The Ricci flow, which is used to compute the canonical metric based on the smooth manifold, was proposed by Hamilton [19] in 1982. The Ricci flow provides an evolution expression of metrics for a Riemannian manifold as follows:

The Einstein metric can be naturally generalized to Ricci solitons which are defined on the Riemannian manifold (M,g∗) [6]. The triplet (g∗,V,ω1) is a Ricci soliton where g∗, V are the Riemannian metric or the pseudo Riemannian metric, and vector field (potential vector field), respectively. The real scalar ω1 is expressed in terms of a Ricci tensor S, and a Lie derivative operator £V, as

If ω1>0, the Ricci solitons is called expanding while it is called shrinking if ω1<0. The case ω1=0 represents the steady Ricci soliton [20]. The Einstein equation can be recovered from Ricci solitons for V=0. The metric expressed by (2) is generally known as quasi-Einstein [7], [8] and is used frequently in physical systems. The fixed Ricci flow points for ∂∂tg∗=−2S which are projected from metrics space onto its diffeomorphic quotient, modulo scaling, are referred to as the compact Ricci solitons. These solitons frequently arise in Ricci flow on compact manifolds for larger limiting cases. Ricci solitons play an interesting role in string theory, the initial aspects of which were discussed by Friedman [17]. The similarity solution of Ricci flow in Riemannian geometry was introduced by [19] as it explores the concept of a singularity. Several authors have studied the geometric properties of Ricci solitons over different manifolds, for instance see [11], [12], [14] and [15],[23].

Cho and Kimura [9] proposed the concept of η∗-Ricci solitons as type of generalized Ricci solitons. Calin and Crasmareanu [5], [6] extended this concept for Hopf hypersurfaces in complex space. The tuple (g∗,V,ω1,ω2) with constants ω1 and ω2 denote the η∗-Ricci solitons with the condition

In the current scenario, η-Ricci solitons are studied by various researchers have considered such η-Ricci solitons, and have found interesting geometric properties in many contexts: on Lorentzian para-Sasakian manifolds [2], [28], gradient η-Ricci solitons [3], on ϵ-para Sasakian manifolds [21] and [4], quasi-Sasakian 3-manifolds [22], 3-dimensional Kenmotsu manifolds [24], Sasakian 3-manifolds [25], para-Sasakian manifolds [26] and para Kenmotsu manifolds [29] and studied Lorentzian Sasakian manifold [27] etc.

The structure of the paper is as follows. The neccessary basic theory about α-LS manifolds is given in Section 2. In Section 3, the geometric properties of η∗-Ricci solitons on α-LS manifolds are investigated. In Section 4, we show that Ricci semisymmetric η∗-Ricci solitons on α-LS manifolds reduce to an η∗-Einstein manifold. In Section 5, we study φ∗-symmetric η∗-Ricci solitons on α-LS manifolds. In Section 6, we show that a φ∗-Ricci symmetric η∗-Ricci soliton on such a manifold is also an η∗-Einstein manifold. Finally, we provide an example of a 3-dimensional α-LS manifold for the existence of such solitons.

A (2n+1)-dimensional differentiable manifold M is said to be an α-LS manifold if it admits a (1,1)-tensor field φ∗, a vector field ζ and 1-form η∗ and Lorentzian metric g_* which satisfy the following conditions:

for any vector fields E,F on M.

Also, α-LS manifolds satisfy [16],

where ∇ has the usual meaning.

Moreover, on α-LS manifolds the following relations hold (see [1]):

where R, S, Q are the Riemannian curvature, Ricci tensor and Ricci operator, respectively while α is a constant. S and Q are related by S(E,F)=g∗(QE,F) for all E,F∈χ(M).

As per the definition of the Lie derivative, we have

Definition 2.1. ([18]) A (2n+1)-dimensional α-LS manifold with constants a, b and vector fields E,F defined on M satisfying the condition

is called an η∗-Einstein manifold.

Definition 2.2. An α-LS manifold M with vector fields E,F defined on M satisfying the condition

is called Ricci semisymmetric.

Definition 2.3. ([13]) An α-LS manifold M with vector fields E,F defined on M satisfying the condition

is called φ∗-Ricci symmetric.

If E and F are orthogonal to ζ, then the manifold is said to be locally φ∗-Ricci symmetric.

Definition 2.4. ([30]) An α-LS manifold M is called φ∗-symmetric if

for all vector fields E,F,G,H on M.

Let (M,φ∗,ζ,η∗,g∗) be an α-LS manifold. The (potential) vector field V spans, and is orthogonal to, ζ, so we only consider the case V=ζ. Using equation (3), we obtain

Because of (2.15), the (3.1) becomes

for any E,F∈χ(M), or equivalently:

for any E,F∈χ(M).

The data (g∗,ζ,ω1,ω2) which satisfy the equation (3.1) is said to be an η∗-Ricci soliton on M (see [9]), in particular, if ω2=0, (g∗,ζ,ω1) is a Ricci soliton [20]. Ricci solitons is said to be expanding, shrinking and steady according as ω1 is positive, negative or zero [10].

From (3.2), we get

Taking F=ζ in (3.2), we have

From (2.11) and (3.4), we obtain

In this section, we investigate Ricci semisymmetric α-LS manifolds on η∗-Ricci solitons . According to Definition 2.2, we get

which implies that

Taking E=ζ in (4.1), we obtain

Using (2.8) in (4.2), we infer

Since α≠0, we obtain

Using (3.2) and (3.4) in (4.4), we infer

Putting K=ζ in (4.5) and using (2.2), we infer

which is equivalent to

Putting G=φ∗G, we obtain

Subtracting (4.7) and (4.8), we get

for any F,G on M and follows ω2=α. From the relation (3.5), we have ω1=α−2nα2.

