Three-dimensional trans-Sasakian manifolds and solitons

Abstract

Purpose

The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.

Design/methodology/approach

The authors have used the tensorial approach to achieve the goal.

Findings

A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.

Originality/value

The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.

Citation

Chaubey, S.K. and De, U.C. (2022), "Three-dimensional trans-Sasakian manifolds and solitons", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 112-125. https://doi.org/10.1108/AJMS-12-2020-0127

1. Introduction

Let the product manifold M~=M×ℜ, known as an almost Hermitian manifold, possesses an almost complex structure J and the product metric G, where M is a (2n+1)-dimensional almost contact metric manifold [1] and ℜ, a real line. In 1980, Gray and Hervella [2] characterized the 16 classes of almost Hermitian manifolds and hence listed many classes of M, such as Sasakian structure, quasi-Sasakian structure, Kenmotsu structure and many structures. During the classification of M~ there appeared a class 4, which is close to the locally conformal Kaehler manifold. In view of the structure in the class 4 on M~, there exists a class of almost contact metric structure, known as trans-Sasakian structure, which generalizes the Sasakian and Kenmotsu structures. The trans-Sasakian structure of type (α, β) coincides with the class C6⊕C5, where α and β are some smooth functions on M. A trans-Sasakian structure of type (0, 0), (α, 0) and (0, β) is said to be a cosymplectic structure, α-Sasakian structure and β-Kenmotsu structure, respectively. Since then, many geometers characterized the geometrical properties of (2n+1)-dimensional trans-Sasakian manifolds. A trans-Sasakian manifold is said to be proper if α and β are nonzero. There was a natural question “Up to which dimension, the proper trans-Sasakian manifold exists”? Marrero [3], in 1992, gave an affirmative answer of this question. He proved that a trans-Sasakian manifold of dimension ≥5 is either a cosymplectic manifold, or an α-Sasakian manifold or a β-Kenmotsu manifold. Since then, the study of three-dimensional trans-Sasakian manifolds attract the researchers. For more details, we refer [4–10] and the references therein.

The Eisenhart problem of finding the parallel tensors (symmetric and skew-symmetric) is an important subject in the differential geometry and its allied areas. In 1923, Eisenhart showed that if a positive definite Riemannian manifold admits a second-order parallel symmetric covariant tensor other than a constant multiple of the metric tensor, then it is reducible [11]. In Ref. [12], Levy presented that a second-order parallel symmetric nondegenerate tensor in a space form is proportional to the metric tensor. The Eisenhart problems of finding the properties of second-order parallel tensors have been locally studied by Eisenhart and Levy, whereas Sharma [13] has solved the same problem globally on complex space form. Since then, many geometers studied the Eisenhart problems on different geometrical structures. For some deep results on this topic, we recommend [14–18] and the references therein.

A Ricci flow:

∂∂tg=−2S,

introduced by Hamilton [19] on a Riemannian manifold M, is used to solve the celebrated Poincaré conjecture [20, 21] and differentiable structure theorem (extension of Hamilton’s sphere theorem) [22]. It has applications in the string theory, thermodynamics, general relativity, cosmology, quantum field theory, etc. Also, the uniformization theorem and geometrization conjecture can be solved with the help of Ricci flow. Here g and S are the Riemannian metric and the Ricci tensor of M, respectively. Let the Ricci flow be governed by a one parameter family of diffeomorphisms and scalings, then its solution is known as a Ricci soliton. A Ricci soliton (g, V, λ) on an n-dimensional Riemannian manifold M is an extension of Einstein soliton, and is defined by

(1.1)LVg+2S+2λg=0,

where LV denotes the Lie derivative along the vector field V, called the soliton vector of the Ricci soliton and λ is a real constant. The soliton (g, V, λ) with λ>0, <0, or =0 is said to be expanding, shrinking or steady, respectively.

A Riemannian metric g of a Riemannian manifold of dimension n is said to be an η-Ricci soliton [23] if it satisfies the equation

LVg+2S+2λg+2μη⊗η=0

for some λ, μ∈ℜ, where ℜ is used for set of real numbers. If we choose λ and μ as smooth functions in the above equations, then the Ricci and η-Ricci solitons are called as almost Ricci and almost η-Ricci solitons, respectively. Many researchers have been studied the geometrical and physical properties of Ricci solitons and η-Ricci solitons. For readers, we suggest to cite [24-27] and the references therein.

