Ricci solitons of the $\mathbb{H}^{2} \times \mathbb{R}$ Lie group

1.   Introduction

The notion of Ricci solitons is introduced by Hamilton in [16], which is a naturel generalization of Einstein metrics. A Ricci solitons is a pseudo-Riemannian metric $g$ on a smooth manifold $\mathcal{M}$ such that there exists a smooth vector field $X$ on $\mathcal{M}$ satisfying the following equation:

where $\mathcal{L}_{X}$ denotes the Lie derivative in the direction of $X,$ $Ric$ denotes the Ricci tensor and $\lambda$ is a real number. A Ricci soliton is said to be a shrinking, steady or expanding, respectively, if $\lambda>0,$ $\lambda = 0$ or $\lambda<0.$ Moreover, we say that a Ricci soliton $(M, g)$ is a gradient Ricci soliton if it admits a vector field $X$ satisfying $X = \mathrm{grad}$ $h$, for some potential function $h$.

The description of Ricci solitons can be regarded as a first step in understanding the Ricci flow, since they are the fixed points of the flow. Moreover, they are important in understanding singularities of the Ricci flow. Under suitable conditions, type $I$ singularity models correspond to shrinking solitons, type $II$ models correspond to steady Ricci solitons, while type $III$ models correspond to expanding Ricci solitons.

In the special case that $\mathcal{M}$ is a Lie group and $g$ is a left-invariant metric, we say that $g$ is a left-invariant Ricci soliton on $\mathcal{M}$ if the above equation (1) holds.

Study of Ricci soliton, over different geometric spaces is one of interesting topics in geometry and mathematical physics. In particular, it has become more important after Grigori Perelman applied Ricci solitons to solve the long standing Poincaré conjecture. Ricci solitons correspond to self-similar solutions of Hamilton's Ricci flow [17], play a fundamental role in the formation of singularities of the flow and have been studied by several authors (see [12], [13]). They can be viewed as fixed points of the Ricci flow, as a dynamical system, on the space of Riemannian metrics modulo diffeomorphisms and scalings. Ricci solitons are of interests to physicists as well and are called quasi-Einstein metrics in physics literature. In fact, Theoretical physicists have been looking into the equation of Ricci solitons in relation with String Theory. A seminal contribution in this direction is due to Friedan ([15]).

Lorentzian Ricci solitons have been intensively studied, showing many essential differences with respect to the Riemannian case (see [2], [4], [3], [8], [23], [22]). In fact, although there exist three-dimensional Riemannian homogeneous Ricci solitons [1], [20], there are no left-invariant Riemannian Ricci solitons on three-dimensional Lie groups [14] (see also [18] and [24]). Moreover, the Lorentzian case is much richer, allowing the existence of expanding, steady and shrinking left-invariant Ricci soliton [7]. In [22] prove that Lorentzian and Riemannian five-dimensional solvable Lie groups admit different vector fields resulting in expanding Ricci solitons. Also has proved that those Ricci solitons are not gradient.

In this paper, we consider the left-invariant Riemannian metric admitted by the three-dimensional Lie group denoted by $\mathbb{H}^{2} \times \mathbb{R}$ and we prove the existence of vector field for which the soliton equation (1) holds. In Section 3, Ricci solitons of three-dimensional Lie group $\mathbb{H}^{2} \times \mathbb{R}$ are characterized via a system of partial differential equations. In particular, we show that three-dimensional Lie group $\mathbb{H}^{2} \times \mathbb{R}$ admit a vector fields in expanding Ricci solitons. Finally, we show that there are not gradient Ricci solitons.

2.   Connection and curvature of the $\mathbb{H}^{2} \times \mathbb{R}$ Lie group

Let $\mathbb{H}^{2}$ be represented by the upper half-plane model $\{(x, y)\in\mathbb{R}^{2}| y>0\}$ equipped with the metric $g_{\mathbb{H}^{2}} = \frac{1}{y^{2}}(dx^{2}+dy^{2}).$ The space $\mathbb{H}^{2},$ with the group structure derived by the composition of proper affine map, is a Lie group and the metric $g_{\mathbb{H}^{2}}$ is left invariant. Therefore the Riemannian product space $\mathbb{H}^{2} \times \mathbb{R}$ is a Lie group with respect to the operation

and the left invariant product metric

With respect to the metric $g$ an orthonormal basis of left invariant vector fields on $\mathbb{H}^{2} \times \mathbb{R}$ is

From this, the Lie brackets are given by

Throughout the paper, we shall endow the three-dimensional Lie group $\mathbb{H}^{2} \times \mathbb{R}$ with left-invariant Riemannian $g.$

We will denote by $\nabla$ the Levi-Civita connection of $(\mathbb{H}^{2} \times \mathbb{R}, g),$ by $R$ its curvature tensor, taken with the sign convention:

and by $Ric$ the Ricci tensor of $(\mathbb{H}^{2} \times \mathbb{R}, g),$ which is defined by

where $\{E_{k}\}_{k = 1, .., 3}$ is an orthonormal basis.

