# Comptes Rendus
## Mathématique

Hui Zhang and Zhiqi Chen

Pseudo-Riemannian Lie groups admitting left-invariant conformal vector fields

Volume 358, issue 2 (2020), p. 143-149

Published online: 15 June 2020

https://doi.org/10.5802/crmath.23

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_2.png" /></figure>

Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l'édition scientifique ouverte
www.centre-mersenne.org

e-ISSN : 1778-3569
Geometry, Dynamical Systems / Géométrie, Systèmes dynamiques

# Pseudo-Riemannian Lie groups admitting
left-invariant conformal vector fields

*Groupes de Lie pseudo-riemanniens admettant des
champs vectoriels conformes invariants à gauche*

Hui Zhang<sup>*a*</sup> and Zhiqi Chen<sup>*b*</sup>

<sup>*a*</sup> School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China

<sup>*b*</sup> Corresponding author. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China.

*E-mails*: 2120160023@mail.nankai.edu.cn, chenzhiqi@nankai.edu.cn.

**Abstract.** Let $G$ be a Lorentzian Lie group or a pseudo-Riemannian Lie group of type $(n-2, 2)$. If $G$ admits a non-Killing left-invariant conformal vector field, then $G$ is solvable.

**Résumé.** Soit $G$ un groupe de Lie lorentzien ou un groupe de Lie pseudo-riemannien de type $(n-2, 2)$. Si $G$ admet un champ vectoriel invariant à gauche non-Killing, alors $G$ est résoluble.

**Keywords.** Conformal vector fields, Killing vector fields, Pseudo-Riemannian Lie groups, Lorenztian Lie groups.

2020 Mathematics Subject Classification. 53C25, 53C30, 22E60.

Manuscript received 26th December 2019, revised and accepted 21st February 2020.

## 1. Introduction

Let $(M, g)$ be a pseudo-Riemannian manifold. The conformal transformation group of $(M, g)$, denoted by $\text{Conf}(M, g)$, is called *essential* if no metric in the conformal class of $g$ is preserved by $\text{Conf}(M, g)$. If $(M, g)$ is Riemannian of dimension $\ge 2$, then $\text{Conf}(M, g)$ is essential if and only if $(M, g)$ is conformally diffeomorphic to the standard sphere $S^n$ or $\mathbb{R}^n$ with the canonical flat metric. This is the famous *Lichnérowicz's Conjecture*, which was finally proved by J. Ferrand in [7]. However, in the pseudo-Riemannian case, the situation is quite different as shown in [1, 8, 11, 14].
Thus it is meaningful to study the structure of a pseudo-Riemannian manifold $(M, g)$ with $\mathrm{Conf}(M, g)$ essential, for example a homogeneous pseudo-Riemannian manifold with a non-Killing conformal vector field. Here a vector field $X$ on $(M, g)$ is said to be conformal, if

$$L_X g = 2\rho g, \qquad (1)$$

where $L_X$ is the Lie derivation and $\rho$ is a smooth function on $M$ called the conformal factor with respect to $g$. If $g$ is a Riemannian metric, the existence of the function $\rho$ might give some information about the topological structure of the Riemannian manifold (see [5, 12]). As a subclass of conformal vector fields, a Yamabe soliton vector field, i.e. a vector field $X$ satisfies $L_X g = 2(\mathrm{scal} - \lambda)g$ where $\mathrm{scal}$ is the scalar curvature of the metric $g$ and $\lambda$ is a constant, plays an important role in exploring Yamabe flow (see [2–4, 6, 9]). If $\rho = 0$, we call $X$ a Killing vector field which provides a close link between the geometry of a manifold $M$ and the algebra of $I(M)$, where $I(M)$ denotes the set of all isometries in $(M, g)$ (see [13]).

Here we restrict $(M, g)$ to be a pseudo-Riemannian Lie group which is a Lie group with a left-invariant pseudo-Riemannian metric. All Lie groups are assumed to be connected. Furthermore the Lie group is called type $(p, q)$ if the signature of the pseudo-Riemannian metric is of $(p, q)$. As we know, there are some studies on Lorentzian Lie groups with non-Killing left-invariant conformal vector fields, i.e. pseudo-Riemannian Lie groups of type $(n-1, 1)$. For example the results in [2, 4, 15, 17].

The paper is organized as follows. First, we recall some facts on non-Killing left-invariant conformal vector fields on pseudo-Riemannian Lie groups in Section 2, and then prove the following Theorem in Section 3.