The above deep studies motivate us to characterize the three-dimensional trans-Sasakian manifolds whose metrics are almost Ricci solitons and η-Ricci solitons. We also find the solutions of the Eisenhart problems and properties of three-dimensional weakly  symmetric trans-Sasakian manifolds. We bind up our work with a nontrivial example of three-dimensional trans-Sasakian manifold.

2. Basic results of trans-Sasakian manifolds and some definitions

A triplet (φ, ξ, η) on a (2n+1)-dimensional differentiable manifold M is said to an almost contact structure on M if it satisfies

(2.1)φ2+I=η⊗ξ, η(ξ)=1

for the identity transformation I. Here φ ξ, η and ⊗ denote a vector field of type (1, 1), a vector field of type (1, 0), a 1-form and tensor product, respectively. The manifold M equipped with the structure (φ, ξ, η) is known as an almost contact manifold [1]. From equation (2.1), we deduce

(2.2)φξ=0, ηoφ=0 and rank φ=2n.

If M admits a Riemannian metric g such that

(2.3)g(X1, X2)=g(φX1, φX2)+η(X1)η(X2), g(X1, ξ)=η(X1)

for all X1, X2∈X(M), where X(M) represents the collection of all smooth vector fields of M, then it is known as an almost contact metric manifold. Additionally, if M holds the following relation for the smooth functions α and β,

(∇X1φ)(X2)=α{g(X1, X2)ξ−η(X2)X1}+β{g(φX1, X2)ξ−η(X2)φX1},

which gives

(2.4)∇X1ξ=−αφX1+β(X1−η(X1)ξ)

for all X1, X2∈X(M), then M is said to be a trans-Sasakian manifold of type (α, β) [28].

The curvature tensor R of a three-dimensional trans-Sasakian manifold satisfies [29]

(2.5)R(X1, X2)X3=(r2+2ξ(β)−2(α2−β2)){g(X2,X3)X1−g(X1, X3)X2}−g(X2, X3){(r2+ξ(β)−3(α2−β2))η(X1)ξ+[X1(β)+φX1(α)]ξ−η(X1)(φgradα−gradβ)}−{[X3(β)+φX3(α)]η(X2)+[X2(β)+φX2(α)]η(X3)+(r2+ξ(β)−3(α2−β2))η(X2)η(X3)}X_{1}+{[X3(β)+φX3(α)]η(X1)+[X1(β)+φX1(α)]η(X3)+(r2+ξ(β)−3(α2−β2))η(X1)η(X3)}X2−g(X1, X3){[X2(β)+φX2(α)]ξ+(r2+ξ(β)−3(α2−β2))η(X2)ξ−η(X2)(φ grad α−grad β)}

for all X1, X2, X3∈X(M), where X1(α)=g(X1, gradα) and grad stands for gradient. Contracting the above equation along the vector field X1, we find

(2.6)S(X2, X3)=(r2+ξ(β)−(α2−β2))g(X2, X3)−(X3(β)+φX3(α))η(X2)−(r2+ξ(β)−3(α2−β2))η(X2)η(X3)−(X2(β)+φX2(α))η(X3).

Setting X3=ξ in Eqn (2.5) and making use of equations (2.1) and (2.2), we lead

R(X1, X2)ξ=(α2−β2){η(X2)X1−η(X1)X2}+2αβ(η(X2)φX1−η(X1)φX2)+X2(α)φX1−X1(α)φX2+X2(β)φ2X1−X1(β)φ2X2,

which takes the form

(2.7)R(ξ, X1)ξ=(α2−β2−ξ(β))(η(X1)ξ−X1).

Also, from equation (2.5) we have

(2.8)R(ξ, X1)X2=(α2−β2){g(X1, X2)ξ−η(X2)X1}+2αβ{g(φX2, X1)ξ+η(X2)φX1}+X2(α)φX1+g(φX2, X1)grad α+X2(β)(X1−η(X1)ξ)−g(φX1, φX2)grad β.

Next, it can be easily verified that M satisfies

2αβ+ξ(α)=0,

(2.9)S(X1, ξ)={2(α2−β2)−ξ(β)}η(X1)−X1(β)−(φX1)α,

which assumes the form

(2.10)Qξ=(2(α2−β2)−ξ(β))ξ−grad β+φ(grad α).

Here r, Q, S and R represents the scalar curvature, Ricci operator, Ricci tensor and the curvature tensor of the manifold, respectively.

