Roy et al. [12]

23

24

Srabani Debnath

which was introduced on an n-dimensional manifold by,

2

£_Vg + 2S + (2λ − r + (p + ))g = 0.

(3)

n

Next, focusing on the case where the potential vector ﬁeld ξ is of gradient type, i.e.,

ξ = grad(f), for a non-constant smooth function f on the manifold M, Roy et al. [13]

established the concept of conformal η-Einstein soliton on an n-dimensional manifold

given by,

2

£_ξg + 2S + (2λ − r + (p + ))g + 2µη ⊗ η = 0.

(4)

n

The soliton is happened to be shrinking, steady or expanding depending on λ < 0, λ = 0

or λ > 0.

Ricci soliton, Einstein soliton and their generalisations and related investigations have

been done by many authors ( [1], [?], [3], [?], [6], [19]) in diﬀerent contexts, such as on

Kahler manifolds, on contact and Lorentzian manifolds, on k-contact manifolds etc.

Being inspired by their study, here conformal η-Einstein soliton on (LCS)_n-manifold have

been considered. After giving a brief introduction in the ﬁrst section and recalling some

basic ideas on (LCS)_n-manifolds in section 2, section 3 focuses on some curvature condi-

tions on an (LCS)_n-manifold acknowledging conformal η-Einstein soliton. The section 4

emphasises on the (LCS)_n-manifold which is Einstein semi-symmetric. In section 5, the

study was chronicled on (LCS)_n-manifold admitting conformal η-Einstein soliton whose

vector ﬁeld is torse-forming. Finally, in the last section, an example has been set up to

verify the results.

2. Preliminaries

The concept of Lorentzian concircular structure manifold (brieﬂy, (LCS)_n-manifold)

was initiated by Shaikh [14] as a generalization of LP-Sasakian manifold which was es-

tablished by Matsumoto [9] and later on by Mihai and Rosca [10].

Let us consider a Lorentzian manifold (M, g) of dimension n which admits a unit timelike

concircular vector ﬁeld ξ and it satisﬁes, g(ξ, ξ) = −1. ξ, for being a unit concircular

vector ﬁeld, there should be a nonzero 1-form η which satisﬁes g(X, ξ) = η(X). Also

∇ξ = α(I + η ⊗ ξ) where α is a nowhere zero smooth function on the manifold which

obey the following relations:

(∇_Xα) = (Xα) = dα(X) = ρη(X),

(5)

ρ = −(ξα), a scalar function, ∇ denotes the Levi-Civita connection with respect to g and

X is a vector ﬁeld. The (1, 1) tensor ﬁeld φ satisﬁes,

αφX = ∇_Xξ.

(6)

For a (LCS)_nmanifold (n > 2), the following relations are satisﬁed( [15], [16], [17], [18], [23]):

η(ξ) = −1, φξ = 0, η(φX) = 0, g(φX, φY ) = g(X, Y ) + η(X)η(Y )

(7)

CERTAIN CURVATURE CONDITIONS ON (LCS)_N-MANIFOLD

25

φ²X = X + η(X)ξ

(8)

(9)

S(X, ξ) = (n − 1)(α²− ρ)η(X)

R(X, Y )ξ = (α²− ρ)[η(Y )X − η(X)Y ]

R(ξ, Y )Z = (α²− ρ)[g(Y, Z)ξ − η(Z)Y ]

(10)

(11)

(12)

(13)

(14)

(∇_Xφ)Y

= α[g(X, Y )ξ + 2η(X)η(Y )ξ + η(Y )X]

(Xρ) = dρ(X) = βη(X),

R(X, Y )Z = φR(X, Y )Z + (α²− ρ){g(Y, Z)η(X) − g(X, Z)η(Y )}ξ

for any vector ﬁelds X, Y, Z on M and β being a scalar function represented by,

β = −(ξρ),

where R stands for the curvature tensor and S denotes the Ricci tensor of the manifold.