**Theorem 1.** Let $G$ be a Lorentzian Lie group or a pseudo-Riemannian Lie group of type $(n-2, 2)$, where $n \ge 4$. If $G$ admits a non-Killing left-invariant conformal vector field, then $G$ is solvable.

But it is unknown for general type $(p, q)$ for $p, q \ge 3$, and we conjecture that Theorem 1 holds for any type $(p, q)$ for $p, q \ge 3$.

In Section 4, we construct a class of pseudo-Riemannian solvable Lie groups of type $(p, q)$ which admit non-Killing left-invariant conformal vector fields, and then prove they are conformally flat. That is, they satisfy Lichnérowicz conjecture in the pseudo-Riemannian case.

## 2. Preliminaries

Let $G$ be a Lie group with the Lie algebra $\mathfrak{g}$ consisting of left-invariant vector fields and let $\langle \cdot, \cdot \rangle$ be a pseudo-Riemannian metric on $G$. Assume that $\nabla$ is the Levi-Civita connection associated with $\langle \cdot, \cdot \rangle$. Then

$$[X, Y] = \nabla_X Y - \nabla_Y X. \qquad (2)$$

If $\langle \cdot, \cdot \rangle$ is left-invariant on $G$, we have

$$\langle \nabla_Z X, Y \rangle + \langle X, \nabla_Z Y \rangle = 0, \qquad (3)$$

for any $X, Y, Z \in \mathfrak{g}$. By (2) and (3),

$$\langle \nabla_X Y, Z \rangle = \frac{1}{2} (\langle [X, Y], Z \rangle - \langle [Y, Z], X \rangle + \langle [Z, X], Y \rangle), \qquad (4)$$

where $X, Y, Z$ are all left-invariant vector fields. Assume that $X \in \mathfrak{g}$ is a conformal vector field, by (1), we have

$$0 = L_X \langle X, X \rangle = 2\rho |X|^2.$$

Furthermore if $\langle \cdot, \cdot \rangle$ is Riemannian, then $\rho = 0$ or $X = 0$. That is, $X$ is Killing or trivial.

For left-invariant pseudo-Riemannian metrics $\langle \cdot, \cdot \rangle$, we have
**Lemma 2 ([2]).** Let $G$ be a unimodular pesudo-Riemannian Lie group. Then any left-invariant conformal vector field on $G$ is a Killing vector field.

If $G$ is a non-unimodular pseudo-Riemannian Lie group, we have the following result.

**Lemma 3 ([2]).** Let $G$ be an $n$-dimensional non-unimodular pseudo-Riemannian Lie group of type $(p, q)$. If $g$ admits a non-Killing conformal vector field, then $\dim C(g) \le \min(p, q)$ and $\dim[g, g] \ge \dim g - \min(p, q)$.

Furthermore, for a Lorentzian Lie group, we have the following Lemma.

**Lemma 4 ([15]).** Let $G$ be a Lorentzian Lie group admitting a non-Killing left-invariant conformal vector field. Then $\dim[g, g] = \dim g - 1$.

**3. The proof of Theorem 1**

In order to prove Theorem 1, we first recall two facts.

**Lemma 5 ([10]).** Let $g$ be a Lie algebra over $\mathbb{R}$. If there is an invertible derivation on $g$, then $g$ is nilpotent.

**Lemma 6.** For any matrix $H \in \mathfrak{so}(n, 1)$, either $H$ has $n-1$ purely imaginary and two non-zero real eigenvalues $\pm r \in \mathbb{R}$, or $H$ has $n+1$ purely imaginary eigenvalues. Here, we consider $0$ as a purely imaginary number and $\mathfrak{so}(n, 1)$ is defined by

$$ \mathfrak{so}(n, 1) = \left\{ \begin{pmatrix} A & C \\ C' & 0 \end{pmatrix} : A = -A' \in \mathbb{R}^{n \times n}, C \in \mathbb{R}^{n \times 1} \right\}, $$

where $A'$ denotes the transpose of $A$.

Let $G$ be a pseudo-Riemannian Lie group whose Lie algebra is $g$ and let $X$ be a non-Killing left-invariant conformal vector field on $G$. Denote by $\langle \cdot, \cdot \rangle$ the pseudo-Riemannian metric of signature $(p, q)$ on $G$. Clearly $\langle X, X \rangle = 0$. By the definition of a conformal vector field (1), we have

$$ \langle [X, U], V \rangle + \langle U, [X, V] \rangle = -2\rho \langle U, V \rangle, \qquad (5) $$

where $U, V \in g$ and $0 \ne \rho$ is a constant.