The notion of quasi-conformal curvature tensor on a Riemannian manifold was introduced by Yano and Sewaki [30]. The quasi conformal curvature tensor C~ on a (2n+1)-dimensional trans-Sasakian manifold is defined by

(2.11)C˜(X1, X2)X3=aR(X1, X2)X3+b{S(X2, X3)X1−S(X1, X3)X2+g(X2, X3)QX1−g(X1, X3)QX2}−r(2n+1)(a2n+2b){g(X2, X3)X1−g(X1, X3)X2}

for all X1,X2, X3∈X(M), where a and b are constants. In particular, if we take a=1 and b=−12n−1 then the quasi-conformal curvature tensor reduces to the Weyl-conformal curvature tensor C [31], defined by

(2.12)C(X1, X2)X3=R(X1, X2)X3−12n−1{S(X2, X3)X1−S(X1, X3)X2+g(X2, X3)QX1−g(X1, X3)QX2}+r2n(2n−1){g(X2, X3)X1−g(X1, X3)X2}

for all X1, X2, X3∈X(M). A trans-Sasakian manifold is said to be quasi-conformally (conformally) flat if C~=0 (C=0).

In 2011, Mantica and Molinari [32] introduced the notion of generalized  tensor on a semi-Riemannian manifold, and is defined by

(2.13)=S+ϕg,

where ϕ is some smooth function. The choice of ϕ=−rn shows that the generalized  tensor reduces to the classical  tensor. The deep results of  tensor have been noticed in [33–39].

A three-dimension trans-Sasakian manifold M is said to be weakly  symmetric [40] if the nonvanishing  tensor satisfies

(2.14)(∇X1)(X2, X3)=A(X1)(X2, X3)+B(X2)(X1, X3)+D(X3)(X1, X2)

for all X1, X2, X3∈X(M), where A, B and D are 1-forms. The weakly  symmetric manifold with ϕ=0 becomes a weakly Ricci symmetric manifold, introduced by Tamássy and Binh [40] within the context of Riemannian manifold. A weakly Ricci symmetric manifold with A=B=D=0 is said to be a Ricci symmetric trans-Sasakian manifold.

A trans-Sasakian manifold M is said to be an η-Einstein manifold if its nonzero Ricci tensor S satisfies

S=ag+bη⊗η,

where a and b are some smooth functions on M. Particularly, if we choose b=0 and a is constant in the above equation, then we get the Einstein manifold.

3. Second-order parallel symmetric tensor on three-dimensional trans-Sasakian manifolds

Let δ be a symmetric tensor of type (0,2), then it is said to be parallel with respect to ∇ if ∇δ=0. This equation together with Ricci identity ∇X1, X22δ(X3, X4)−∇X1, X22δ(X4, X3)=0 give

(3.1)δ(R(X1, X2)X3, X4)+δ(X3, R(X1, X2)X4)=0.

Setting X1=X4=ξ in Eqn (3.1) and making use of equations (2.7) and (2.8), we obtain

(α2−β2−ξ(β))δ(X2, X3)={(α2−β2)g(X2, X3)+2αβg(φX3, X2)−X3(β)η(X2)}δ(ξ, ξ)+{2αβη(X3)+X3(α)}δ(φX2,ξ)+{X3(β)−(α2−β2)η(X3)}δ(X2, ξ)+g(φX3, X2)δ(grad α, ξ)−g(φX3, φX2)δ(grad β, ξ)+(α2−β2−ξ(β))η(X2)δ(X3, ξ).

Again changing X3 with ξ in the above equation and using equations (2.1) and (2.2), we have

(3.2)δ(X2, ξ)=g(X2, ξ)δ(ξ, ξ),

provided α2−β2−ξ(β)≠0. Differentiating Eqn (3.2) covariantly along the vector field X1, we infer

(3.3)δ(X2, ∇X1ξ)=g(X2, ∇X1ξ)δ(ξ, ξ)+2δ(∇X1ξ, ξ)g(X2, ξ).

Replacing X2 with ∇X1ξ in Eqn (3.2) and then using the foregoing equation in (3.3), we find

δ(X2, ∇X1ξ)=[g(X2, ∇X1ξ)+2g(∇X1ξ, ξ)g(X2, ξ)]δ(ξ, ξ).

In consequence of Eqns (2.2)–(2.4) and (3.2), the above equation assumes the form

(3.4)−α{δ(X2, φX1)−g(X2, φX1)δ(ξ, ξ)}+β{δ(X1, X2)−g(X1, X2)δ(ξ, ξ)}=0.

