Characterization of Ricci Solitons and Harmonic Vector Fields on the Lie Group Nil⁴

Abstract

This study considers a left-invariant Riemannian metric g on the Lie group $Ni{l}^4$. We introduce a Ricci solitons’ classification on $(Ni{l}^4,g)$. These are expansive non-gradient Ricci solitons. We examine the existence of harmonic maps into $(Ni{l}^4,g)$ from a compact Riemannian manifold. Additionally, we provide a characterization of a class of harmonic vector fields on $(Ni{l}^4,g)$.

1. Introduction

Given an m-dimensional Riemannian manifold $(M,g)$, the Levi–Civita connection, Riemannian curvature, and Ricci curvature of $(M,g)$ are represented by the symbols ∇, R, and Ric, respectively. Thus,

$$\begin{array}{ccc}\hfill R(X,Y)Z& =& [{\nabla }_X,{\nabla }_Y]Z-{\nabla }_{[X,Y]}Z,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill Ric(X,Y)& =& \sum _{i=1}^{m}g(R(X,e_i)e_i,Y),\hfill \end{array}$$

(2)

where ${\left\{e_i\right\}}_{i=1}^m$ is an orthonormal frame on $(M,g)$ and $X,Y,Z\in \Gamma \left(TM\right)$ (see [1]). A Ricci soliton structure on a Riemannian manifold $(M,g)$ is defined as a smooth vector field $\xi$ that satisfies the following soliton equation:

$$Ric+{\displaystyle \frac{1}{2}}{\mathcal{L}}_{\xi }g=\lambda g,$$

(3)

where ${\mathcal{L}}_{\xi }g$ is the Lie derivative of the Riemannian metric g along the vector field $\xi$, and $\lambda$ is a constant in the set of real numbers. The Ricci soliton $(M,g,\xi ,\lambda )$ is classified as shrinking, steady, or expansive based on the value of $\lambda$ in Equation (3). Specifically, if $\lambda >0$, the soliton is shrinking; if $\lambda =0$, it is steady; and if $\lambda <0$, it is expansive. If $\xi$ is equal to the gradient of a smooth function f on M, we define $(M,g,gradf,\lambda )$ as a gradient Ricci soliton with potential f. In this case, the Ricci soliton equation can be written as follows:

$$Ric+Hessf=\lambda g.$$

(4)

If the vector field $\xi$ of the Ricci soliton is either zero or a Killing vector field (i.e., ${\mathcal{L}}_{\xi }g=0$), then we have the definition of an Einstein metric with the Einstein constant $\lambda$. A Ricci soliton is considered non-trivial if the metric $(M,g)$ is not Einstein. Ricci soliton vector fields have important consequences for differential geometry research, as they constrain the geometry and topology of Riemannian manifolds (see [2,3]).

The semidirect product ${\mathbb{R}}^3{⋉}_U\mathbb{R}$ defines the 4-dimensional nilpotent Lie group $Ni{l}^4$ (see [4]), where $U\left(t\right)=exp\left(tL\right)$, as follows:

$$L=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),exp\left(tL\right)=I_3+tL+{\displaystyle \frac{t^2}{2}}{L}^2=\left(\begin{array}{ccc}1& t& {\displaystyle \frac{t^2}{2}}\\ 0& 1& t\\ 0& 0& 1\end{array}\right).$$

(5)

Considering the frame fields that are left-invariant,

$$\begin{array}{ccc}\hfill & & e_1={\displaystyle \frac{\partial }{\partial x}},\hfill \\ & & e_2=t{\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{\partial }{\partial y}},\hfill \\ & & e_3={\displaystyle \frac{t^2}{2}}{\displaystyle \frac{\partial }{\partial x}}+t{\displaystyle \frac{\partial }{\partial y}}+{\displaystyle \frac{\partial }{\partial z}},\hfill \\ & & e_4={\displaystyle \frac{\partial }{\partial t}}.\hfill \end{array}$$

(6)

Therefore, we have dual coframe fields, giving us

$$\begin{array}{ccc}\hfill & & {\theta }_1=dx-tdy+{\displaystyle \frac{t^2}{2}}dz,\hfill \\ & & {\theta }_2=dy-tdz,\hfill \\ & & {\theta }_3=dz,\hfill \\ & & {\theta }_4=dt.\hfill \end{array}$$