The Levi-Civita connection $\nabla$ of the $\mathbb{H}^{2} \times \mathbb{R}$ Lie group with respect to this frame (3) is

The non-vanishing curvature tensor $R$ components are computed as

The Ricci curvature components $\{Ric_{ij}\}$ are computed as

The scalar curvature $\tau$ of the $\mathbb{H}^{2} \times \mathbb{R}$ Lie group is constant and we have

3.   Ricci solitons of 3-dimensional $\mathbb{H}^{2} \times \mathbb{R}$ Lie group

In this section we analyze the existence of Ricci solitons on three-dimensional Riemannian Lie group $(\mathbb{H}^{2} \times \mathbb{R}, g)$ equipped with the left-invariant Riemannian metric (2).

Let $X = f_{1}E_{1}+f_{2}E_{2}+f_{3}E_{3}$ be an arbitrary vector field on $(\mathbb{H}^{2} \times \mathbb{R}, g)$, where $f_{1}, .., f_{3}$ are smooth functions of the variables $x, y, z$. We will denote the coordinate basis $\left\{ \frac{\partial }{\partial x}, \frac{ \partial }{\partial y}, \frac{\partial }{\partial z}\right\}$ by $\left\{ \partial _{x}, \partial _{y}, \partial _{z}\right\}$.

The Lie derivative of the metric (2) with respect to $X$ is given by :

Thus, by using (2), (6) and (8) in (1), a standard calculation gives that the three-dimensional Lie group $\left(\mathbb{H}^{2} \times \mathbb{R}, g\right)$ is a Ricci soliton if and only if the following system holds,

We deriving the third equation in (9) with respect to $z,$ we get

The equation (10) yields that

where $\varphi$ and $\psi$ are smooth functions.

Deriving the first equation with respect to $y$ and used the fourth equation in (9), we find

Next, deriving the second equation with respect to $y$ in (9) and used equation in (12), we find

Replacing $f_{1}$ in equations (12) and (13), we get

By derivation of the equations (14) with respect to $z,$ we get

We deriving the second equation with respect to $x$ and used the fourth equation in (15), we find

Integrating the first and second equations in (15), we find

where $\alpha_{i}\in\mathbb{R},$ and $\varphi_{1}, \psi_{1}$ are smooth functions with only one variable $x.$ Thus

Next, we replace $f_{1}$ in the second equation in (9), we get

We derive equation (19) with respect to $z,$ we get

By integration of (20) with respect to $x,$ we find that

where $\alpha_{i}\in\mathbb{R}.$ Thus

From the first equation in (9), gives

The last equation in (9) gives

where $\xi$ is a smooth function depending to $x$ and $y.$ We replacing $f_{1}, f_{2},$ and $f_{3}$ in the third and fifth equations in (9), we get

Integrating the first equation in (25) with respect to $x,$ we get

and replace $\xi$ in the second equation in (25), we find that

finally for arbitrary reals constants $\alpha_{i}$ we have

Thus, it easily follows that the vector field $X =$ $f_{1}E_{1}+f_{2}E_{2}+f_{3}E_{3}$ where $f_{1}, .., f_{3}$ are given by (26) satisfies (9). Note that $\lambda = -1$. Summarizing, we proved that the three-dimensional Lie group $\mathbb{H}^{2} \times \mathbb{R}$ admits appropriate vector fields for which (1) holds, obtaining the following result.

Theorem 3.1. Let $(\mathbb{H}^{2} \times \mathbb{R}, g)$ be the three-dimensional Lie group equipped with the left-invariant Riemannian metric $g$ given by (2). Then, $\left(\mathbb{H}^{2} \times \mathbb{R}, g\right)$ is an Expanding Ricci soliton.

Now, let $X = \mathrm{grad}$ $h$ be an arbitrary gradient vector field on $\left( \mathbb{H}^{2} \times \mathbb{R}, g\right)$ with potential function $h$, $X$ is then given by

From (26) it follows that $\left( \mathbb{H}^{2} \times \mathbb{R}, g\right)$ is gradient soliton if and only if the potential function $h$ satisfy the following systems

Hence, with direct integration we prove that

But the function $h$ does not verify the first equation in (28), we proved the following result.

Corollary 1. Let $(\mathbb{H}^{2} \times \mathbb{R}, g)$ be the three-dimensional Lie group equipped with the left-invariant Riemannian metric $g$ given by (2). Then, $\left(\mathbb{H}^{2} \times \mathbb{R}, g\right)$ is not gradient Ricci soliton.

4.   Conclusion

Our study on the Ricci solitons of three-dimensional Lie group denoted by $\mathbb{H}^{2} \times \mathbb{R}$, equipped with left-invariant Riemannian metric.The existence of non-trivial (i.e., not Einstein) Ricci solitons on three-dimensional Lie group $\mathbb{H}^{2} \times \mathbb{R}$ is proved. More precisely, we proved that there are not gradient Ricci solitons.

Acknowledgments

We would like to thank the referee for valuable suggestions regarding both the contents and exposition of this article.