Changing X1 with φX1 in Eqn (3.4) and then using equations (2.1) and (3.2), we conclude that

(3.5)α[δ(X1, X2)−g(X1, X2)δ(ξ, ξ)]+β[δ(φX1, X2)−g(φX1, X2)δ(ξ, ξ)]=0.

We suppose that α and β are nonzero and α≠±β on M, then from equations (3.4) and (3.5) we lead to

(3.6)δ(X1, X2)=g(X1, X2)δ(ξ, ξ), ∀ X1, X2∈X(M).

This fact together with equation (3.2) reflect that ∇X3δ(ξ, ξ)=2δ(ξ, ξ)g(∇X3ξ, ξ)=0. Since X3 is an arbitrary vector field of M and therefore δ(ξ, ξ) is constant on M. Hence we are in position to state the following:

Theorem 3.1.

A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g. In other words, the almost contact metric on a regular three-dimensional trans-Sasakian manifold is irreducible.

Next, we consider that the regular three-dimensional trans-Sasakian manifold is Ricci symmetric, that is, ∇S=0, and therefore from Theorem 3.1 we obtain S=S(ξ,ξ)g. In view of equations (2.1), (2.2) and (2.9), we conclude that S(ξ,ξ)=2{α2−β2−ξ(β)}. Thus, we have

(3.7)S(X1,X2)=2(α2−β2−ξ(β))g(X1,X2).

This shows that the three-dimensional Ricci symmetric trans-Sasakian manifold is an Einstein manifold, provided α2−β2−ξ(β) is nonzero. Thus we state:

Corollary 3.2.

Every three-dimensional regular Ricci symmetric trans-Sasakian manifold M is an Einstein manifold, provided the ξ-sectional curvature tensor of M is non-zero.

Particularly, if α=1, β=0 and α=0, β=1 then the trans-Sasakian manifolds reduce to the Sasakian and the Kenmotsu manifolds, respectively. Hence, we can state:

Corollary 3.3.

Every three-dimensional Ricci symmetric Sasakian (or Kenmotsu) manifold is Einstein.

For a three-dimensional Kenmotsu manifold, Corollary 3.3 has been proved by De and Pathak [41].

If possible, we suppose that the three-dimensional trans-Sasakian manifold (M,g) is Ricci symmetric and therefore it satisfies equation (3.7). Hence we have

Qξ=2{α2−β2−ξ(β)}ξ.

This equation together with equation (2.10) reflect that

grad β−φ(grad α)=ξ(β)ξ,

which is equivalent to

(3.8) X1(β)+φX1(α)=ξ(β)η(X1)

for all vector field X1 on M. This tell us the following lemma.

Lemma 3.4.

Every three-dimensional Ricci symmetric trans-Sasakian manifold satisfies ξ(β)ξ=grad β−φ(grad α).

If we take α=0 and β≠0 (or α≠0 and β=0), then the trans-Sasakian manifold reduces to β-Kenmotsu (or α-Sasakian) manifold. Thus from Lemma 3.4, we have the following corollary.

Corollary 3.5.

A three-dimensional Ricci symmetric β-Kenmotsu (or α-Sasakian) manifold holds the relation grad(β)=ξ(β)ξ (or φ(gradα)=0).

Let the three-dimensional trans-Sasakian manifold be Ricci symmetric, then it satisfies equation (3.8). In consequence of equations (2.5) and (3.8), we have

(3.9)R(X1, X2)X3=(r2−2(α2−β2)+2ξ(β)){g(X2, X3)X1−g(X1, X3)X2}−(r2−3(α2−β2−ξ(β))){g(X2, X3)η(X1)−g(X1, X3)η(X2)}ξ−(r2−3(α2−β2−ξ(β))){η(X2)X1−η(X1)X2}η(X3)

for all vector fields X1, X2 and X3 on M. If {ei}i=13 denotes a set of orthonormal vector fields of M, then for X1=X2=ei, 1≤ i ≤3, equation (3.7) assumes the form

(3.10)r=6(α2−β2−ξ(β)).

In consequence of equations (3.9) and (3.10), we find that

(3.11)R(X1, X2)X3=(α2−β2−ξ(β)){g(X2, X3)X1−g(X1, X3)X2}.

This shows that the trans-Sasakian manifold M of dimension three possesses a space of constant scalar curvature α2−β2−ξ(β).

Next, we suppose that M has a space of constant scalar curvature α2−β2−ξ(β) and therefore equation (3.11) holds on M. The contraction of (3.11) along the vector field X1 gives equation (3.7) and we have

(3.12)QX2=2(α2−β2−ξ(β))X2.