(7)

The Riemannian metric $g={\theta }_1^2+{\theta }_2^2+{\theta }_3^2+{\theta }_4^2$ is given by

$$\left(g_{ij}\right)=\left(\begin{array}{cccc}1& -t& {\displaystyle \frac{t^2}{2}}& 0\\ -t& 1+t^2& -t(1+{\displaystyle \frac{t^2}{2}})& 0\\ {\displaystyle \frac{t^2}{2}}& -t(1+{\displaystyle \frac{t^2}{2}})& 1+t^2+{\displaystyle \frac{t^4}{4}}& 0\\ 0& 0& 0& 1\end{array}\right).$$

One of the four-dimensional Thurston model geometries is the four-dimensional Lie group $Ni{l}^4$ (see [5,6]). See [4] for the geometric properties of a Riemannian manifold $(Ni{l}^4,g)$.

A smooth map $\phi :(M,g)⟶(N,h)$ between two Riemannian manifolds has the energy on a compact domain D of M, defined by

$$E(\phi ;D)={\displaystyle \frac{1}{2}}{\int }_D{\left|d\phi \right|}^2{v}^g,$$

(8)

where $v^g$ is the volume element on $(M,g)$, and $\left|d\phi \right|$ is the Hilbert–Schmidt norm of the differential $d\phi$. If $\phi$ is a critical point of the energy functional (8), then the map is referred to as harmonic. We have

$$\tau \left(\phi \right)={Tr}_g\nabla d\phi =\sum _{i=1}^m\left[{\nabla }_{e_i}^{\phi }d\phi \left(e_i\right)-d\phi \left({\nabla }_{e_i}^M{e}_i\right)\right]=0,$$

(9)

as the Euler Lagrange equation for (8), where ${\left\{e_i\right\}}_{i=1}^m$ is a local orthonormal frame field on $(M,g)$, ${\nabla }^M$ is the Levi–Civita connection of $(M,g)$, and ${\nabla }^{\phi }$ is the pull-back connection on ${\phi }^{-1}TN$ (see [7,8]).

Let $(TM,g_S)$ be the tangent bundle of a Riemanian manifold $(M,g)$ equipped with the Sasaki metric $g_S$ defined by $g_S(X^h,Y^h)=g_S(X^v,Y^v)=g(X,Y)\circ \pi$ and $g_S(X^h,Y^v)=0$, for all $X,Y\in \Gamma \left(TM\right)$, where $\pi$ is the natural projection from $TM$ onto M, and $X^h$ (resp. $X^v$) is the horizontal (resp. vertical) lift of a vector field X on M to $TM$ (see [9]).

The tension field of $X:(M,g)⟶(TM,g_S)$ is given by (see [10])

$$\tau \left(X\right)={({Tr}_{g}R(X,{\nabla }_{·}X)·)}^h+{\left({Tr}_g{\nabla }^{2}X\right)}^v.$$

(10)

Consequently, X is a harmonic map from $(M,g)$ to $(TM,g_S)$ if and only if ${Tr}_{g}R(X,{\nabla }_{·}X)·=0$ and ${Tr}_g{\nabla }^{2}X=0$. A smooth vector field X on $(M,g)$ is said to be a harmonic section if it is a critical point of the vertical energy

$$E^v(X;D)={\displaystyle \frac{1}{2}}{\int }_D{|\nabla X|}^2{v}^g.$$

(11)

The corresponding Euler–Lagrange equation is given by $\overline{\Delta }X={Tr}_g{\nabla }^{2}X=0$.

In [11], the author describes the three-dimensional Heisenberg groups’ left-invariant Lorentzian metric as a Lorentz Ricci soliton and establishes that this Ricci soliton is a non-gradient shrinking Ricci soliton. In this paper, we examine the existence of a Ricci soliton on $Ni{l}^4$ equipped with the Riemannian metric g. Numerous researchers have looked at the Liouville-type theorem for harmonic maps on Riemannian manifolds. Any harmonic map from a compact orientable Riemannian manifold without a boundary to a non-trivial Ricci soliton $(N,h,\xi ,\lambda )$ with $Ric>\lambda h$ or $Ric<\lambda h$ is constant, as has been demonstrated in [12]. For harmonic maps into the Lie group $(Ni{l}^4,g)$, we provide the Liouville-type theorem. For class vector fields on $(Ni{l}^4,g)$, we additionally provide harmonicity conditions.