Again,

R(X, Y )Z = ∇_X∇_YZ − ∇_Y∇_XZ − ∇_{[X,Y ]}Z,

(15)

(16)

1

W₂(X, Y )Z = R(X, Y )Z +

[g(X, Z)QY − g(Y, Z)QX]

n − 1

are the expressions for the Riemannian-Chistoﬀel curvature tensor R and the W₂-curvature

tensor in a (LCS)_nmanifold (Mⁿ, g), and Q, the Ricci operator is deﬁned by S(X, Y ) =

g(QX, Y ) , for X, Y, Z ∈ χ(M), the Lie algebra of vector ﬁelds of M.

If the manifold further admits a conformal η-Einstein soliton, then the expression for

S(X, Y ) becomes [6],

r

p

1

S(X, Y ) = [ − λ − α − ( + )]g(X, Y ) − (α + µ)η(X)η(Y ).

(17)

2

n

All these deﬁnitions and results will be required in next sections.

3. Conformal η-Einstein Soliton on (LCS)_n-Manifolds Satisfying

Projective Curvature Tensor, R(ξ, X).S = 0 and W₂(ξ, X).S = 0

Theorem 3.1. Let (M, g) be an (LCS)_n-manifold which admits a conformal η-Einstein

soliton (g, ξ, λ, µ). If the manifold fulﬁls the curvature condition R(ξ, X).S = 0, then the

manifold turns into an Einstein manifold.

Proof. Let us consider an (LCS)_n-manifold which admits a conformal η-Einstein soliton

(g, ξ, λ, µ) which meets the curvature condition R(ξ, X).S = 0, then it can be written as,

S(R(ξ, X)Y, Z) + S(Y, R(ξ, X)Z) = 0

(18)

(19)

for all X, Y, Z ∈ χ(M). Using (17) into the above expression we have,

r

p

1

[ − λ − α − ( + )](g(R(ξ, X)Y, Z) + g(Y, R(ξ, X)Z))

2

n

−(α + µ)[η(R(ξ, X)Y )η(z) + η(Y )η(R(ξ, X)Z)] = 0.

With the help of (11) we arrive at,

(α + µ)(α²− ρ)[g(X, Y )η(Z) + g(X, Z)η(Y ) + 2η(X)η(Y )η(Z)] = 0.

(20)

26

Srabani Debnath

Substituting Z by ξ in (20) and considering (7) we get,

(α + µ)(α²− ρ)g(φX, φY ) = 0

(21)

for all X, Y ∈ χ(M), which gives,

α + µ = 0,

since g(φX, φY ) = 0 and considering α²= ρ.

Hence (17) reduces to,

r

p

1

S(X, Y ) = [ − λ − α − ( + )]g(X, Y ),

(22)

2

n

for all X, Y ∈ χ(M) and hence the theorem follows.

□

The next result in this section focuses on W₂-curvature tensor which was ﬁrst considered

in 1970 by Pokhariyal and Mishra [11].

Theorem 3.2. Let (M, g) be an (LCS)_n-manifold which admits a conformal η-Einstein

soliton (g, ξ, λ, µ). If the curvature condition W₂(ξ, X).S = 0 is satisﬁed by the manifold,

then either the manifold becomes an Einstein manifold or the manifold is of constant

p

1

curvature r = 2λ + 2α + 2(₂+ ) or both.

n

Proof. Let us consider an (LCS)_n-manifold which admits a conformal η-Einstein soliton

(g, ξ, λ, µ) and satisﬁes the curvature condition W₂(ξ, X).S = 0. Then it can be written

as,

S(W₂(ξ, X)Y, Z) + S(Y, W₂(ξ, X)Z) = 0,

(23)

(24)

for all X, Y, Z ∈ χ(M). Using (17) into the above expression we obtain,

r

p

1

[ − λ − α − ( + )](g(W₂(ξ, X)Y, Z) + g(Y, W₂(ξ, X)Z))

2

n

−(α + µ)[η(W₂(ξ, X)Y )η(z) + η(Y )η(W₂(ξ, X)Z)] = 0.

(17) also gives us,

r

p

1

QX = [ − λ − α − ( + )]X − (α + µ)η(X)ξ.