The covariant derivative of (3.7) gives ∇S=0, provided α2−β2−ξ(β)=constant. Thus we can state:

Theorem 3.6.

A three-dimensional trans-Sasakian manifold M is Ricci symmetric if and only if it is a space of constant scalar curvature α2−β2−ξ(β).

Again from equation (3.11), we can state the following:

Corollary 3.7.

A three-dimensional Ricci symmetric Kenmotsu manifold is locally isometric to the hyperbolic space ℍ3(−1).

Corollary 3.8.

A Ricci symmetric Sasakian manifold of dimension three is locally isometric to the sphere S3(1).

In the light of equations (2.11), (3.7) and (3)–(3.12), we conclude that C~=0. This shows that M is quasi-conformally flat. Thus, we can say that every three-dimensional Ricci symmetric trans-Sasakian manifold is quasi-conformally flat. From the straight forward calculations, we can easily show that a three-dimensional quasi-conformally flat trans-Sasakian manifold M is not Ricci symmetric. Hence, we have

Corollary 3.9.

A three-dimensional Ricci symmetric trans-Sasakian manifold M is quasi-conformally flat, but the converse is not true.

We suppose that a three-dimensional trans-Sasakian manifold is quasi-conformally flat, that is, C~=0. Hence, from equation (2.11), for a three-dimensional trans-Sasakian manifold, we have

R(X1, X2)X3=−ba{S(X2, X3)X1−S(X1, X3)X2+g(X2, X3)QX1−g(X1, X3)QX2}+r3a(a2+2b){g(X2, X3)X1−g(X1, X3)X2},

which is equivalent to

g(R(X1,X2)X3,X4)=−ba{S(X2, X3)g(X1,X4)−S(X1, X3)g(X2, X4)+g(X2, X3)S(X1,X4)−g(X1,X3)S(X2,X4)}+r3a(a2+2b){g(X2, X3)g(X1, X4)−g(X1, X3)g(X2, X4)},

provided a≠0. Changing X1 and X3 with the characteristic vector field ξ in the above equation and then using Eqns (2.1)–(2.3), (2.7) and (2.9), we obtain

(3.13) bS(X2, X4)=(r6(a+4b)−(a+2b)(α2−β2−ξ(β)))g(X2, X4)+((a+4b)(α2−β2)−(a+2b)ξ(β)−r6(a+4b))η(X2)η(X4)−b{(X2(β)+φX2(α))η(X4)+(X4(β)+φX4(α))η(X2)}.

With the help of equations (2.6) and (3.13), we find

(a+b)[r−6(α2−β2−ξ(β))]{g(X2, X4)−η(X2)η(X4)}=0.

Since g(φX2, φX4)≠0 on an almost contact metric manifold, therefore the above equation reflects that either b=−a(≠0) or r=6(α2−β2−ξ(β)). If b=−a and r≠6(α2−β2−ξ(β)), then from equations (2.11) and (2.12), we have C~=a C. This shows that the quasi-conformal and Weyl conformal curvature tensors on a three-dimensional trans-Sasakian manifold are linearly dependent. Next, we assume that a+b≠0 and r=6(α2−β2−ξ(β)), then equation (3.13) takes the form

(3.14) S(X2, X4)=2(α2−β2−ξ(β))g(X2, X4)+2(ξ(β))η(X4)η(X2)−[{X2(β)+φX2(α)}η(X4)+{X4(β)+φX4(α)}η(X2)].

By considering equations (2.3), (2.4), g(φX1, X2)+g(X1, φX2)=0 and the definition of Lie derivative, we conclude that

(3.15) (Lξg)(X1, X2)=g(∇X1ξ, X2)+g(X1, ∇X2ξ)=2β{g(X1, X2)−η(X1)η(X2)}.

From equations (3.14) and (3.15), we notice that

(3.16) 12(Lξg)(X1, X2)+S(X1, X2)+λg(X1, X2)+μη(X1)η(X2)=2ξ(β)η(X1)η(X2)−[(X2(β)+φX2(α))η(X1)+(X1(β)+φX1(α))η(X2)],

where λ=2(β2+ξ(β)−α2)−β and μ=β. Hence, with help of equations (2.9), (3.14) and (3.16), we can state the following results.

Theorem 3.10.

Let M be a three-dimensional quasi-conformally flat trans-Sasakian manifold, then either M

1. is conformally flat or

2. possesses an almost η-Ricci soliton (g,ξ,λ,μ) if and only if ξ(β)ξ=grad β−φ(grad α).

