Trans-Sasakian Structures with Certain Restrictions

Abstract

We find restrictions on a trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ so that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. In that, first we show that if the vector $\mathbf{u}$ of the trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ is an affine conformal vector with affine potential $\alpha \ne 0$ and the condition $\mathbf{u}\left(\alpha \right)=-{\beta }^2$ holds, necessarily implies $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. Similarly, it is shown that if the vector $\mathbf{u}$ of the trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ is a projective vector and the sectional curvatures of the plane sections containing $\mathbf{u}$ are positive constant, then $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. Finally, we find certain generic conditions on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ possessing a trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ so that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold.

1. Introduction

In Ref. [1], author proved that an almost contact structure $(F,\mathbf{u},\gamma )$, where F is a $\left(1,1\right)$-tensor field, $\mathbf{u}$ a vector field and $\gamma$ a 1-form on a $\left(2n+1\right)$-dimensional Riemannian manifold $\left(M^{2n+1},g\right)$ satisfying

$$F^2=-I+\gamma \otimes \mathbf{u},F\left(\mathbf{u}\right)=0,\gamma \left(\mathbf{u}\right)=1$$

is a trans-Sasakian structure if and only if it is normal and

$$d\mathsf{\Omega }=2\beta \gamma \wedge \mathsf{\Omega },d\gamma =\alpha \mathsf{\Omega },$$

where $\mathsf{\Omega }$ is the fundamental 2-form defined by $\mathsf{\Omega }\left(X,Y\right)=g\left(X,FY\right)$ for vector fields $X,Y$ on M and $\alpha ={\displaystyle \frac{1}{2n}}\delta \mathsf{\Omega }\left(\mathbf{u}\right)$ and $\beta ={\displaystyle \frac{1}{2n}}div\left(\mathbf{u}\right)$. The structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ is called the trans-Sasakian structure on $\left(M^{2n+1},g\right)$. The local structure of trans-Sasakian manifolds of dimension $n\ge 5$ has been completely characterized by Marrero (cf. [2]), thereby showing that only in dimension 3 a Riemannian manifold $\left(M^3,g\right)$ can possess proper trans-Sasakian structures $\left(F,\mathbf{u},\gamma F,\mathbf{u},\gamma ,\alpha ,\beta \right)$, that is, with both $\alpha ,\beta$ nonzero.

Notice that a 3-dimensional unit sphere $\left(S^3,g\right)$ inherits a Sasakian structure $\left(F,\xi ,\gamma \right)$ as embedded hypersurface of the Euclidean space $R^4$ with complex structure J and Hermitian Euclidean metric $\overline{g}$, where g is the induced canonical metric on $S^3$ (cf. [3]). If we choose a positive function $\rho$ on $S^3$ and deform the metric g as $g^{*}=\rho g+\left(1-\rho \right)\gamma \otimes \gamma$, then it follows that $\left(F,\xi ,\gamma ,{\displaystyle \frac{1}{\rho }},{\displaystyle \frac{1}{2}}\xi \left(ln\rho \right)\right)$ is a trans-Sasakian structure on the Riemannian manifold $\left(S^3,g^{*}\right)$, which is a proper Trans-Sasakian structure.

We shall abbreviate a trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ as a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$. There has been interesting results on geometry of a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ admitting a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ obtained by several authors (cf. [1,2,4,5,6,7,8,9,10]). It is worth noting in an interesting recent article (cf. [7]), authors have introduced generalized transS-structure on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$.

Geometrization conjecture is an analogue of the uniformization theorem in dimension two, which states that each simply connected surface acquires one of the three geometries namely, Euclidean, spherical or hyperbolic. However, in dimension three, it is not always possible to assign a single geometry to a whole manifold. Thus, the geometrization conjecture states that every closed 3-dimensional Riemannian manifold can be decomposed into pieces that each have one of eight types of geometric structure (cf. [11,12]).

Also, there is an interesting article studying hyperbolic Ricci solitons on trans-Sasakian manifolds enriching the geometry of 3-dimensional Riemannian manifolds admitting a transS-structure [13]. Since, in this article as we focus on obtaining sufficient conditions on a Riemannian manifold admitting a transS-structure, it is worth noting that in [14] authors have studied Riemann solitons on Sasakian 3-manifolds and obtained interesting results. Moreover, in [15] authors investigate trans-Sasakian and almost trans-Sasakian structures on Riemannian manifolds and obtained profound results relating to the structure equations and geometry of these manifolds.

It is worth noting that owing to Geometrization conjecture, geometry of 3-dimensional Riemannian manifold $\left(M^3,g\right)$ admitting a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ is significantly important. In that, the Three out of the Eight geometries in Geometrization conjecture being Sasakian manifold, finding conditions under which a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ admitting a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ is homothetic to a Sasakian manifold is an important question. In this article, we address this question and the article is arranged as follows:

In Section 2, we have recalled known results on transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$, which are used in subsequent sections. In Section 3, in the first result we show that the vector $\mathbf{u}$ of the transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$, is an affine conformal vector with affine potential $\alpha \ne 0$ such that $\mathbf{u}\left(\beta \right)=-{\beta }^2$ necessarily implies that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold (see Theorem 2). In the second result, we show that the vector $\mathbf{u}$ of the transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ is an affine conformal vector with affine potential $\beta$ with an additional condition that the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ is a positive constant, necessarily implies that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold (see Theorem 3).