**Lemma 7 ([2]).** Let $G$ be a Lorentzian Lie group admitting a non-Killing left-invariant conformal vector field. Then the restriction of $\langle \cdot, \cdot \rangle$ on $[g, g]$ is degenerate.

In fact, this lemma holds for any pseudo-Riemannian Lie group.

**Lemma 8.** Let $G$ be a pseudo-Riemannian Lie group admitting a non-Killing left-invariant conformal vector field $X$. Then the restriction $\langle \cdot, \cdot \rangle$ on $[g, g]$ is degenerate.

**Proof.** Assume that the restriction $\langle \cdot, \cdot \rangle$ on $[g, g]$ is non-degenerate. Then the restriction $\langle \cdot, \cdot \rangle$ on $[g, g]^\perp$ is non-degenerate. Thus there is a vector field $U \in [g, g]^\perp$ such that $\langle U, U \rangle \ne 0$. By (5),

$$ 0 = \langle [X, U], U \rangle + \langle U, [X, U] \rangle = -2\rho \langle U, U \rangle. $$

Thus $\rho = 0$, it is a contradiction. So the restriction $\langle \cdot, \cdot \rangle$ on $[g, g]$ is degenerate.

<figure><img src="image_3.png" /></figure>

**Proposition 9.** Assume that $G$ is a pseudo-Riemannian Lie group of type $(p, q)$. If $G$ admits a non-Killing left-invariant conformal vector field and $\dim[g, g] = \dim g - \min(p, q)$, then $G$ is solvable.
**Proof.** Assume that $X \in \mathfrak{g}$ is a non-Killing conformal vector field. Let $V_0 = \{y \in \mathfrak{g} | adX(y) = 0\}$. By (5), for any $u, v \in V_0$ we have

$$0 = \langle adX(u), v \rangle + \langle u, adX(v) \rangle = -2\rho \langle u, v \rangle.$$

Since $\rho \neq 0$, it follows $\langle u, v \rangle = 0$ for any $u, v \in V_0$. So $V_0$ is an isotropy subspace of $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. Let $k = \dim V_0$. Since $adX(\mathfrak{g}) \subset [\mathfrak{g}, \mathfrak{g}]$, we have

$$k \geq \dim \mathfrak{g} - \dim[\mathfrak{g}, \mathfrak{g}] = \min(p, q).$$

Let $m$ denote the multiplicity of the eigenvalue $0$ of $adX$. Clearly $m \geq k$. We claim $k = m$. Otherwise, $m > k$. By the Jordan canonical form theory for a nilpotent matrix, there are non-zero vectors $w \in \mathfrak{g}, v_0 \in V_0$ satisfying $adX(w) = v_0$. Obviously, $w \notin V_0$. Since $V_0$ is an isotropy subspace, for any $v \in V_0$, we have

$$0 = \langle adX(w), v \rangle + \langle w, adX(v) \rangle = -2\rho \langle w, v \rangle,$$

which implies $\langle w, v \rangle = 0$ since $\rho \neq 0$. It follows that

$$0 = \langle adX(w), w \rangle + \langle w, adX(w) \rangle = -2\rho \langle w, w \rangle,$$

which implies $\langle w, w \rangle = 0$. Then we have an isotropy subspace of dimension $\geq \min(p, q) + 1$ spanned by $V_0$ and $w$, which is impossible. That is, $k = m$.

If $adX$ isn't invertible on $[\mathfrak{g}, \mathfrak{g}]$, then we have

$$k = m \geq \min(p, q) + 1.$$

Namely $\dim V_0 \geq \min(p, q) + 1$. It is a contradiction since $V_0$ is an isotropy subspace of $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. Hence $adX$ must be invertible on $[\mathfrak{g}, \mathfrak{g}]$. By Lemma 5, we know that $[\mathfrak{g}, \mathfrak{g}]$ is nilpotent which implies the solvability of $\mathfrak{g}$. $\square$

**The proof of Theorem 1.** For the Lorentzian case, by Lemma 4, $\dim[\mathfrak{g}, \mathfrak{g}] = \dim \mathfrak{g} - 1$. Thus $G$ is solvable by Proposition 9.