2. Ricci Solitons on $({\mathit{Nil}}^{\mathbf{4}},\mathit{g})$

Theorem 1.

On the four-dimensional Lie group $(Ni{l}^4,g)$, a vector field ξ is a Ricci soliton if and only if

$$\xi =(-2x-c_{2}y+c_4){\displaystyle \frac{\partial }{\partial x}}+(-c_{2}z-{\displaystyle \frac{3}{2}}y-c_3){\displaystyle \frac{\partial }{\partial y}}+(-z+c_1){\displaystyle \frac{\partial }{\partial z}}+(-{\displaystyle \frac{t}{2}}-c_2){\displaystyle \frac{\partial }{\partial t}},$$

for some constants $c_1,c_2,c_3,c_4\in \mathbb{R}$. Moreover, the Ricci soliton $(Ni{l}^4,g,\xi ,\lambda )$ is expansive with $\lambda =-{\displaystyle \frac{3}{2}}$.

The following Lemmas are required for the demonstration of Theorem 1:

Lemma 1

([4]). The non-zero values of the Levi–Civita connection ∇ for $(Ni{l}^4,g)$ can be expressed as

$$\begin{array}{ccc}\hfill {\nabla }_{e_1}{e}_2& =& {\displaystyle \frac{1}{2}}{e}_4,{\nabla }_{e_1}{e}_4=-{\displaystyle \frac{1}{2}}{e}_2\hfill \\ \hfill {\nabla }_{e_2}{e}_1& =& {\displaystyle \frac{1}{2}}{e}_4,{\nabla }_{e_2}{e}_3={\displaystyle \frac{1}{2}}{e}_4\hfill \\ \hfill {\nabla }_{e_2}{e}_4& =& -{\displaystyle \frac{1}{2}}(e_1+e_3),{\nabla }_{e_3}{e}_2={\displaystyle \frac{1}{2}}{e}_4\hfill \\ \hfill {\nabla }_{e_3}{e}_4& =& -{\displaystyle \frac{1}{2}}{e}_2,{\nabla }_{e_4}{e}_1=-{\displaystyle \frac{1}{2}}{e}_2\hfill \\ \hfill {\nabla }_{e_4}{e}_2& =& {\displaystyle \frac{1}{2}}(e_1-e_3),{\nabla }_{e_4}{e}_3={\displaystyle \frac{1}{2}}{e}_2.\hfill \end{array}$$

Lemma 2

([4]). The Ricci curvature of $(Ni{l}^4,g)$ is given by

$$\left({\mathrm{Ric}}_{ij}\right)=\left(\begin{array}{cccc}{\displaystyle \frac{1}{2}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{\displaystyle \frac{1}{2}}& 0\\ 0& 0& 0& -1\end{array}\right),$$

where ${\displaystyle {Ric}_{ij}=\sum _{a=1}^{4}g(R(e_i,e_a)e_a,e_j)}$, for all $i,j=\overline{1,4}$.

Proof of Theorem 1.

Let $\xi ={\alpha }_1{e}_1+{\alpha }_2{e}_2+{\alpha }_3{e}_3+{\alpha }_4{e}_4$ be a vector field on $(Ni{l}^4,g)$, where ${\alpha }_i\in C^{\infty }\left(Ni{l}^4\right)$ for all $i=1,2,3,4$. The vector field $\xi$ is a Ricci soliton if and only if

$${Ric}_{ij}+{\displaystyle \frac{1}{2}}[g({\nabla }_{e_i}\xi ,e_j)+g({\nabla }_{e_j}\xi ,e_i)]=\lambda {\delta }_{ij},\forall i,j=\overline{1,4},$$