In Section 4, we study the impact of the vector $\mathbf{u}$ of the transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ as a projective vector field, such that the sectional curvatures of plane sections containing $\mathbf{u}$ are positive constant. It is observed that in this case $\left(M^3,g\right)$ is homothetic to a Sasakian manifold (see Theorem 4). In Section 5, first we consider a less restrictive condition namely $S\mathbf{u}=3^{-1}\tau \mathbf{u}$, than being an Einstein condition $S=3^{-1}\tau I$, where S is the Ricci operator, $\tau$ is the scalar curvature of a 3-dimensional compact and connected Riemannian manifold $\left(M^3,g\right)$ possessing a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ and seek requirement that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. We answer this question by assuming that $\tau \ne 0$ and the function $\beta$ is a constant along the integral curves of the vector field $\mathbf{u}$ (see Theorem 5). In the next result of this section, we consider a 3-dimensional compact and simply connected Riemannian manifold $\left(M^3,g\right)$ possessing a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ and show that $∥\mathsf{\nabla }\alpha ∥$ is a constant and the Hessian operator ${\mathit{H}}_{\alpha }$ is invariant under $\mathsf{\nabla }\alpha$, necessarily imply that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold (see Theorem 6). Finally, in this section, it is shown that a 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ with a transS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ satisfying (i) ${\left(\mathbf{u}\left(\alpha \right)\right)}^2+{\alpha }^2=c$ for a constant $c\ne 0$ and (ii) $Hess\left(\alpha \right)\left(\mathbf{u},\mathbf{u}\right)\ne -\alpha$, is necessarily homothetic to a Sasakian manifold (see Theorem 7).

2. Preliminaries

A Trans-Sasakian structure abbreviated as TransS-structure on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$, is the quintuple $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$, where F is a $(1,1)$ tensor field, $\mathbf{u}$ a unit vector field, $\gamma$ a 1-form dual to $\mathbf{u}$ and $\alpha$, $\beta$ are smooth functions on $M^3$ satisfying (cf. [4,5,6,16])

$$F^2=-I+\gamma \otimes \mathbf{u},F\left(\mathbf{u}\right)=0,\gamma \circ F=0,g\left(FX,FY\right)=g\left(X,Y\right)-\gamma \left(X\right)\gamma \left(Y\right)$$

(1)

and

$$\left({\mathsf{\nabla }}_{X}F\right)\left(Y\right)=\alpha \left(g\left(X,Y\right)\mathbf{u}-\gamma \left(Y\right)X\right)+\beta \left(g\left(FX,Y\right)\mathbf{u}-\gamma \left(Y\right)FX\right),$$

(2)

for $X,Y\in \mathsf{\Gamma }\left(T{M}^3\right)$, where $\mathsf{\Gamma }\left(T{M}^3\right)$ is the set of smooth sections of the tangent bundle $T{M}^3$ and ∇ is the Riemannian connection on $\left(M^3,g\right)$. The Ricci tensor $Ric$ of $\left(M^3,g\right)$ is a symmetric tensor given by

$$Ric\left(X,Y\right)=\sum _{k}g\left(R\left(e_k,X\right)Y,e_k\right),X,Y\in \mathsf{\Gamma }\left(T{M}^3\right),$$

(3)

where $R\left(X,Y\right)Z$ is the curvature tensor and $\left\{e_1,e_2,e_3\right\}$ is a local orthonormal frame on $\left(M^3,g\right)$. The Ricci operator S of $\left(M^3,g\right)$ is also a symmetric operator given by

$$Ric\left(X,Y\right)=g\left(SX,Y\right),X,Y\in \mathsf{\Gamma }\left(T{M}^3\right).$$

(4)

We shall summarize the known results about the TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ as follows (cf. [4,5,6,17]):

 Lemma 1. 

Let $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ be a TransS-structure on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$. Then the following hold

$$\left(\mathrm{i}\right)\mathbf{u}\left(\alpha \right)=-2\alpha \beta \left(\mathrm{ii}\right){\mathsf{\nabla }}_X\mathbf{u}=-\alpha FX+\beta \left(X-\gamma \left(X\right)\mathbf{u}\right)\left(\mathrm{iii}\right)div\mathbf{u}=2\beta$$

$$\left(\mathrm{iv}\right)S\mathbf{u}=F\left(\mathsf{\nabla }\alpha \right)-\mathsf{\nabla }\beta +2\left({\alpha }^2-{\beta }^2\right)\mathbf{u}-\mathbf{u}\left(\beta \right)\mathbf{u}\left(\mathrm{v}\right)div\left(F\mathbf{u}\right)=0.$$

Note that (v) in the above Lemma follows from (iii) and the Equation (2) in computing

$$\begin{array}{ccc}\hfill div\left(F\mathbf{u}\right)& =& \sum _{k}g\left(\mathsf{\nabla }{e}_{k}F\mathbf{u},e_k\right)\hfill \\ & =& \sum _{k}g\left(\left(\mathsf{\nabla }{e}_{k}F\right)\left(\mathbf{u}\right)+F\left(-\alpha F{e}_k+\beta \left(e_k-\gamma \left(e_k\right)\mathbf{u}\right)\right),e_k\right)\hfill \\ & =& -\sum _{k}g\left(\mathbf{u},\left(\mathsf{\nabla }{e}_{k}F\right)\left(e_k\right)\right)-\sum _{k}g\left(-\alpha F{e}_k+\beta \left(e_k-\gamma \left(e_k\right)\mathbf{u}\right),F{e}_k\right)\hfill \\ & =& -2\alpha +2\alpha =0.\hfill \end{array}$$

Note that for a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ with $\alpha$ a nonzero constant and $\beta =0$, using (ii) in Lemma 1, we get

$${\mathsf{\nabla }}_Y\mathbf{u}=-\alpha FY$$

(5)

and further differentiating it and using Equation (2), we get

$${\mathsf{\nabla }}_X{\mathsf{\nabla }}_Y\mathbf{u}=-\alpha \left({\mathsf{\nabla }}_{X}F\right)\left(Y\right)-\alpha F\left({\mathsf{\nabla }}_{X}Y\right)=-{\alpha }^2\left(g\left(X,Y\right)\mathbf{u}-\gamma \left(Y\right)X\right)-\alpha F\left({\mathsf{\nabla }}_{X}Y\right),$$

that is, on using Equation (5), we have

$${\alpha }^{-2}\left({\mathsf{\nabla }}_X{\mathsf{\nabla }}_Y\mathbf{u}-{\mathsf{\nabla }}_{{\mathsf{\nabla }}_{X}Y}\mathbf{u}\right)=g\left(Y,\mathbf{u}\right)X-g\left(X,Y\right)\mathbf{u}.$$

The above equation together with the result in [18] gives a conclusion that can be summarized as:

 Theorem 1.  

A 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ admitting a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ with α a nonzero constant and $\beta =0$ is homothetic to a Sasakian manifold.