In consequence of Theorem 3.10, we can state the following corollary.

Corollary 3.11.

A three-dimensional quasi-conformally flat trans-Sasakian manifold M with a+b≠0 possesses the constant scalar curvature.

Again, in view of equation (3.16) and Theorem 3.10, we can state the following:

Corollary 3.12.

The metric of a three-dimensional quasi-conformally flat β-Kenmotsu manifold with a+b≠0 is an almost η-Ricci soliton (g,ξ,λ,μ) if and only if ξ(β)ξ=grad β.

Corollary 3.13.

A three-dimensional quasi-conformally flat α-Sasakian manifold with a+b≠0 admits an almost η-Ricci soliton (g, ξ, λ, μ) if and only if φ grad α=0.

Corollary 3.14.

A three-dimensional quasi-conformally flat Sasakian manifold satisfying a+b≠0 is a shrinking Ricci soliton with λ=−2.

Corollary 3.15.

Every quasi-conformally flat Kenmotsu manifold of dimension three with a+b≠0 possesses an expanding type η-Ricci soliton.

In view of equation (3.14), we can state:

Corollary 3.16.

Suppose a three-dimensional quasi-conformally flat trans-Sasakian manifold M satisfies a+b≠0. Then M is an η-Einstein manifold if and only if grad β=φ(grad α).

From equation (3.16), we have λ=2(β2+ξ(β)−α2)−β. This shows that the almost η-Ricci soliton under consideration is expanding, shrinking or steady if β2+ξ(β)>, <, or =α2+β2, respectively. Thus, we can state:

Theorem 3.17.

Let a three-dimensional quasi-conformally flat trans-Sasakian manifold M satisfies a+b≠0 and grad β−φ(grad α)=ξ(β)ξ. Then the almost η-Ricci soliton (g,ξ,λ,μ) on M is shrinking, expanding or steady if β2+ξ(β)<, >, or =α2+β2, respectively.

With the help of Theorem 3.17, we can state the following:

Corollary 3.18.

If a three-dimensional quasi-conformally flat β-Kenmotsu manifold M satisfies a+b≠0 and gradβ=ξ(β)ξ, then the η-Ricci soliton (g, ξ, λ, μ) on M is shrinking, expanding or steady if β2+ξ(β)<, >, or =β2, respectively.

Corollary 3.19.

Suppose a three-dimensional quasi-conformally flat α-Sasakian manifold M satisfies a+b≠0 and grad α+ξ(α)ξ=0. Then the almost η-Ricci soliton (g,ξ,λ,μ) on M is always shrinking.

Let us suppose that δ1=Lξg+2S,  Lξg is parallel and the regular three-dimensional trans-Sasakian manifold is Ricci symmetric. Then in view of Eqns (3.6), (3.15) and Theorem 3.1, we find

(3.17) δ1(X1, X2)=δ1(ξ, ξ)g(X1, X2),

where δ1(ξ, ξ)=(Lξg)(ξ, ξ)+2S(ξ, ξ)=2(α2−β2−ξ(β)). In consequence of equations (1.1) and (3.17), we observe that λ=−12δ1(ξ, ξ)=−(α2−β2−ξ(β)). Thus, we discuss our results in the following corollary:

Theorem 3.20.

Let Lξg is parallel on a three-dimensional Ricci symmetric trans-Sasakian manifold M. Then the almost Ricci soliton (ξ,g,λ) on M is shrinking, expanding, or steady if the ξ-sectional curvature of M is positive, negative or zero, respectively.

It is well-known that a three-dimensional trans-Sasakian manifold of type (α,β) reduces to a Sasakian; Kenmotsu or cosymplectic manifolds if α=1, β=0; α=0, β=1 and α=0, β=0, respectively. Thus, we can state the following corollaries.

Corollary 3.21.

If Lξg is parallel on a three-dimensional Ricci symmetric Sasakian manifold M, then the Ricci soliton (ξ, g, λ) on M is always shrinking.

Corollary 3.22.

A Ricci soliton (ξ,g,λ) on a three-dimensional Ricci symmetric Kenmotsu manifold with a parallel vector field Lξg is expanding.

Corollary 3.23.

Suppose Lξg is parallel on a three-dimensional Ricci symmetric cosymplectic manifold M, then (ξ,g,λ) on M is steady.

Now, we define the following definitions as:

Definition 3.24.

A vector field X1∈X(M) on a Riemannian manifold is said to be an affine Killing vector field if ∇LX1g=0.

Definition 3.25.