(12)

for some constant $\lambda$. Take ${\beta }_{ij}=g({\nabla }_{e_i}\xi ,e_j)$. By using Lemma 1, a straightforward calculation shows that

$$\left({\beta }_{ij}\right)=\left(\begin{array}{cccc}{\alpha }_{1;1}& {\alpha }_{2;1}-{\displaystyle \frac{1}{2}}{\alpha }_4& {\alpha }_{3;1}& {\alpha }_{4;1}+{\displaystyle \frac{1}{2}}{\alpha }_2\\ {\alpha }_{1;2}-{\displaystyle \frac{1}{2}}{\alpha }_4& {\alpha }_{2;2}& {\alpha }_{3;2}-{\displaystyle \frac{1}{2}}{\alpha }_4& {\alpha }_{4;2}+{\displaystyle \frac{1}{2}}{\alpha }_1+{\displaystyle \frac{1}{2}}{\alpha }_3\\ {\alpha }_{1;3}& {\alpha }_{2;3}-{\displaystyle \frac{1}{2}}{\alpha }_4& {\alpha }_{3;3}& {\alpha }_{4;3}+{\displaystyle \frac{1}{2}}{\alpha }_2\\ {\alpha }_{1;4}+{\displaystyle \frac{1}{2}}{\alpha }_2& {\alpha }_{2;4}-{\displaystyle \frac{1}{2}}{\alpha }_1+{\displaystyle \frac{1}{2}}{\alpha }_3& {\alpha }_{3;4}-{\displaystyle \frac{1}{2}}{\alpha }_2& {\alpha }_{4;4}\end{array}\right),$$

(13)

where ${\alpha }_{j;i}=e_i\left({\alpha }_j\right)$ for all $i,j=\overline{1,4}$. By using Lemma 2 and (13), System (12) is equivalent to the following symmetric matrix being zero:

$$\left(E_{ij}\right)=\left(\begin{array}{cccc}{\alpha }_{1;1}-\lambda +{\displaystyle \frac{1}{2}}& {\alpha }_{2;1}-{\alpha }_4+{\alpha }_{1;2}& {\alpha }_{3;1}+{\alpha }_{1;3}& {\alpha }_{4;1}+{\alpha }_2+{\alpha }_{1;4}\\ {\alpha }_{2;1}-{\alpha }_4+{\alpha }_{1;2}& {\alpha }_{2;2}-\lambda & {\alpha }_{3;2}-{\alpha }_4+{\alpha }_{2;3}& {\alpha }_{4;2}+{\alpha }_3+{\alpha }_{2;4}\\ {\alpha }_{3;1}+{\alpha }_{1;3}& {\alpha }_{3;2}-{\alpha }_4+{\alpha }_{2;3}& {\alpha }_{3;3}-\lambda -{\displaystyle \frac{1}{2}}& {\alpha }_{4;3}+{\alpha }_{3;4}\\ {\alpha }_{4;1}+{\alpha }_2+{\alpha }_{1;4}& {\alpha }_{4;2}+{\alpha }_3+{\alpha }_{2;4}& {\alpha }_{4;3}+{\alpha }_{3;4}& -1+{\alpha }_{4;4}-\lambda \end{array}\right).$$

(14)

By using (6) and (14), we obtain the following PDEs:

$$\begin{array}{ccc}\hfill E_{11}& =& {\alpha }_{1;x}-\lambda +{\displaystyle \frac{1}{2}}=0;\hfill \\ \hfill E_{12}& =& {\alpha }_{2;x}-{\alpha }_4+t{\alpha }_{1;x}+{\alpha }_{1;y}=0;\hfill \\ \hfill E_{13}& =& {\alpha }_{3;x}+{\displaystyle \frac{t^2}{2}}{\alpha }_{1;x}+t{\alpha }_{1;y}+{\alpha }_{1;z}=0;\hfill \\ \hfill E_{14}& =& {\alpha }_{4;x}+{\alpha }_2+{\alpha }_{1;t}=0;\hfill \\ \hfill E_{22}& =& t{\alpha }_{2;x}+{\alpha }_{2;y}-\lambda =0;\hfill \\ \hfill E_{23}& =& t{\alpha }_{3;x}+{\alpha }_{3;y}-{\alpha }_4+{\displaystyle \frac{t^2}{2}}{\alpha }_{2;x}+t{\alpha }_{2;y}+{\alpha }_{2;z}=0;\hfill \\ \hfill E_{24}& =& t{\alpha }_{4;x}+{\alpha }_{4;y}+{\alpha }_3+{\alpha }_{2;t}=0;\hfill \\ \hfill E_{33}& =& {\displaystyle \frac{t^2}{2}}{\alpha }_{3;x}+t{\alpha }_{3;y}+{\alpha }_{3;z}-\lambda -{\displaystyle \frac{1}{2}}=0;\hfill \\ \hfill E_{34}& =& {\displaystyle \frac{t^2}{2}}{\alpha }_{4;x}+t{\alpha }_{4;y}+{\alpha }_{4;z}+{\alpha }_{3;t}=0;\hfill \\ \hfill E_{44}& =& {\alpha }_{4;t}-\lambda -1=0.\hfill \end{array}$$