Note that given two Riemannian manifolds $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$ a diffeomorphism $f:M_1\to M_2$ is said to be a conformal transformation if $f^{*}\left(g_2\right)=e^{\rho }{g}_1$, where $\rho$ is a smooth function on $M_1$. If $\rho$ is a constant, then f is called a homothety and we say $\left(M_1,g_1\right)$ is homothetic to $\left(M_2,g_2\right)$.

For a smooth function $f:M^3\to R$ on a Riemannian manifold $\left(M^3,g\right)$, the Hessian operator ${\mathit{H}}_f$ of the function f and the Laplace operator $\mathsf{\Delta }$ acting on f, is given by

$${\mathit{H}}_{f}X={\mathsf{\nabla }}_X\mathsf{\nabla }f,\mathsf{\Delta }f=Tr{\mathit{H}}_f.$$

(6)

Also, the Laplace operator $\mathsf{\Delta }$ acting on a smooth vector field $\xi$ on a Riemannian manifold $\left(M^3,g\right)$ is given by

$$\mathsf{\Delta }\xi =\sum _k\left({\mathsf{\nabla }}_{e_k}{\mathsf{\nabla }}_{e_k}\xi -{\mathsf{\nabla }}_{{\mathsf{\nabla }}_{e_k}{e}_k}\xi \right).$$

(7)

Recall that a vector field $\xi$ on a Riemannian manifold $\left(M^3,g\right)$ is said to be an affine conformal vector with affine conformal potential f (cf. [19,20]), if

$$\left({\pounds }_{\xi }\mathsf{\nabla }\right)\left(X,Y\right)=X\left(f\right)Y+Y\left(f\right)X-g\left(X,Y\right)\mathsf{\nabla }f,X,Y\in \mathsf{\Gamma }\left(T{M}^3\right),$$

(8)

where

$$\left({\pounds }_{\xi }\mathsf{\nabla }\right)\left(X,Y\right)=R\left(\xi ,X\right)Y+{\mathsf{\nabla }}_X{\mathsf{\nabla }}_Y\xi -{\mathsf{\nabla }}_{{\mathsf{\nabla }}_{Y}Y}\xi .$$

(9)

Also, a vector field $\xi$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ is said to be a projective vector (cf. [7,10,17]), if

$$\left({\pounds }_{\xi }\mathsf{\nabla }\right)\left(X,Y\right)={\displaystyle \frac{1}{4}}\left\{X\left(div\xi \right)Y+Y\left(div\xi \right)X\right\},X,Y\in \mathsf{\Gamma }\left(T{M}^3\right).$$

(10)

All Riemannian manifolds considered in this article are without boundary.

3. TransS-Structure $\left(\mathit{F},\mathbf{u},\mathit{\gamma },\mathit{\alpha },\mathit{\beta }\right)$ with $\mathbf{u}$ an Affine Conformal Vector

In this section, we consider a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that the vector field $\mathbf{u}$ is an affine conformal vector with affine conformal potential f. Then by Equation (8), we have

$$\left({\pounds }_{\mathbf{u}}\mathsf{\nabla }\right)\left(X,Y\right)=X\left(f\right)Y+Y\left(f\right)X-g\left(X,Y\right)\mathsf{\nabla }f,X,Y\in \mathsf{\Gamma }\left(T{M}^3\right).$$

(11)

Note that by equation (ii) in Lemma 1, we have

$${\mathsf{\nabla }}_Y\mathbf{u}=-\alpha FY+\beta Y-\beta \gamma \left(Y\right)\mathbf{u}$$

and again differentiating above equation and using (ii) in Lemma 1, we have

$$\begin{array}{ccc}\hfill {\mathsf{\nabla }}_X{\mathsf{\nabla }}_Y\mathbf{u}& =& -X\left(\alpha \right)FY-\alpha \left({\mathsf{\nabla }}_{X}F\right)\left(Y\right)-\alpha F\left({\mathsf{\nabla }}_{X}Y\right)+X\left(\beta \right)Y+\beta {\mathsf{\nabla }}_{X}Y\hfill \\ & & -X\left(\beta \right)\gamma \left(Y\right)\mathbf{u}-\beta \gamma \left({\mathsf{\nabla }}_{X}Y\right)\mathbf{u}-\beta g\left(Y,{\mathsf{\nabla }}_X\mathbf{u}\right)\mathbf{u}-\beta \gamma \left(Y\right){\mathsf{\nabla }}_X\mathbf{u}\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill {\mathsf{\nabla }}_X{\mathsf{\nabla }}_Y\mathbf{u}-{\mathsf{\nabla }}_{{\mathsf{\nabla }}_{Y}Y}\mathbf{u}& =-X\left(\alpha \right)FY-\alpha \left({\mathsf{\nabla }}_{X}F\right)\left(Y\right)+X\left(\beta \right)Y-X\left(\beta \right)\gamma \left(Y\right)\mathbf{u}\hfill \\ \hfill & -\beta g\left(Y,{\mathsf{\nabla }}_X\mathbf{u}\right)\mathbf{u}-\beta \gamma \left(Y\right){\mathsf{\nabla }}_X\mathbf{u}.\hfill \end{array}$$

(12)

 Theorem 2.  

A 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that the vector field $\mathbf{u}$ is an affine conformal vector with affine conformal potential $\alpha \ne 0$ such that $\mathbf{u}\left(\beta \right)=-2{\beta }^2$ holds, then $\left(M^3,g\right)$ is homothetic to a Sasakian manifold.

 Proof. 