Now from above, we are able to state our results.

Theorem 4.1. If (M,φ∗,ζ,η∗,g∗) is a Ricci semisymmetric α-LS manifold, (g∗,ζ,ω1,ω2) is an η∗-Ricci soliton on M, then ω2=α and ω1=α−2nα2.

In case ω2=0, we derive the following.

Corollary 4.2. If α-LS manifolds (M,φ∗,ζ,η∗,g∗) satisfy the condition R(ζ,F)⋅S=0, then there does not exist Ricci solitons with potential vector field ζ.

From (3.2), (3.5) and (4.7), we obtain

As a consequence, we can state following proposition.

Proposition 4.3. Let (M,φ∗,ζ,η∗,g∗) be an α-LS manifold. If M is Ricci semisymmetric and (g∗,ζ,ω1,ω2) is an η∗-Ricci soliton on M, then the manifold is an η∗-Einstein manifold.

Consider φ∗-symmetric η∗-Ricci solitons on α-LS manifolds. Then from definition 2.3, we have

Using (2.1) and (5.1), we get

Taking inner product in (5.2) with G, we have

which implies

After simplification, we obtain

Putting F=ζ in (5.5), we have

Using (2.5) and (3.3) in (5.6), we infer

Taking G=φ∗G in (5.7), we get

Since α≠0, we get

Now, using (3.3) in (5.9), we infer

Taking E=φ∗E in (5.10), we have

Subtracting (5.10) from (5.11), we obtain

for any E,G and follows ω2=α. From the relation (3.5), we obtain ω1=α−2nα2.

Now, we are ready to state the following results.

Theorem 5.1. If (M,φ∗,ζ,η∗,g∗) is φ∗-symmetric on α-LS manifolds, (g∗,ζ,ω1,ω2) is an η∗-Ricci soliton on M, then ω2=α and ω1=α−2nα2.

In case ω2=0, we can state next result.

Corollary 5.2. If α-LS manifolds (M,φ∗,ζ,η∗,g∗) satisfy the condition φ∗2((∇EQ)(ζ))=0, then there does not exist Ricci solitons with potential vector field ζ.

From (3.2), (3.4) and (5.10) we have

This leads to the following proposition.

Proposition 5.3. Let (M,φ∗,ζ,η∗,g∗) be an α-LS manifold. If M is a φ∗-symmetric and (g∗,ζ,ω1,ω2) is an η∗-Ricci soliton on M, then the manifold is an η∗-Einstein manifold.

This section is devoted to the study of φ∗-Ricci symmetric η∗-Ricci solitons on α-LS manifolds.

Consider a φ∗-Ricci symmetric η∗-Ricci solitons on α-LS manifolds. Then from definition 2.4, we have

Using (2.1), we infer

Taking inner product with U in (57), we obtain

Let {σi}, i=1,2,...,n, be an orthonormal basis of the tangent space at any point of the manifold. Then by putting E=U=σi in (53) and taking summation over i , 1≤i≤n, we get

Putting G=ζ in (6.4), we get

The second term of (6.5), takes the form

and we obtain

The equations (6.5) and (6.7) imply that

which gives

In view of (2.5) and (3.4), we have

Putting F=φ∗F in (6.8), we infer

Using (2.4), (2.7) and (3.3) in (6.9), we get

Since α≠0, we infer

Putting F=φ∗F, we have

Subtracting (6.11) from (6.12), we get

for any F,H it follows ω2=α. Using (3.5), we obtain ω1=α−2nα2.

Hence, we can state the following results.

Theorem 6.1. If (M,φ∗,ζ,η∗,g∗) is a φ∗-Ricci symmetric on an α-LS manifold and (g∗,ζ,ω1,ω2) is an η∗-Ricci soliton, then ω2=α and ω1=α−2nα2.

In case ω2=0, we deduce

Corollary 6.2. If α-LS manifolds (M,φ∗,ζ,η∗,g∗) satisfy the condition φ∗2((∇HR)(E,F)ζ)=0, then there does not exist Ricci solitons with potential vector field ζ.

Using (3.2), (3.5) and (6.11) we have

This leads to the following proposition.

Proposition 6.3. Let (M,φ∗,ζ,η∗,g∗) be an α-LS manifold. If M is a φ∗-Ricci symmetric and (g∗,ζ,ω1,ω2) is an η∗-Ricci soliton on M, then the manifold is an η∗-Einstein manifold.

Example 6.4. Now, we assume the 3-dimensional manifold

where p,q,r are the standard coordinates in R³.

The vector fields

are linearly independent at each point of M and α is a constant.

Let g∗ be the Lorentzian metric defined as

Let σ3=ζ. Then Lorentzian metric on M is given below

Let η∗ be the 1-form defined as

for any vector field G on M.

Let φ∗ be the (1,1)-tensor field defined as

Then, using the linearity of φ∗ and g∗, we get

for any vector field G,H on M.

It is easy to observe

Thus for σ3=ζ, the structure (φ∗,ζ,η∗,g∗) defines a Lorentzian almost contact metric structure on M.

Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g∗. Then we have

Using Koszul's formula

One can easily obtain

Now, we see that the manifold is an α-LS manifold.

Also, the Riemannian curvature tensor R is given by

Then

Then, the Ricci tensor S is given by

From (3.2), we obtain S(σ1,σ1)=S(σ2,σ2)=α−ω1 and S(σ3,σ3)=ω1−ω2, therefore ω1=α−2α2 and ω2=α. The data (g∗,ζ,ω1,ω2) for ω1=α−2α2 and ω2=α provides an η∗-Ricci soliton on an α-LS manifold.