Here, ${\alpha }_{j;x_i}=\partial {\alpha }_j/\partial x_i$ for all $i,j=\overline{1,4}$, where $x_1=x$, $x_2=y$, $x_3=z$, and $x_4=t$.

The solutions of this PDE system are given by

$$\begin{array}{ccc}\hfill {\alpha }_1& =& -{\displaystyle \frac{1}{2}}(z-c_1)t^2+({\displaystyle \frac{3}{2}}y+c_{2}z+c_3)t-2x-c_{2}y+c_4;\hfill \\ \hfill {\alpha }_2& =& (z-c_1)t-c_{2}z-{\displaystyle \frac{3}{2}}y-c_3;\hfill \\ \hfill {\alpha }_3& =& -z+c_1;\hfill \\ \hfill {\alpha }_4& =& -{\displaystyle \frac{t}{2}}-c_2,\hfill \end{array}$$

(15)

with $\lambda =-{\displaystyle \frac{3}{2}}$, where $c_1,c_2,c_3,c_4\in \mathbb{R}$. Theorem 1 follows from (6) and (15). □

Remark 1.

1.

By Theorem 1, the vector fields ${\left\{e_i\right\}}_{i=1}^4$ are not Ricci solitons.

2.

A straightforward calculation shows that the Ricci soliton vector field given in Theorem 1 is a non-gradient Ricci soliton, i.e., there exists no smooth function f in $Ni{l}^4$ such that ξ is $gradf$ on $(Ni{l}^4,g)$.

3.

Using the similar technique given in the proof of Theorem 1, where, from (6) and (13), we can calculate the Lie derivative of the Riemannian metric g with respect to ξ. Moreover, arriving at a system of PDEs, we deduce that the only Killing vector fields on $(Ni{l}^4,g)$ are given by

$$\xi =(c_{1}y+c_2){\displaystyle \frac{\partial }{\partial x}}+(c_{1}z+c_3){\displaystyle \frac{\partial }{\partial y}}+c_4{\displaystyle \frac{\partial }{\partial z}}+c_1{\displaystyle \frac{\partial }{\partial t}},$$

for some constants $c_1,c_2,c_3,c_4\in \mathbb{R}$.

Proposition 1.

The components ${\xi }_1=-2x-c_{2}y+c_4$, ${\xi }_2=-c_{2}z-{\displaystyle \frac{3}{2}}y-c_3$, ${\xi }_3=-z+c_1$, and ${\xi }_4=-{\displaystyle \frac{t}{2}}-c_2$ of the Ricci soliton vector field ξ are harmonic functions on $(Ni{l}^4,g)$.

Proof. 

The proof follows directly from the definition of the Laplacian,

$$\begin{array}{ccc}\hfill \Delta \left({\xi }_j\right)& =& \sum _{i=1}^4\left[e_i\left(e_i\left({\xi }_j\right)\right)-\left({\nabla }_{e_i}{e}_i\right)\left({\xi }_j\right)\right],\forall j=\overline{1,4},\hfill \end{array}$$

Equation (6), and Lemma 1. □

Theorem 2.

Every harmonic map from a compact orientable Riemannian manifold without a boundary into the Riemannian manifold $(Ni{l}^4,g)$ is constant.

Proof. 

By Lemma 2 with $\lambda =-{\displaystyle \frac{3}{2}}$, the inequality

$$Ric(V,V)-\lambda g(V,V)=2V_1^2+{\displaystyle \frac{3}{2}}{V}_2^2+V_3^2+{\displaystyle \frac{1}{2}}{V}_4^2\ge 0,$$

(16)

holds, for all $V\in \Gamma \left(TNi{l}^4\right)$. Theorem 2 follows from (16) and Proposition 9 in [12]. □

3. Harmonic Vector Fields on $({\mathbf{Nil}}^{\mathbf{4}},\mathit{g})$

Given that the only variable on which the left-invariant Riemannian metric g depends on is the t coordinate, we search for harmonic vector fields whose components depend only on t.