If $\mathbf{u}$ is an affine conformal vector with affine conformal potential $\alpha$ on $\left(M^3,g\right)$, then Equation (11) implies

$$\left({\pounds }_{\mathbf{u}}\mathsf{\nabla }\right)\left(X,Y\right)=X\left(\alpha \right)Y+Y\left(\alpha \right)X-g\left(X,Y\right)\mathsf{\nabla }\alpha ,X,Y\in \mathsf{\Gamma }\left(T{M}^3\right)$$

and taking $X=Y=e_k$ in above equation for a local frame $\left\{e_1,e_2,e_3\right\}$ and summing the resulting equation, we have

$$\sum _k\left({\pounds }_{\mathbf{u}}\mathsf{\nabla }\right)\left(e_k,e_k\right)=-\mathsf{\nabla }\alpha .$$

(13)

Note that by Equation (3), we have

$$Ric\left(X,Y\right)=\sum _{k}g\left(R\left(e_k,X\right)Y,e_k\right)=\sum _{k}g\left(R\left(X,e_k\right)e_k,Y\right),$$

that is, in view of Equation (4), we conclude

$$SX=\sum _{k}R\left(X,e_k\right)e_k.$$

(14)

Now, combining Equations (7) and (9), in view of Equation (14), yields

$$\sum _k\left({\pounds }_{\mathbf{u}}\mathsf{\nabla }\right)\left(e_k,e_k\right)=S\mathbf{u}+\mathsf{\Delta }\mathbf{u}.$$

(15)

Consequently, inserting above equation in Equation (13) reveals

$$S\mathbf{u}+\mathsf{\Delta }\mathbf{u}=-\mathsf{\nabla }\alpha .$$

(16)

Using Equation (2), we have

$$\sum _k\left({\mathsf{\nabla }}_{e_k}F\right)\left(e_k\right)=2\alpha \mathbf{u}$$

and therefore, using Equations (7) and (12), we get

$$\mathsf{\Delta }\mathbf{u}=-F\left(\mathsf{\nabla }\alpha \right)-2{\alpha }^2\mathbf{u}+\mathsf{\nabla }\beta -\mathbf{u}\left(\beta \right)\mathbf{u}-\beta div\mathbf{u}-\beta {\mathsf{\nabla }}_{\mathbf{u}}\mathbf{u}.$$

Using Lemma 1 and ${\mathsf{\nabla }}_{\mathbf{u}}\mathbf{u}=0$ (an outcome of Lemma 1) in above equation, we get

$$\mathsf{\Delta }\mathbf{u}=-F\left(\mathsf{\nabla }\alpha \right)-2\left({\alpha }^2+{\beta }^2\right)\mathbf{u}+\mathsf{\nabla }\beta -\mathbf{u}\left(\beta \right)\mathbf{u}.$$

(17)

Next, on using (iv) of Lemma 1 and above equation, we conclude

$$S\mathbf{u}+\mathsf{\Delta }\mathbf{u}=-4{\beta }^2\mathbf{u}-2\mathbf{u}\left(\beta \right)\mathbf{u}$$

(18)

and combining it with Equation (16), reveals

$$\mathsf{\nabla }\alpha =2\left(\mathbf{u}\left(\beta \right)+2{\beta }^2\right)\mathbf{u}$$

Then, using the statement in above equation on a connected $M^3$, we conclude that $\alpha$ is constant. Thus, by (i) in Lemma 1, we have

$$\alpha \beta =0.$$

Since, the constant $\alpha \ne 0$, we must have $\beta =0$ and consequently, the requirements of the Theorem 1 are satisfied. Thus, $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. □

In the next result, we study the impact of $\mathbf{u}$ being an affine conformal vector with affine conformal potential $\beta$ on the geometry of a 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$. Indeed we prove:

 Theorem 3.  

A 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that the vector field $\mathbf{u}$ is an affine conformal vector with affine conformal potential β having Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ is a positive constant is homothetic to a Sasakian manifold.

 Proof. 

If $\mathbf{u}$ is an affine conformal vector with affine conformal potential $\beta$ on $\left(M^3,g\right)$, then Equation (11) implies

$$\left({\pounds }_{\mathbf{u}}\mathsf{\nabla }\right)\left(X,Y\right)=X\left(\beta \right)Y+Y\left(\beta \right)X-g\left(X,Y\right)\mathsf{\nabla }\beta ,X,Y\in \mathsf{\Gamma }\left(T{M}^3\right)$$

and taking $X=Y=e_k$ in above equation for a local frame $\left\{e_1,e_2,e_3\right\}$ and summing the resulting equation, we have

$$\sum _k\left({\pounds }_{\mathbf{u}}\mathsf{\nabla }\right)\left(e_k,e_k\right)=-\mathsf{\nabla }\beta .$$

(19)

Using Equation (15), in above equation yields

$$S\mathbf{u}+\mathsf{\Delta }\mathbf{u}=-\mathsf{\nabla }\beta$$

and combining it with Equation (18), we have

$$\mathsf{\nabla }\beta =4{\beta }^2\mathbf{u}+2\mathbf{u}\left(\beta \right)\mathbf{u}.$$

(20)

On taking the inner product with $\mathbf{u}$ in above equation gives $\mathbf{u}\left(\beta \right)=-4{\beta }^2$ and accordingly above equation changes to

$$\mathsf{\nabla }\beta =-4{\beta }^2\mathbf{u}.$$

(21)

Differentiating above equation and using Equation (6) with Lemma 1, we get

$${\mathit{H}}_{\beta }X=-8\beta X\left(\beta \right)\mathbf{u}-4{\beta }^2\left(-\alpha FX+\beta X-\beta \gamma \left(X\right)\mathbf{u}\right),$$

that is, on using Equation (21) in the form $X\left(\beta \right)=-4{\beta }^2\gamma \left(X\right)$ in above equation yields

$$\left({\mathit{H}}_{\beta }X-36{\beta }^3\gamma \left(X\right)\mathbf{u}+4{\beta }^{3}X\right)=4\alpha {\beta }^{2}FX,X\in \mathsf{\Gamma }\left(T{M}^3\right).$$

(22)

Taking the inner product in above equation with Y, yields

$$g\left({\mathit{H}}_{\beta }X,Y\right)-36{\beta }^3\gamma \left(X\right)\gamma \left(Y\right)+4{\beta }^{3}g\left(X,Y\right)=4\alpha {\beta }^{2}g\left(FX,Y\right),$$

which on interchanging X and Y, gives

$$g\left({\mathit{H}}_{\beta }Y,X\right)-36{\beta }^3\gamma \left(X\right)\gamma \left(Y\right)+4{\beta }^{3}g\left(X,Y\right)=4\alpha {\beta }^{2}g\left(FY,X\right).$$

