Three-dimensional trans-Sasakian manifolds and solitons Open Access

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:

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

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

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.

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

for the identity transformation I. Here $φξ,η$ 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

If M admits a Riemannian metric g such that

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,

which gives

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]

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

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

which takes the form

Also, from equation (2.5) we have

Next, it can be easily verified that M satisfies

which assumes the form

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

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

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

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

for all $X1,X2,X3∈X(M)$⁠, where $A,BandD$ 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

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.

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

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

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

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

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

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

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

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

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:

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

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:

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.

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

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

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

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

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

Now, we define the following definitions as:

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

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:

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

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