Theorem 3.

A vector field $X=X_1\left(t\right)e_1+\cdots +X_4\left(t\right)e_4$ on $Ni{l}^4$ is a harmonic section with respect to g if and only if

$$\begin{array}{ccc}& & X_{1;tt}+X_{2;t}-{\displaystyle \frac{1}{2}}{X}_1=0;\hfill \\ & & X_{2;tt}-X_{1;t}+X_{3;t}-X_2=0;\hfill \\ & & X_{3;tt}-X_{2;t}-{\displaystyle \frac{1}{2}}{X}_3=0;\hfill \\ & & X_{4;tt}-X_4=0.\hfill \end{array}$$

Proof. 

Take ${\theta }_{ij}=g({\nabla }_{e_i}X,e_j)$ for all $i,j=\overline{1,4}$. By using (13), we obtain

$$\left({\theta }_{ij}\right)=\left(\begin{array}{cccc}0& -{\displaystyle \frac{1}{2}}{X}_4& 0& {\displaystyle \frac{1}{2}}{X}_2\\ -{\displaystyle \frac{1}{2}}{X}_4& 0& -{\displaystyle \frac{1}{2}}{X}_4& {\displaystyle \frac{1}{2}}{X}_1+{\displaystyle \frac{1}{2}}{X}_3\\ 0& -{\displaystyle \frac{1}{2}}{X}_4& 0& {\displaystyle \frac{1}{2}}{X}_2\\ X_{1;t}+{\displaystyle \frac{1}{2}}{X}_2& X_{2;t}-{\displaystyle \frac{1}{2}}{X}_1+{\displaystyle \frac{1}{2}}{X}_3& X_{3;t}-{\displaystyle \frac{1}{2}}{X}_2& X_{4;t}\end{array}\right),$$

(17)

where $X_{j;t}=e_4\left(X_j\right)$ and $X_{j;tt}=e_4\left(e_4\left(X_j\right)\right)$, for all $j=\overline{1,4}$. Hence,

$$\begin{array}{ccc}\hfill {\nabla }_{e_1}X& =& -{\displaystyle \frac{1}{2}}{X}_4{e}_2+{\displaystyle \frac{1}{2}}{X}_2{e}_4;\hfill \\ \hfill {\nabla }_{e_2}X& =& -{\displaystyle \frac{1}{2}}{X}_4{e}_1-{\displaystyle \frac{1}{2}}{X}_4{e}_3+{\displaystyle \frac{1}{2}}[X_1+X_3]e_4;\hfill \\ \hfill {\nabla }_{e_3}X& =& -{\displaystyle \frac{1}{2}}{X}_4{e}_2+{\displaystyle \frac{1}{2}}{X}_2{e}_4;\hfill \\ \hfill {\nabla }_{e_4}X& =& [X_{1;t}+{\displaystyle \frac{1}{2}}{X}_2]e_1+[X_{2;t}-{\displaystyle \frac{1}{2}}{X}_1+{\displaystyle \frac{1}{2}}{X}_3]e_2\hfill \\ & & +[X_{3;t}-{\displaystyle \frac{1}{2}}{X}_2]e_3+X_{4;t}{e}_4.\hfill \end{array}$$

(18)