Subtracting the last equation from previous one, while noticing that ${\mathit{H}}_{\beta }$ is symmetric and F is skew symmetric, yields $8\alpha {\beta }^{2}g\left(FX,Y\right)=0$, $X,Y\in \mathsf{\Gamma }\left(T{M}^3\right)$, that is,

$$\alpha {\beta }^{2}FX=0,X\in \mathsf{\Gamma }\left(T{M}^3\right),$$

which on taking the inner product with $FX$ in above equation yields

$$\alpha {\beta }^2{∥FX∥}^2=0.$$

Summing above equation over a frame gives

$$\alpha {\beta }^2=0.$$

(23)

Now, by Lemma 1, we have

$$Ric\left(\mathbf{u},\mathbf{u}\right)=-2\mathbf{u}\left(\beta \right)+2\left({\alpha }^2-{\beta }^2\right),$$

which in view of (21) implies

$$Ric\left(\mathbf{u},\mathbf{u}\right)=6{\beta }^2+2{\alpha }^2.$$

(24)

If $\alpha =0$ and as $Ric\left(\mathbf{u},\mathbf{u}\right)$ is a constant, above equation would imply ${\beta }^2$ is a constant and by Equation (21) would imply $\beta =0$, that is, the above equation reads $Ric\left(\mathbf{u},\mathbf{u}\right)=0$. This is contrary to the assumption $Ric\left(\mathbf{u},\mathbf{u}\right)$ is a nonzero constant. Hence, $\alpha \ne 0$ and by Equation (23), we have $\beta =0$. This makes $Ric\left(\mathbf{u},\mathbf{u}\right)=2{\alpha }^2$ and consequently $\alpha$ is a nonzero constant. Thus, by Theorem 1 we get the result. □

4. TransS-Structure $\left(\mathit{F},\mathbf{u},\mathit{\gamma },\mathit{\alpha },\mathit{\beta }\right)$ with $\mathbf{u}$ a Projective Vector

In this section, we are interested in studying the impact of the vector field $\mathbf{u}$ of a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ being a projective vector on the geometry of $\left(M^3,g\right)$. We prove the following:

 Theorem 4.  

A 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that the vector field $\mathbf{u}$ is a projective vector and the sectional curvatures of the plane sections containing $\mathbf{u}$ are a positive constant is homothetic to a Sasakian manifold.

 Proof. 

Let $\mathbf{u}$ be a projective vector. Then, using Lemma 1 and Equation (10), we have

$$\left({\pounds }_{\xi }\mathsf{\nabla }\right)\left(X,Y\right)={\displaystyle \frac{1}{2}}\left\{X\left(\beta \right)Y+Y\left(\beta \right)X\right\},X,Y\in \mathsf{\Gamma }\left(T{M}^3\right).$$

(25)

Also, using Equations (2) and (9) and Lemma 1 in computing

$$\begin{array}{ccc}\hfill \left({\pounds }_{\xi }\mathsf{\nabla }\right)\left(X,Y\right)& =& R\left(\mathbf{u},X\right)Y+{\mathsf{\nabla }}_X{\mathsf{\nabla }}_Y\mathbf{u}-{\mathsf{\nabla }}_{{\mathsf{\nabla }}_{X}Y}\mathbf{u}\hfill \\ & =& R\left(\mathbf{u},X\right)Y-X\left(\alpha \right)FY-{\alpha }^2\left(g\left(X,Y\right)\mathbf{u}-\gamma \left(Y\right)X\right)\hfill \\ & & -\alpha \beta \left(g\left(FX,Y\right)\mathbf{u}-\gamma \left(Y\right)FX\right)-X\left(\beta \right)\gamma \left(Y\right)\mathbf{u}\hfill \\ & & -\beta g\left(Y,-\alpha FX+\beta X-\beta \gamma \left(X\right)\mathbf{u}\right)\mathbf{u}\hfill \\ & & -\beta \gamma \left(Y\right)\left(-\alpha FX+\beta X-\beta \gamma \left(X\right)\mathbf{u}\right).\hfill \end{array}$$

Simplifying above equation and inserting in Equation (25), we conclude

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\left\{X\left(\beta \right)Y+Y\left(\beta \right)X\right\}& =R\left(\mathbf{u},X\right)Y-X\left(\alpha \right)FY-\left({\alpha }^2+{\beta }^2\right)g\left(X,Y\right)\mathbf{u}\hfill \\ \hfill & +\left({\alpha }^2-{\beta }^2\right)\gamma \left(Y\right)X+2\alpha \beta \gamma \left(Y\right)FX\hfill \\ \hfill & -X\left(\beta \right)\gamma \left(Y\right)\mathbf{u}+2{\beta }^2\gamma \left(X\right)\gamma \left(Y\right)\mathbf{u}.\hfill \end{array}$$

(26)

Taking $X=Y=e_k$ for a local frame $\left\{e_1,e_2,e_3\right\}$ in above equation and summing the resulting equation and using Equation (14), we conclude

$$\mathsf{\nabla }\beta =S\mathbf{u}-F\left(\mathsf{\nabla }\alpha \right)-3\left({\alpha }^2+{\beta }^2\right)\mathbf{u}+\left({\alpha }^2-{\beta }^2\right)\mathbf{u}-\mathbf{u}\left(\beta \right)\mathbf{u}+2{\beta }^2\mathbf{u},$$

that is,

$$S\mathbf{u}=F\left(\mathsf{\nabla }\alpha \right)+2\left({\alpha }^2+{\beta }^2\right)\mathbf{u}+\mathbf{u}\left(\beta \right)\mathbf{u}+\mathsf{\nabla }\beta .$$

Comparing above equation with (iv) in Lemma 1, we arrive at

$$\mathsf{\nabla }\beta +\left(\mathbf{u}\left(\beta \right)+2{\beta }^2\right)\mathbf{u}=0$$

(27)

and taking the inner product with $\mathbf{u}$ in above equation yields

$$\mathbf{u}\left(\beta \right)=-{\beta }^2.$$

(28)

Inserting above equation in Equation (27), we have

$$\mathsf{\nabla }\beta =\mathbf{u}\left(\beta \right)\mathbf{u}.$$

(29)