For the pseudo-Riemannian Lie group of type $(n-2, 2)$, by Lemma 3, $\dim[\mathfrak{g}, \mathfrak{g}] \geq \dim \mathfrak{g} - 2$. By Lemma 2, we must have $\dim[\mathfrak{g}, \mathfrak{g}] = \dim \mathfrak{g} - 2$ or $\dim[\mathfrak{g}, \mathfrak{g}] = \dim \mathfrak{g} - 1$. If $\dim[\mathfrak{g}, \mathfrak{g}] = \dim \mathfrak{g} - 2$, then $G$ is solvable by Proposition 9. If $\dim[\mathfrak{g}, \mathfrak{g}] = n-1$, by Lemma 8, there is a basis $\{e_1, e_2, \dots, e_{n-1}, e_n\}$ of $\mathfrak{g}$ such that $[\mathfrak{g}, \mathfrak{g}] = \operatorname{span}\{e_1, e_2, \dots, e_{n-1}\}$ and the metric matrix associated with this basis is defined by

$$\begin{pmatrix} I_{n-3} & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{pmatrix}. \qquad (6)$$

Set $adX(e_j) = \sum_{i=1}^{n} a_{ij} e_i$, $a_{ij} \in \mathbb{R}$, $1 \leq i, j \leq n$. By (5), we know the matrix of $adX$ associated with this basis is represented by

$$\begin{pmatrix} H - \rho I_{n-3} & \mathbf{0} & \alpha \\ \beta & -2\rho & 0 \\ \mathbf{0}' & 0 & 0 \end{pmatrix}.$$

where $H \in \mathfrak{so}(n-3, 1)$, $\alpha \in \mathbb{R}^{(n-2) \times 1}$ and $\beta \in \mathbb{R}^{1 \times (n-2)}$. By Lemma 6, we know the eigenvalues of $(adX)|_{[\mathfrak{g}, \mathfrak{g}]}$ are of the forms:

$$-\rho, -2\rho, -\rho \pm \lambda, -\rho + i\alpha \quad (0 \neq a \in \mathbb{R}).$$

If $\lambda \neq \pm\rho$, then $(adX)|_{[\mathfrak{g}, \mathfrak{g}]}$ is invertible. By Lemma 5, $[\mathfrak{g}, \mathfrak{g}]$ is nilpotent which implies the solvability of $\mathfrak{g}$. If $\lambda = \rho$ or $\lambda = -\rho$, then the eigenvalues of $(adX)|_{[\mathfrak{g}, \mathfrak{g}]}$ are of the forms:

$$-\rho, -2\rho \text{ (of multiplicity 2), } 0 \text{ (of multiplicity 1), } -\rho + i\alpha \quad (0 \neq a \in \mathbb{R}).$$
Assume that $g = s \times t$ is a Levi decomposition of $g$ with $s \neq 0$. Let $X = X_s + X_t$ be the corresponding decomposition of $X$. Then we have

$$ adX = \begin{pmatrix} (adX_s)|_s & 0 \\ * & (adX)|_t \end{pmatrix} : s \times t \to s \times t. $$

In particular, the eigenvalues of $(adX_s)|_s$ would be also eigenvalues of $adX$. It contradicts to $\text{tr}(adX_s)|_s = 0$. So $s = 0$, i.e. $g$ is solvable.

That is, we have Theorem 1.

<figure><img src="image_4.png" /></figure>

In general, we have the following conjecture.

**Conjecture 10.** Let $G$ be a pseudo-Riemannian Lie group of type $(p, q)$ where $p, q \ge 3$. If $G$ admits a non-Killing left-invariant conformal vector field, then $G$ is solvable.

## 4. Conformally flat pseudo-Riemannian Lie groups

The following example generalizes the Lorentzian case in [2] to type $(p, q)$.

**Example 11.** Consider the Lie algebra $g$ defined by

$$ [e_n, e_i] = -\rho e_i, \quad 1 \le i \le n-2, \quad [e_n, e_{n-1}] = -2\rho e_{n-1}, \qquad (7) $$

where $\{e_1, e_2, \dots, e_n\}$ is a basis of $g$, and $\rho$ is a non-zero constant. Clearly, $g$ is a non-umimodular solvable Lie algebra with abelian derived algebra $[g, g] = \text{span}\{e_1, e_2, \dots, e_{n-1}\}$. Define an inner product $\langle \cdot, \cdot \rangle$ of signature $(p, q)$ $(p, q \ge 1)$ on $g$ associated with the basis $\{e_1, e_2, \dots, e_n\}$ by

$$ \begin{pmatrix} I_{p-1} & & \\ & -I_{q-1} & \\ & & 0 & 1 \\ & & 1 & 0 \end{pmatrix}. \qquad (8) $$