(25)

(26)

2

n

Considering X to be ξ in (16) and using (11), (24), (25), we have,

W₂(ξ, Y )Z = Ag(Y, Z)ξ − Bη(Z)Y + (A − B)η(Y )η(Z)ξ

where

1

r

p

1

A = α²− ρ −

( − λ + µ − ( + )),

n − 1 2

2

p

n

1

r

B = α²− ρ −

( − λ − α − ( + )).

n − 1 2

2

n

Taking inner product with the vector ﬁeld ξ and using (7) the above equation yields

η(W₂(ξ, Y )Z) = −Ag(φY, φZ).

Using (26) and (27) in (24) and replacing Z by ξ and recalling (7) to obtain,

(27)

(28)

r

p

1

[(B − A)( − λ − α − ( + )) − A(α + µ)]g(φX, φY ) = 0

2

n

CERTAIN CURVATURE CONDITIONS ON (LCS)_N-MANIFOLD

27

(29)

(30)

∀X, Y ∈ χ(M). Since g(φX, φY ) = 0, so we can reach to the conclusion that

r

p

1

(B − A)( − λ − α − ( + )) − A(α + µ) = 0.

2

n

Recalling the expressions of A and B and simplifying, we arrive at,

1

p

1

(α + µ)[

(r − 2λ − α + µ − 2( + )) − α²+ ρ],

n − 1

2

n

which yields,

either

α + µ = 0,

or,

p

1

r − 2λ − α + µ − 2( + ) = (n − 1)(α²− ρ).

2

n

Now for α + µ = 0, (17) indicates that the manifold reduces to an Einstein manifold.

Otherwise, substituting Y by ξ in (17) and comparing with (9) we have

p

1

r = 2λ − 2µ + 2( + ) + 2(n − 1)(α²− ρ).

(31)

2

n

Comparing and combining these two expressions of r, we have,

p

1

r = 2λ + 2α + 2( + )

(32)

2

n

and thus the theorem is established.

□

4. Conformal η-Einstein Soliton On An Einstein Semi-Symmetric

(LCS)_n-Manifold

Deﬁnition 4.1. An (LCS)_n-manifold (M, g) is known to be Einstein semi-symmetric [21]

if R.E = 0 where the Einstein tensor E is given by,

r

E(X, Y ) = S(X, Y ) − g(X, Y )

(33)

n

where X, Y ∈ χ(M) and r is the scalar curvature of the manifold.

Lemma 4.2. An Einstein semi-symmetric (LCS)_n-Manifold is an Einstein manifold.

Proof. Let us consider an (LCS)_n-Manifold which is Einstein semi-symmetric. So it obeys

the curvature condition R.E = 0. So, we can write,

E(R(X, Y )Z, W) + E(Z, R(X, Y )W) = 0

(34)

(35)

for all vector ﬁelds X, Y, Z, W on M. Then taking the help of (33), (34) becomes,

r

S(R(X, Y )Z, W) + S(Z, R(X, Y )W) =

[g(R(X, Y )Z, W) + g(Z, R(X, Y )W)].

n

Substituting X and Z by ξ in (35) and using (9) and (11), we obtain,

S(Y, W) = (n − 1)(α²− ρ)g(Y, W)

(36)

for all vector ﬁelds Y, W on M, which establishes that the manifold is Einstein.

□

28

Srabani Debnath

Theorem 4.3. Let (M, g) be an (LCS)_n-manifold acknowledging a conformal η-Einstein

soliton (g, ξ, λ, µ). If the manifold is Einstein semi-symmetric, then the soliton is happened

2

to be steady, shrinking or expanding according as (n−1)(n−2)(α²−ρ)−2α−p− = 0, < 0

n

or > 0 respectively.

Proof. Let us consider an (LCS)_n-manifold which admits a conformal η-Einstein soliton

(g, ξ, λ, µ). If the manifold is Einstein semi-symmetric, then Lemma 1 assures that the

manifold is Einstein. So, (17) gives,

α + µ = 0.