Now, taking $Y=\mathbf{u}$ in Equation (26), we have

$$\begin{array}{ccc}\hfill -R\left(X,\mathbf{u}\right)\mathbf{u}& =& \left({\alpha }^2+{\beta }^2\right)\gamma \left(X\right)\mathbf{u}+{\displaystyle \frac{1}{2}}\left[\mathbf{u}\left(\beta \right)-2\left({\alpha }^2-{\beta }^2\right)\right]X\hfill \\ & & +{\displaystyle \frac{3}{2}}X\left(\beta \right)\mathbf{u}-2\alpha \beta FX,\hfill \end{array}$$

and on taking the inner product with Y in above equation

$$\begin{array}{cc}\hfill -R\left(X,\mathbf{u};\mathbf{u},Y\right)& =\left({\alpha }^2+{\beta }^2\right)\gamma \left(X\right)\gamma \left(Y\right)+{\displaystyle \frac{1}{2}}\left[\mathbf{u}\left(\beta \right)-2\left({\alpha }^2-{\beta }^2\right)\right]g\left(X,Y\right)\hfill \\ \hfill & +{\displaystyle \frac{3}{2}}X\left(\beta \right)\gamma \left(Y\right)-2\alpha \beta g\left(FX,Y\right).\hfill \end{array}$$

(30)

Interchanging X and Y in above equation, we have

$$\begin{array}{cc}\hfill -R\left(Y,\mathbf{u};\mathbf{u},X\right)& =\left({\alpha }^2+{\beta }^2\right)\gamma \left(X\right)\gamma \left(Y\right)+{\displaystyle \frac{1}{2}}\left[\mathbf{u}\left(\beta \right)-2\left({\alpha }^2-{\beta }^2\right)\right]g\left(X,Y\right)\hfill \\ \hfill & +{\displaystyle \frac{3}{2}}Y\left(\beta \right)\gamma \left(X\right)-2\alpha \beta g\left(FY,X\right).\hfill \end{array}$$

and subtracting this equation from Equation (30), we conclude

$${\displaystyle \frac{3}{2}}\left[X\left(\beta \right)\gamma \left(Y\right)-Y\left(\beta \right)\gamma \left(X\right)\right]=4\alpha \beta g\left(FX,Y\right).$$

Thus, we have

$$4\alpha \beta FX={\displaystyle \frac{3}{2}}\left(X\left(\beta \right)\mathbf{u}-\gamma \left(X\right)\mathsf{\nabla }\beta \right),$$

which in view of Equation (29) implies

$$4\alpha \beta FX={\displaystyle \frac{3}{2}}\left(X\left(\beta \right)\mathbf{u}-\gamma \left(X\right)\mathbf{u}\left(\beta \right)\right)\mathbf{u}.$$

Operating F on above equation, we get

$$4\alpha \beta F^{2}X=0$$

and taking trace in above equation, yields

$$\alpha \beta =0.$$

(31)

Now, Equation (30) in view of Equation (31) for X orthogonal to $\mathbf{u}$, gives

$$-R\left(X,\mathbf{u};\mathbf{u},X\right)={\displaystyle \frac{1}{2}}\left[\mathbf{u}\left(\beta \right)-2\left({\alpha }^2-{\beta }^2\right)\right]{∥X∥}^2$$

and using (28) in above equation, we conclude

$$R\left(X,\mathbf{u};\mathbf{u},X\right)={\displaystyle \frac{1}{2}}\left(2{\alpha }^2-{\beta }^2\right){∥X∥}^2.$$

(32)

If $\alpha =0$ above expression will imply sectional curvatures of plane sections containing $\mathbf{u}$ are not positive and this is contrary to our assumption in the statement. Hence, $\alpha \ne 0$ and combining it with Equation (31), gives

$$R\left(X,\mathbf{u};\mathbf{u},X\right)={\alpha }^2$$

for unit vector X orthogonal to $\mathbf{u}$. As the sectional curvatures of plane sections containing $\mathbf{u}$ are positive constant, we get $\alpha$ is a nonzero constant. Thus, requirements of Theorem 1 are met and we confirm $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. □

5. TransS-Structure $\left(\mathit{F},\mathbf{u},\mathit{\gamma },\mathit{\alpha },\mathit{\beta }\right)$ with Generic Restrictions

Let $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ be a TransS-structure on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ and $\tau$ be the scalar curvature of $\left(M^3,g\right)$. In this section, first we seek the impact of the condition

$$S\left(\mathbf{u}\right)={\displaystyle \frac{\tau }{3}}\mathbf{u}$$

on the geometry of $\left(M^3,g\right)$. We prove the following:

 Theorem 5.

A 3-dimensional compact and connected Riemannian manifold $\left(M^3,g\right)$ of nonzero scalar curvature τ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that β is constant along the integral curves of the vector field $\mathbf{u}$ and the condition

$$S\left(\mathbf{u}\right)={\displaystyle \frac{\tau }{3}}\mathbf{u}$$

holds, is homothetic to a Sasakian manifold.

 Proof. 

Using Lemma 1 and the condition in the statement, we get

$$F\left(\mathsf{\nabla }\alpha \right)-\mathsf{\nabla }\beta +2\left({\alpha }^2-{\beta }^2\right)\mathbf{u}-\mathbf{u}\left(\beta \right)\mathbf{u}={\displaystyle \frac{\tau }{3}}\mathbf{u}.$$

(33)

Since, $\beta$ is constant along the integral curves of $\mathbf{u}$, we have $\mathbf{u}\left(\beta \right)=0$. Taking the inner product in (33) with $\mathbf{u}$, we get

$$\tau =6\left({\alpha }^2-{\beta }^2\right),$$

(34)

which together with $\mathbf{u}\left(\beta \right)=0$ used in Equation (33) gives

$$F\left(\mathsf{\nabla }\alpha \right)=\mathsf{\nabla }\beta .$$

(35)

We wish to compute $div\left(F\left(\mathsf{\nabla }\alpha \right)\right)$ and get

$$div\left(F\left(\mathsf{\nabla }\alpha \right)\right)=\sum _{k}g\left({\mathsf{\nabla }}_{e_k}F\left(\mathsf{\nabla }\alpha \right),e_k\right)=\sum _{k}g\left(\left({\mathsf{\nabla }}_{e_k}F\right)\left(\mathsf{\nabla }\alpha \right)+F\left({\mathit{H}}_{\alpha }{e}_k\right),e_k\right).$$