A vector field X1∈X(M) on a Riemannian manifold is said to be a Killing vector field if LX1g=0.

From the Definition 3.24 and Definition 3.25, it is clear that if a vector field is Killing then it is affine Killing, but converse is not, in general, true. Here we prove that the converse is true in a three-dimensional trans-Sasakian manifold, provided α2−β2 is nonzero constant.

Let us suppose that X1 is an affine Killing vector field on a three-dimensional trans-Sasakian manifold M, then Theorem 3.1 and Definition 3.24 reflect that

(LX1g)(X2, X3)=cg(X2, X3),

where c=(LX1g)(ξ, ξ). The covariant derivative of Qξ=[2(α2−β2)−ξ(β)]ξ along the vector field X1 gives (LX1Q)(ξ)=0 and hence (LX1S)(ξ, ξ)=0, provided α2−β2=constant. Also, we have (LX1S)(ξ, ξ)=−2S(LX1ξ, ξ)=−2[2(α2−β2)−ξ(β)]g(LX1ξ, ξ). Since (LX1S)(ξ, ξ)=0 and therefore g(LX1ξ, ξ)=0 on M. By considering this fact, we have c=(LX1g)(ξ, ξ)=−2g(LX1ξ, ξ)=0. Thus, with the help of Definition 3.25, we can say that the vector field X1 is Killing on a three-dimensional trans-Sasakian manifold, provided α2−β2 is a nonzero constant. Thus, we can state the following:

Corollary 3.26.

An affine Killing vector field on a three-dimensional Kenmotsu (or Sasakian) manifold is Killing.

It is obvious that the metric tensor g is covariantly constant, that is, ∇g=0, which implies that ∇2λg=0, where λ is a real constant. This fact together with equation (1.1) give ∇(LVg+2S)=0 and therefore Theorem 3.1 reflects that LVg+2S=ag, where a=(LVg)(ξ, ξ)+2S(ξ, ξ)=−2λ is constant. Thus we can state:

Corollary 3.27.

A Ricci soliton (g,V,λ) on a three-dimensional trans-Sasakian manifold is expanding, shrinking or steady if a<0, >0 or =0, respectively.

4. Three-dimensional weakly  symmetric trans-Sasakian manifolds and Ricci flow

This section is dedicated to study the properties of three-dimensional weakly  symmetric trans-Sasakian manifold M. Since the Ricci tensor S and the Riemannian metric g of M are parallel with respect to the Levi-Civita connection, therefore  is also symmetric. From equation (2.14), we conclude that

(4.1)E(X3)(X1, X2)=E(X2)(X1, X3),

where E=B−D≠0. In view of equations (2.13) and (4.1), we observe that

E(X3)S(X1, X2)+ϕE(X3)g(X1, X2)−E(X2)S(X1, X3)−ϕE(X2)g(X1, X3)=0.

Substituting X3=ξ in the above equation, we find

(4.2)E(ξ)S(X1, X2)+ϕE(ξ)g(X1, X2)−E(X2)S(X1,ξ)−ϕE(X2)g(X1, ξ)=0.

Again, replacing X1 with ξ in equation (4.2) and making use of equations (2.1), (2.3) and (2.9), we lead to

(4.3)E(X2)=E(ξ)[(2(α2−β2)−ξ(β)+ϕ)η(X2)−X2(β)−φX2(α)]ϕ+2(α2−β2−ξ(β)),

provided ϕ+2(α2−β2−ξ(β))≠0. Substituting the value of E from Eqn (4.3) in equation (4.2), we observe that

(4.4)S(X1, X2)=[(2(α2−β2)−ξ(β)+ϕ)η(X2)−X2(β)−φX2(α)]ϕ+2(α2−β2−ξ(β))×[(2(α2−β2)−ξ(β)+ϕ)η(X1)−X1(β)−φX1(α)]−ϕg(X1, X2).

By considering the above discussions, we can state the following:

Theorem 4.1.

The Ricci tensor of a three-dimensional weakly  symmetric trans-Sasakian manifold with B−D≠0 is given by (4).

If possible, we suppose that X1(β)+φX1(α)=0⇔grad β=φ grad α, then equation (4) assumes the form

(4.5) S(X1,X2)=bη(X2)η(X1)+ag(X1,X2),

where b=(2(α2−β2)−ξ(β)+ϕ)2ϕ+2(α2−β2−ξ(β))≠0 and a=−ϕ. The above equation shows that the three-dimensional weakly  symmetric manifold is an η-Einstein manifold, provided B≠D and grad β=φ grad α. Thus, we can state the following:

Corollary 4.2.