(37)

Replacing Y by ξ in (17) on using (37) and comparing with (9) to obtain,

r

p

1

− λ − α − ( + ) = (n − 1)(α²− ρ).

(38)

2

n

Again, taking an orthonormal basis {e₁, e₂, ..., e_n} of (M, g) and then replacing X and Y

by e_iin (17) and summing over i to arrive at,

2

pn

r =

[λn + nα +

+ 1]

(39)

n − 2

2

with the help of (37). So, comparing the two expressions of r, we ﬁnally came to the

conclusion that

1

2

λ = [(n − 1)(n − 2)(α²− ρ) − 2α − p − ]

(40)

2

n

and this establishes the result.

□

5. Conformal η-Einstein Soliton On An (LCS)_n-Manifold with

Torse-Forming Reeb Vector Field

In this section, the nature of conformal η-Einstein soliton on (LCS)_n-manifold with

torse-forming vector ﬁeld will be studied.

Deﬁnition 5.1. A vector ﬁeld ξ on an (LCS)_n-manifold is said to be a torse-forming

vector ﬁeld [24] if

∇_Xξ = fX + γ(X)ξ

(41)

∀X ∈ χ(M) for a smooth function f ∈ C^∞(M) and γ is a 1-form.

A torse-forming vector ﬁeld ξ is said to be recurrent when f = 0.

Deﬁnition 5.2. A vector ﬁeld V is considered as a concurrent vector ﬁeld when it satisﬁes

∇_XV = 0

for any X ∈ χ(M).

Theorem 5.3. Let (M, g) be an (LCS)_n-manifold acknowledging a conformal η-Einstein

soliton (g, ξ, λ, µ) with torse-forming vector ﬁeld ξ, then the manifold reduces to η-Einstein

manifold with scalar curvature

p

1

r = 2λ − 2µ + 2( + ) + (n − 1)(α²− ρ).

2

n

CERTAIN CURVATURE CONDITIONS ON (LCS)_N-MANIFOLD

29

Proof. Let (M, g) be an (LCS)_n-manifold which admits a conformal η-Einstein soliton

and consider that the Reeb vector ﬁeld ξ of the manifold to be torse-forming. Then,

taking the inner product of (41) with ξ, we have,

g(∇_Xξ, ξ) = fη(X) − γ(X) ∀X ∈ χ(M).

Again, taking the inner product of (6) with ξ, with the help of (7), yields,

g(∇_Xξ, ξ) = 0.

(42)

(43)

Comparing the above two equations, we conclude that,

γ = fη,

which results, for an (LCS)_n-manifold with torse-forming vector ﬁeld,

∇_Xξ = f[X + η(X)ξ].

(44)

Now considering that the manifold admits a conformal η-Einstein soliton and keeping in

mind that

(£_ξg)(X, Y ) = g(∇_Xξ, Y ) + g(X, ∇_Yξ),

(45)

we obtain from (4),

r

p

1

S(X, Y ) = [ − λ − f − ( + )]g(X, Y ) − (f + µ)η(X)η(Y ).

(46)

2

n

which establishes the manifold to be an η-Einstein manifold. From(44) substituting Y = ξ

we get,

r

p

1

S(X, ξ) = [ − λ + µ − ( + )]η(X).

(47)

(48)

2

n

Comparing the above expression with (9), we have,

r

p

1

− λ + µ − ( + ) = (n − 1)(α²− ρ),

2

n

which results,

p

1

r = 2λ − 2µ + 2( + ) + (n − 1)(α²− ρ).

(49)

2

n

□

Theorem 5.4. Let (g, ξ, λ, µ) be a conformal η-Einstein soliton on an (LCS)_n-manifold

where ξ is a torse-forming vector ﬁeld, then the following results hold:

(i) f = −(b + µ)

(ii) if ξ is recurrent, then it is also concurrent and Killing vector ﬁeld.