Now, using the facts that F is skew symmetric and ${\mathit{H}}_{\alpha }$ is symmetric in above equation, we get

$$div\left(F\left(\mathsf{\nabla }\alpha \right)\right)=-\sum _{k}g\left(\mathsf{\nabla }\alpha ,\left({\mathsf{\nabla }}_{e_k}F\right)\left(e_k\right)\right),$$

which on using Equation (2) and Lemma 1, gives

$$div\left(F\left(\mathsf{\nabla }\alpha \right)\right)=-g\left(\mathsf{\nabla }\alpha ,2\alpha \mathbf{u}\right)=-2\alpha \mathbf{u}\left(\alpha \right)=4{\alpha }^2\beta .$$

Taking divergence in Equation (35) and using above equation, we conclude

$$\mathsf{\Delta }\beta =4{\alpha }^2\beta ,$$

that is,

$$\beta \mathsf{\Delta }\beta =4{\alpha }^2{\beta }^2.$$

Integrating above equation by parts and noticing that the Riemannian manifold $\left(M^3,g\right)$ is without boundary, we arrive at

$$-\underset{M^3}{\int }{∥\mathsf{\nabla }\beta ∥}^2=4\underset{M^3}{\int }{\alpha }^2{\beta }^2$$

and we conclude

$$\mathsf{\nabla }\beta =0\mathrm{and}\alpha \beta =0.$$

(36)

Note that, $\mathsf{\nabla }\beta =0$ implies $\beta$ is a constant, which together with (iii) of Lemma 1 namely $divr\mathbf{u}=2\beta$, which integrates to give the constant $\beta =0$. Also, Equation (35) becomes

$$F\left(\mathsf{\nabla }\alpha \right)=0$$

and operating F on above equation gives

$$\mathsf{\nabla }\alpha =\mathbf{u}\left(\alpha \right)\mathbf{u}=-2\alpha \beta \mathbf{u}=0.$$

that is, $\alpha$ is a constant and Equation (34) shows $\tau =6{\alpha }^2$ and this proves $\alpha$ is a nonzero constant. This finishes the proof. □

In the next result, we shall use the notion that the Hessian operator ${\mathit{H}}_{\alpha }$ of the function $\alpha$ is invariant under the vector field $\mathsf{\nabla }\alpha$, which requires that ${\mathit{H}}_{\alpha }$ commutes with the differential of the local flow $\left\{{\varphi }_t\right\}$ of the vector field $\mathsf{\nabla }\alpha$. Thus, ${\mathit{H}}_{\alpha }$ is invariant under $\mathsf{\nabla }\alpha$ is equivalent to

$${\pounds }_{\mathsf{\nabla }\alpha }{\mathit{H}}_{\alpha }=0,$$

(37)

where ${\pounds }_{\mathsf{\nabla }\alpha }$ is the Lie derivative with respect to $\mathsf{\nabla }\alpha$. We use this notion to prove the following:

 Theorem 6.  

A 3-dimensional compact and simply connected non-negatively curved Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that $∥\mathsf{\nabla }\alpha ∥$ is a constant and the Hessian operator ${\mathit{H}}_{\alpha }$ is invariant under $\mathsf{\nabla }\alpha$, is homothetic to a Sasakian manifold.

 Proof. 

Since, ${\mathit{H}}_{\alpha }$ is invariant under $\mathsf{\nabla }\alpha$, Equation (37) implies

$$\left[\mathsf{\nabla }\alpha ,{\mathit{H}}_{\alpha }X\right]={\mathit{H}}_{\alpha }\left[\mathsf{\nabla }\alpha ,X\right],X\in \mathsf{\Gamma }\left(T{M}^3\right),$$

that is,

$${\mathsf{\nabla }}_{\mathsf{\nabla }\alpha }{\mathit{H}}_{\alpha }X-{\mathsf{\nabla }}_{{\mathit{H}}_{\alpha }X}\mathsf{\nabla }\alpha ={\mathit{H}}_{\alpha }\left({\mathsf{\nabla }}_{\mathsf{\nabla }\alpha }X-{\mathit{H}}_{\alpha }X\right),$$

which gives

$$\left({\mathsf{\nabla }}_{\mathsf{\nabla }\alpha }{\mathit{H}}_{\alpha }\right)\left(X\right)={\mathit{H}}_{\alpha }^{2}X-{\mathit{H}}_{\alpha }^{2}X=0.$$

(38)

Now, as $∥\mathsf{\nabla }\alpha ∥$ is a constant, we have $X\left({∥\mathsf{\nabla }\alpha ∥}^2\right)=0$, which implies

$$g\left({\mathit{H}}_{\alpha }X,\mathsf{\nabla }\alpha \right)=0,X\in \mathsf{\Gamma }\left(T{M}^3\right),$$

that is,

$${\mathit{H}}_{\alpha }\left(\mathsf{\nabla }\alpha \right)=0.$$

(39)

Differentiating above equation gives

$$\left({\mathsf{\nabla }}_X{\mathit{H}}_{\alpha }\right)\left(\mathsf{\nabla }\alpha \right)+{\mathit{H}}_{\alpha }^{2}X=0.$$

(40)

Now, using the identity

$$R\left(X,\mathsf{\nabla }\alpha \right)\mathsf{\nabla }\alpha =\left({\mathsf{\nabla }}_X{\mathit{H}}_{\alpha }\right)\left(\mathsf{\nabla }\alpha \right)-\left({\mathsf{\nabla }}_{\mathsf{\nabla }\alpha }{\mathit{H}}_{\alpha }\right)\left(X\right)$$

and Equations (38) and (40), we conclude

$$R\left(X,\mathsf{\nabla }\alpha \right)\mathsf{\nabla }\alpha =-{\mathit{H}}_{\alpha }^{2}X.$$