A three-dimensional weakly  symmetric manifold with B≠D and grad β=φ grad α is η-Einstein.

We know that a weakly  symmetric trans-Sasakian manifold with ϕ=0 is a weakly Ricci symmetric trans-Sasakian manifold. Now, we suppose that the weakly Ricci symmetric trans-Sasakian manifold of dimension three satisfies grad β=φ grad α, then equation (4.5) turns into the equation

(4.6) S(X1, X2)=rη(X2)η(X1),

where r=b=(2(α2−β2)−ξ(β))22(α2−β2−ξ(β))≠0. It is well-known that a semi-Riemannian manifold is said to be Ricci simple if its nonvanishing Ricci tensor satisfies the relation S=r η⊗η. Thus, we have the following:

Corollary 4.3.

Every weakly Ricci symmetric trans-Sasakian manifold of dimension three with B≠D and grad β=φ grad α is Ricci simple.

In view of equations (3.15) and (4.5), we obtain

Lξg+2S+(2ϕ−2β)g−(2b−2β)η⊗η=0,

which shows that the three-dimensional weakly  symmetric manifold under consideration is an almost η-Ricci soliton (g, ξ, λ, μ) with λ=ϕ−β and μ=β−b≠0. It is obvious that the soliton (g, ξ, λ, μ) is expanding, shrinking or steady if ϕ>,<,or=β, respectively. Thus, we conclude our result as:

Corollary 4.4.

The metric of a three-dimensional weakly  symmetric manifold with B≠D and grad β=φ grad α is an almost η-Ricci soliton. Also, the soliton (g, ξ, λ, μ) is shrinking, expanding or steady if ϕ<, >, or=β, respectively.

Again from equations (3.15) and (4.6), we have

Lξg+2S−2βg−2(r−β)η⊗η=0.

This reflects that a three-dimension weakly Ricci symmetric trans-Sasakian manifold with B≠D and grad β=φ grad α is an η-Ricci soliton and λ=−β. Thus, we can write the following:

Corollary 4.5.

A three-dimensional weakly Ricci symmetric trans-Sasakian manifold with B≠D and grad β=φ grad α is an almost η-Ricci soliton.

5. Example of proper three-dimensional trans-Sasakian manifold

Example 5.1.

Let M3={(x, y, z)∈ℜ3:|z|>0} be a differentiable manifold of dimension three with the standard coordinate system (x, y, z) of ℜ3. Define a (1,1)-tensor ϕ on M3 as: φ(ϱ1)=−ϱ2, φ(ϱ2)=ϱ1, φ(ϱ3)=0, where ϱ1=z(∂∂x+y∂∂z), ϱ2=z∂∂y, ϱ3=∂∂z=ξ are the linear independent vector fields at each point of M3 and therefore it forms a basis of tangent space at each point of M3. Now, if the Riemannian metric g of M3 is defined by g(ϱi, ϱj)=δij, ∀ i, j=1, 2, 3, then we can easily verify that the relations

η(ϱ3)=1, φ2ϱi=−ϱi+η(ϱi)ϱ3

hold for i=1,2,3 and η(ϱi)=g(ϱi,ϱ3), where η is the 1-form. By the definition of Lie bracket ([X1, X2]f=X1X2(f)−X2X1(f) for some smooth function f), we get the following nonzero components of Lie bracket as:

[ϱ1, ϱ2]=yϱ2−z2ϱ3, [ϱ3, ϱ2]=1zϱ2, [ϱ3,ϱ1]=1zϱ1.

The Koszul's formula along with above results give

∇ϱ1ϱ1=1zϱ3, ∇ϱ1ϱ2=−z22ϱ3, ∇ϱ1ϱ3=−1zϱ1+z22ϱ2,∇ϱ2ϱ1=z22ϱ3−yϱ2, ∇ϱ2ϱ2=yϱ1+1zϱ3, ∇ϱ2ϱ3=−1zϱ2−z22ϱ1,∇ϱ3ϱ1=z22ϱ2, ∇ϱ3ϱ2=−z22ϱ1, ∇ϱ3ϱ3=0.

By the straightforward calculations, we can write

∇ϱ3=−αφI+β(I−η⊗ϱ3),

where α=z22 and β=−1z. This shows that the structure (φ, ϱ3, η, g) is a trans-Sasakian structure on M3 and the manifold M3 equipped with (φ, ϱ3, η, g) is a trans-Sasakian manifold with α=z22 and β=−1z.