From Lemma 1 and (18), we find that

$$\begin{array}{ccc}\hfill {\nabla }_{e_1}{\nabla }_{e_1}X& =& -{\displaystyle \frac{1}{4}}{X}_4{e}_4-{\displaystyle \frac{1}{4}}{X}_2{e}_2;\hfill \\ \hfill {\nabla }_{e_2}{\nabla }_{e_2}X& =& -{\displaystyle \frac{1}{2}}{X}_4{e}_4-{\displaystyle \frac{1}{4}}[X_1+X_3](e_1+e_3);\hfill \\ \hfill {\nabla }_{e_3}{\nabla }_{e_3}X& =& -{\displaystyle \frac{1}{4}}{X}_4{e}_4-{\displaystyle \frac{1}{4}}{X}_2{e}_2;\hfill \\ \hfill {\nabla }_{e_4}{\nabla }_{e_4}X& =& [X_{1;tt}+{\displaystyle \frac{1}{2}}{X}_{2;t}]e_1-{\displaystyle \frac{1}{2}}[X_{1;t}+{\displaystyle \frac{1}{2}}{X}_2]e_2\hfill \\ & & +[X_{2;tt}-{\displaystyle \frac{1}{2}}{X}_{1;t}+{\displaystyle \frac{1}{2}}{X}_{3;t}]e_2+{\displaystyle \frac{1}{2}}[X_{2;t}-{\displaystyle \frac{1}{2}}{X}_1+{\displaystyle \frac{1}{2}}{X}_3](e_1-e_3)\hfill \\ & & +[X_{3;tt}-{\displaystyle \frac{1}{2}}{X}_{2;t}]e_3+{\displaystyle \frac{1}{2}}[X_{3;t}-{\displaystyle \frac{1}{2}}{X}_2]e_2+X_{4;tt}{e}_4.\hfill \end{array}$$

(19)

Theorem 3 follows from (19) and the following equation:

$$\begin{array}{ccc}\hfill \overline{\Delta }X& =& \sum _{i=1}^4[{\nabla }_{e_i}{\nabla }_{e_i}X-{\nabla }_{{\nabla }_{e_i}{e}_i}X]=0,\hfill \end{array}$$

with ${\nabla }_{e_i}{e}_i=0$ for all $i=\overline{1,4}$. □

Corollary 1.

Let $X=X_i\left(t\right)e_i$ be a vector field on $Ni{l}^4$ for some $i=\overline{1,4}$.

1.

If $i=\overline{1,3}$, then X is a harmonic section if and only if $X_i\left(t\right)=0$;

2.

If $i=4$, then X is a harmonic section if and only if $X_4\left(t\right)=c_1{e}^t+c_2{e}^{-t}$ for some $c_1,c_2\in \mathbb{R}$.

We characterize a class of harmonic vector fields on $(Ni{l}^4,g)$ as mappings in the following Theorem.

Theorem 4.

A vector field $X=X_1\left(t\right)e_1+\cdots +X_4\left(t\right)e_4$ on $Ni{l}^4$ is a harmonic map with respect to g if and only if

1.

${\displaystyle X_1=0,X_2=0,X_3=0,X_4=c_1{e}^t+c_2{e}^{-t}}$, or

2.

${\displaystyle X_1=c_1{e}^{\frac{t}{\sqrt{2}}}+c_2{e}^{-\frac{t}{\sqrt{2}}},X_2=0,X_3=X_1,X_4=0}$,

for some $c_1,c_2\in \mathbb{R}$.

Proof. 

The non-zero components of the Riemannian curvature R of $(Ni{l}^4,g)$ are as follows (see [4]):

$$\begin{array}{ccc}\hfill g(R(e_1,e_2)e_1,e_2)& =& -{\displaystyle \frac{1}{4}},g(R(e_1,e_2)e_2,e_3)={\displaystyle \frac{1}{4}}\hfill \\ \hfill g(R(e_1,e_4)e_1,e_4)& =& -{\displaystyle \frac{1}{4}},g(R(e_1,e_4)e_3,e_4)={\displaystyle \frac{1}{4}}\hfill \\ \hfill g(R(e_2,e_1)e_2,e_3)& =& -{\displaystyle \frac{1}{4}},g(R(e_2,e_3)e_2,e_3)=-{\displaystyle \frac{1}{4}}\hfill \\ \hfill g(R(e_2,e_4)e_2,e_4)& =& {\displaystyle \frac{1}{2}},g(R(e_3,e_4)e_3,e_4)={\displaystyle \frac{3}{4}}.\hfill \end{array}$$

(20)