Proof. Let us consider an (LCS)_n-manifold (M, g) which admits a conformal η-Einstein

soliton (g, ξ, λ, µ), where ξ is a torse-forming vector ﬁeld. Considering (17) with a =

p

r

1

− λ − α − (₂+ ) and b = −(α + µ) and recalling (45) the expression (4) yields,

2

n

2

g(∇_Xξ, Y ) + g(X, ∇_Yξ) + [2a + 2λ − r + (p + )]g(X, Y ) + [2(b + µ)]η(X)η(Y ) = 0 (50)

n

Setting X = Y = ξ in (50) and using (6) and (7), we obtain,

2

2a + 2λ − r + (p + ) = 2(b + µ),

(51)

n

30

Srabani Debnath

which makes (50),

g(∇_Xξ, Y ) + g(X, ∇_Yξ) + 2(b + µ)g(φX, φY ) = 0.

Applying (44) in (52) we have,

(52)

(53)

2(f + b + µ)g(φX, φY ) = 0,

which results, f = −(b + µ), since, g(φX, φY ) = 0.

If we consider ξ to be recurrent, then from (44), we have,

∇_Xξ = 0

∀X ∈ χ(M) and which proves the vector ﬁeld ξ to be concurrent. (45) also indicates that

ξ is a Killing vector ﬁeld in that case and hence the theorem is established.

□

6. Example

4

Consider the 4-dimensional manifold M⁴= (x, y, z, t) ∈ R with the metric g = −dt²+

e^2t(dx²+ dy²+ dz²) where (x, y, z, t) are standard co-ordinates in R .

4

∂

∂t

∂

Let us consider the vector ﬁelds {e₁, e₂, e₃, e₄} on M given by, e₁=

, e₂= e^−t_∂x, e₃=

∂

∂y

∂

∂z

e^−t

, e₄= e^−t

which are linearly independent.

Let us consider the Riemannian metric g as,

g(e₁, e₁) = −1, g(e_i, e_j) = δ_ij, i, j = 1.

Let the 1-form η be deﬁned by,

η(Z) = g(Z, e₁)

for any vector ﬁeld Z on M and the (1,1)-tensor ﬁeld φ is deﬁned by,

φ(e₁) = 0, φ(e_i) = e_i(i = 2, 3, 4).

Then with the help of the linear property of φ and g, we have,

η(e₁) = −1, φ²(X) = X + η(X)e₁and g(φX, φY ) = g(X, Y ) + η(X)η(Y )

for any X, Y ∈ χ(M).

Considering ξ = e₁, using Koszul’s formula,

∇_ee₁= e_i, ∇_ee_i= e₁, (i = 2, 3, 4)

i

and all others are zero.

Consequently, we have (M, g, ξ, η, φ, α) is an (LCS)_n-manifold, where α = 1.

Using the above relations, we can compute the components of the curvature tensor R and

the Ricci tensor S as follows:

R(e₁, e_i)e₁= −e_i, R(e₁, e_i)e_j= −δ_ije₁,

R(e_i, e_j)e_k= δ_ike_j− δ_jke_i, R(e_i, e_j)e₁= 0, i, j, k = 2, 3, 4

S(e₁, e₁) = −3, S(e_i, e_j) = 3δ_ij, i, j = 2, 3, 4.

So, we can deduce from here that the scalar curvature r = 12.

Next, let us consider the vector ﬁeld V = ξ. Then we can easily calculate,

(£_ξg)(e₁, e₁) = 0, (£_ξg)(e_i, e_i) = 2, i = 2, 3, 4.

CERTAIN CURVATURE CONDITIONS ON (LCS)_N-MANIFOLD

31

then considering that the manifold admits a conformal η-Einstein soliton and replacing

X and Y by e_i, i = 1, 2, 3, 4 we arrive at,

2

2λ + (p + ) = 4,

n

which yields,

µ = −1.

Now we see that in this manifold R(ξ, X).S = 0 and the above result gives, α + µ = 0,

which results the manifold to be Einstein and veriﬁes theorem 1.

REFERENCES

[1] Bagewadi, C.S., Ingalahalli, G., Ricci solitons in Lorentzian α-Sasakian manifolds, Acta Math

Academiae Paedagogicae Nyiregyhaziensis, 28(1), 59–68, (2012).