Above equation implies

$$R\left(X,\mathsf{\nabla }\alpha ;\mathsf{\nabla }\alpha .X\right)=-{∥{\mathit{H}}_{\alpha }X∥}^2$$

and since $\left(M^3,g\right)$ is non-negatively curved, we must have ${\mathit{H}}_{\alpha }X=0$ which implies, $\mathsf{\Delta }\alpha =0$. Thus, $\alpha$ is a constant. We claim that constant $\alpha \ne 0$, for if $\alpha =0$, by (ii) of Lemma 1, the 1-form $\gamma$ is closed and as $\left(M^3,g\right)$ is simply connected, $\gamma =d\rho$ for a smooth function $\rho$ on $M^3$. Thus, $\mathbf{u}=\mathsf{\nabla }\rho$ and since $M^3$ compact there is a point $p\in M^3$ such that $\mathsf{\nabla }\rho \left(p\right)=0$ either where $\rho$ is maximum or minimum. This will imply $\mathbf{u}\left(p\right)=0$, which is a contradiction as $\mathbf{u}$ is a unit vector. Hence, constant $\alpha \ne 0$ and by (i) of Lemma 1, we have $\beta =0$ and combining these two outcomes with Theorem 1, it confirms that the Riemannian manifold $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. □

Recall that for a smooth function f on a Riemannian manifold $\left(M^3,g\right)$, the Hessian of f $Hess\left(f\right)$ is defined by

$$Hess\left(f\right)\left(X,Y\right)=g\left({\mathit{H}}_{f}X,Y\right),X,Y\in \mathsf{\Gamma }\left(T{M}^3\right).$$

(41)

Finally, we prove the following:

 Theorem 7.  

A 3-dimensional connected Riemannian manifold $\left(M^3,g\right)$ that admits a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ such that (i) ${\left(\mathbf{u}\left(\alpha \right)\right)}^2+{\alpha }^2=c$, for a constant $c\ne 0$, and (ii) $Hess\left(\alpha \right)\left(\mathbf{u},\mathbf{u}\right)\ne -\alpha$, is homothetic to a Sasakian manifold.

 Proof. 

Since, ${\left(\mathbf{u}\left(\alpha \right)\right)}^2+{\alpha }^2=c$, for a constant $c\ne 0$. On operating the vector field $\mathbf{u}$ on this equation yields

$$\mathbf{u}\left(\alpha \right)\left(\mathbf{uu}\left(\alpha \right)\right)+\alpha \mathbf{u}\left(\alpha \right)=0.$$

(42)

Using ${\mathsf{\nabla }}_{\mathbf{u}}\mathbf{u}=0$ an outcome of (ii) in Lemma 1, we see through Equation (41) that

$$Hess\left(\alpha \right)\left(\mathbf{u},\mathbf{u}\right)=g\left({\mathsf{\nabla }}_{\mathbf{u}}\mathsf{\nabla }\alpha ,\mathbf{u}\right)=\mathbf{u}g\left(\mathsf{\nabla }\alpha ,\mathbf{u}\right)=\mathbf{uu}\left(\alpha \right)$$

and the Equation (42) becomes

$$\mathbf{u}\left(\alpha \right)\left(Hess\left(\alpha \right)\left(\mathbf{u},\mathbf{u}\right)+\alpha \right)=0.$$

Now, using the condition (ii) in the statement with the above equation on connected $M^3$, it confirms $\mathbf{u}\left(\alpha \right)=0$ and therefore, the condition (i) in the statement implies ${\alpha }^2=c$, for $c\ne 0$, that is, $\alpha$ is nonzero constant. Note that the possibility that $\mathbf{u}\left(\alpha \right)\ne 0$ is excluded owing to above equation and the condition (ii) in the statement. Also, by (i) in Lemma 1 with $\alpha$ a constant implies $\alpha \beta =0$, which confirms $\beta =0$. Thus, by Theorem 1, we see that the Riemannian manifold $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. □

6. Conclusions

Differential geometry of a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ is immensely important because of Geometrization conjecture (cf. [11,12]). This conjecture classifies the geometry of 3-dimensional Riemannian manifolds in eight geometries. Three of the important categories in these eight geometries is the spherical geometry $S^3$, the Euclidean geometry $E^3$ and the special linear group $SL\left(2,R\right)$ are Sasakian manifolds. There are other 3-dimensional Sasakian manifolds other than $S^3$, $E^3$ and $SL\left(2,R\right)$ namely, the unit tangent bundle $U{S}^2$ of the sphere $S^2$, the special unitary group $SU\left(2\right)$ and the Heisenberg group $H^3$ (cf. [16]). It is for this reason, results containing conditions under which a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ admitting a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ is homothetic to a Sasakian manifold are of significance.

In Section 5, we have considered a connected 3-dimensional Riemannian manifold $\left(M^3,g\right)$ admitting a TransS-structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ and used one of the following combinations such as

(i)$\mathbf{u}\left(\beta \right)=0$, $S\left(\mathbf{u}\right)={\displaystyle \frac{\tau }{3}}\mathbf{u}$, with $\tau \ne 0$ and $\left(M^3,g\right)$ compact,

(ii) $∥\mathsf{\nabla }\alpha ∥=c$, c a constant and ${\mathit{H}}_{\alpha }$ is invariant under $\mathsf{\nabla }\alpha$ with $\left(M^3,g\right)$ compact and simply connected,

(iii) ${\left(\mathbf{u}\left(\alpha \right)\right)}^2+{\alpha }^2=c$, c a nonzero constant and $Hess\left(\alpha \right)\left(\mathbf{u},\mathbf{u}\right)\ne -\alpha$,

to ensure that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold.

Naturally, it will be interesting to see if above three conditions could be replaced with the following:

(a) $\mathbf{u}\left(\alpha \right)=0$, $S\left(\mathbf{u}\right)={\displaystyle \frac{\tau }{3}}\mathbf{u}$, with $\tau \ne 0$ and $\left(M^3,g\right)$ compact, (b) $∥\mathsf{\nabla }\beta ∥=c$, c a constant and ${\mathit{H}}_{\beta }$ is invariant under $\mathsf{\nabla }\beta$ with $\left(M^3,g\right)$ compact and simply connected, (c) ${\left(\mathbf{u}\left(\beta \right)\right)}^2+{\beta }^2=c$, c a constant and $Hess\left(\beta \right)\left(\mathbf{u},\mathbf{u}\right)\ne -\beta$ and having the similar conclusions.