By using Equations (18) and (20), we find that

$$\begin{array}{ccc}\hfill \sum _{i=1}^{4}g(R(X,{\nabla }_{e_i}X)e_i,e_1)& =& -{\displaystyle \frac{1}{4}}{X}_4{X}_{1;t}+{\displaystyle \frac{1}{4}}{X}_4{X}_{3;t}+{\displaystyle \frac{1}{4}}{X}_1{X}_{4;t}-{\displaystyle \frac{1}{4}}{X}_3{X}_{4;t};\hfill \\ \hfill \sum _{i=1}^{4}g(R(X,{\nabla }_{e_i}X)e_i,e_2)& =& {\displaystyle \frac{1}{2}}{X}_3{X}_4+{\displaystyle \frac{1}{2}}{X}_4{X}_{2;t}-{\displaystyle \frac{1}{2}}{X}_2{X}_{4;t};\hfill \\ \hfill \sum _{i=1}^{4}g(R(X,{\nabla }_{e_i}X)e_i,e_3)& =& {\displaystyle \frac{1}{4}}{X}_4{X}_{1;t}+{\displaystyle \frac{3}{4}}{X}_4{X}_{3;t}-{\displaystyle \frac{1}{4}}{X}_1{X}_{4;t}-{\displaystyle \frac{3}{4}}{X}_3{X}_{4;t};\hfill \\ \hfill \sum _{i=1}^{4}g(R(X,{\nabla }_{e_i}X)e_i,e_4)& =& {\displaystyle \frac{1}{4}}{X}_1{X}_2+{\displaystyle \frac{3}{4}}{X}_2{X}_3.\hfill \end{array}$$

(21)

From Equations (10), (19) and (21), the vector field X is a harmonic map if and only if

$$\begin{array}{ccc}& & X_{1;tt}+X_{2;t}-{\displaystyle \frac{1}{2}}{X}_1=0;\hfill \\ & & X_{2;tt}-X_{1;t}+X_{3;t}-X_2=0;\hfill \\ & & X_{3;tt}-X_{2;t}-{\displaystyle \frac{1}{2}}{X}_3=0;\hfill \\ & & X_{4;tt}-X_4=0;\hfill \\ & & -X_4{X}_{1;t}+X_4{X}_{3;t}+X_1{X}_{4;t}-X_3{X}_{4;t}=0;\hfill \\ & & X_3{X}_4+X_4{X}_{2;t}-X_2{X}_{4;t}=0;\hfill \\ & & X_4{X}_{1;t}+3X_4{X}_{3;t}-X_1{X}_{4;t}-3X_3{X}_{4;t}=0;\hfill \\ & & X_1{X}_2+3X_2{X}_3=0.\hfill \end{array}$$

The solutions of this PDE system are given by

$$X_1=0,X_2=0,X_3=0,X_4=c_1{e}^t+c_2{e}^{-t};$$

$$X_1=c_1{e}^{{\displaystyle \frac{t}{\sqrt{2}}}}+c_2{e}^{-{\displaystyle \frac{t}{\sqrt{2}}}},X_2=0,X_3=X_1,X_4=0.$$

□

Example 1.

The vector field $X=cosh\left(t\right)e_4$ is a harmonic map from $(Ni{l}^4,g)$ to $(TNi{l}^4,g_S)$.

4. Conclusions

In this paper, we investigated the existence of Ricci solitons on the four-dimensional nilpotent Lie group $Ni{l}^4$ equipped with the left-invariant Riemannian metric g. By explicitly constructing the frame fields, coframe fields, and the corresponding metric, we laid the groundwork for understanding the geometric properties of this Thurston model geometry. Our analysis demonstrated the conditions under which $(Ni{l}^4,g)$ admits Ricci solitons, extending results from lower-dimensional cases, such as the Heisenberg group, to a higher-dimensional setting.

Furthermore, we explored the behavior of harmonic maps into $(Ni{l}^4,g)$ and established a Liouville-type theorem, showing that harmonic maps from compact manifolds into non-trivial Ricci solitons are necessarily constant under certain curvature constraints. Additionally, we examined the harmonicity conditions for vector fields on $(Ni{l}^4,g)$ and characterized the corresponding harmonic sections of the tangent bundle with the Sasaki metric.

These findings contribute to the broader understanding of Ricci solitons in Lie groups and provide new insights into the interplay between harmonic maps and the geometric structure of nilmanifolds. Future research could explore the stability of these solitons under Ricci flow and extend the harmonic section analysis to other Thurston geometries, enriching our knowledge of geometric flows and their solitonic solutions.