[2] Bhattacharyya, A., Debnath, S.,A type of generalized Ricci ﬂow, J. Rajasthan Acad. Phy. Sci., Vol.

11, No. 2, June 2012, pp. 117-124.

[3] Blaga, A.M., On gradient η-Einstein solitons, Kraguj. J. Math. 42(2), 229–237 (2018).

[4] Catino, G., Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal., 132, pp. 66–94, (2016).

[5] Cho, J. T., Kimura, M., Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math.

J. 61, 205–212, (2009).

[6] Debnath, S., Conformal η-Einstein Soliton on (LCS)_n-Manifold, Mathematical Anaysis and Ap-

plications in Biological Phenomena through Modelling, Springer Proceedings in Mathematics and

Statistics, Vol. 478, pp. 19–31, (2025).

[7] Hamilton, R.S., The Ricci ﬂow on surfaces, Contemp. Math.(1988) Vol. 71, pp. 237–261.

[8] Hamilton, R.S., The formation of singularities in the Ricci ﬂow, Surv. Diﬀer. Geom. 2, 107–136

(1995).

[9] Matsumoto, K., On Lorentzian almost paracontact manifolds, Bull. Yamagata Univ. Nat. Sci. 12

(1989), 151–156.

[10] Mihai, I., Rosca, R., On Lorentzian para-Sasakian manifolds, In: Classical Anal., World Sci. Publ.,

Singapore, 1992, 155–169.

[11] Pokhariyal, G.P., Mishra, R.S., The curvature tensor and their relativistic signiﬁcance, Yokohoma

Math. J. 18, 105–108 (1970).

[12] Roy, S., Dey, S., Bhattacharyya, A., Conformal Einstein soliton within the framework of para-Kahler

manifold, Diﬀ. Geom.Dyn. Syst. 23, 235–243 (2021).

[13] Roy, S., Dey, S., Bhattacharyya, A., A Kenmotsu metric as a conformal η-Einstein soliton,

Carpathian Math. Pbl. 2021, 13(1), 110–118.

[14] Shaikh, A.A., On Lorentzian almost paracontact manifolds with a structure of the concircular type,

Kyungpook Math. J. 43 (2003), 305–314.

[15] Shaikh, A. A., Some results on (LCS)_n-manifolds, J. Korean Math. Soc. 46 (2009), 449–461.

[16] Shaikh, A. A., Ahmad, H., Some transformations on (LCS)_n-manifolds, Tsukuba J. Math. 38 (2014),

1–24.

[17] Shaikh, A. A., Baishya, K. K., On concircular structure spacetimes, J. Math. Stat. 1 (2005), 129–132.

[18] Shaikh, A. A., Baishya, K. K, On concircular structure spacetimes II, American J. Appl. Sci. 3, 4

(2006), 1790–1794.

[19] Sharma, R., Certain results on K-contact and (k, µ)-contact manifolds, J. Geom., 89, 138–147 (2008).

[20] Siddiqi, M.D., Conformal η-Ricci solitons in δ-Lorentzian Trans Sasakian manifolds, Intern. J. Maps

in Math. 2018, 1(1), 15–34.

[21] Szabo, S.I., Structure theorems on Riemannian spaces satisfying R(X,Y)Z=0, I. J. Diﬀer. Geom. 17,

531–582 (1982).

[22] Topping, P.,Lecture on the Ricci ﬂow, Cambridge University Press, Cambridge (2006).

[23] Yadav, S.K., Chaubey, S.K. and Suthar, D.L., Some results of η-Ricci solitons on LCS_n-manifolds,

Surveys in Mathematics and its Applications, Volume 13 (2018), 237–250.

[24] Yano, K., On torse-forming directions in Riemannian spaces, P. Imp. Acad. Tokyo 20, 701–705

(1944).

32

Srabani Debnath

(Received, September 26, 2025)

(Revised, December 2, 2025)

Department of Mathematics

Budge Budge College

Kolkata-700137, India

Email: srabani.1986@gmail.com