Let $G$ denote the simply connected Lie group with the Lie algebra $g$, and we also use the symbol $\langle \cdot, \cdot \rangle$ to denote the induced left-invariant pseudo-Riemannian metric on $G$. Then $X \in g$ is a non-Killing conformal vector field on $(G, \langle \cdot, \cdot \rangle)$ if and only if

$$ \langle [X, e_i], e_j \rangle + \langle e_i, [X, e_j] \rangle = -2c \langle e_i, e_j \rangle, \qquad (9) $$

where $1 \le i, j \le n$, and $c$ is a non-zero constant. By a straightforward computation, we know $X = e_n$ is a left-invariant non-Killing conformal vector field on $(G, \langle \cdot, \cdot \rangle)$ satisfying

$$ L_X \langle \cdot, \cdot \rangle = 2\rho \langle \cdot, \cdot \rangle. \qquad (10) $$

The following is to prove that $(G, \langle \cdot, \cdot \rangle)$ is conformally flat. We first recall some definitions and a theorem of Weyl. For a pseudo-Riemannian manifold $(M, g)$, denote by $\nabla$, $R$ and scal the Levi-Civita connection, the Riemann curvature tensor, the Ricci tensor and the scalar curvature respectively. For symmetric (0,2)-type tensor fields $h, k$ on $(M, g)$, define the *Kulkarni-Nomizu* product as the (0,4)-type tensor field by

$$ h \circ k(v_1, v_2, v_3, v_4) = \frac{1}{2} (h(v_1, v_4)k(v_2, v_3) + h(v_2, v_3)k(v_1, v_4)) \\ - \frac{1}{2} (h(v_1, v_3)k(v_2, v_4) + h(v_2, v_4)k(v_1, v_3)). $$

The *Schouten tensor* for $n > 2$ is given by

$$ P = \frac{2}{n-2} \text{Ric} - \frac{\text{scal}}{(n-1)(n-2)} g, \qquad (11) $$
and the Weyl conformal curvature tensor W is defined by

$$R = P \circ g + W, \qquad (12)$$

where R is the (0,4)-type Riemann curvature tensor and ◦ is the Kulkarni–Nomizu product. The following result of Weyl is well-known.

**Theorem 12** ([16]). A pseudo-Riemannian manifold M of dimension ≥ 4 is conformally flat if and only if the Weyl conformal curvature tensor W vanishes identically.

Let the notations as Example 11 for $n \ge 4$. Denote by B the left-invariant pseudo-Riemannian metric $\langle \cdot, \cdot \rangle$ and denote by ∇ the Levi-Civita connection as usual. Let

$$B_{ij} = B(e_i, e_j), [e_i, e_j] = C_{ij}{}^k e_k, \nabla_{e_i} e_j = \Gamma_{ji}{}^k e_k, \\ R(e_i, e_j)e_k = R_{ijkl}{}^l e_l, R_{ijkl} = R_{ijk}{}^s B_{sl}, H_{ijkl} = (P \circ B)(e_i, e_j, e_k, e_l).$$

Using the Koszul's formula, we have

$$\Gamma_{ij}{}^k = \frac{1}{2} (-C_{ij}{}^s B_{st} - C_{jt}{}^s B_{si} + C_{ti}{}^s B_{sj}) B^{tk}.$$

where $(B^{ij})$ denotes the inverse matrix of $(B_{ij})$. Furthermore, we have

$$R_{ijk}{}^l = \Gamma_{kj}{}^s \Gamma_{si}{}^l - \Gamma_{ki}{}^s \Gamma_{sj}{}^l - C_{ij}{}^s \Gamma_{ks}{}^l$$

By the definition of g in Example 11, the non-zero $C_{ij}{}^k$ are

$$C_{in}{}^i = \rho = -C_{ni}{}^i, i \le n-2; C_{n-1,n}{}^{n-1} = 2\rho = -C_{n,n-1}{}^{n-1}.$$

Set $\epsilon_i = B(e_i, e_i) \in \{\pm 1\}$ for $i \le n-2$. By a straightforward computation, the non-zero $\Gamma_{ij}{}^k$ are

$$\Gamma_{ii}{}^{n-1} = -\rho\epsilon_i, \Gamma_{ni}{}^i = \rho, i \le n-2; \Gamma_{n-1,n}{}^{n-1} = -2\rho, \Gamma_{nn}{}^n = 2\rho,$$

and by the symmetry of R corresponding to subscript, the fundamental non-zero $R_{ijkl}{}^l$ are

$$R_{ini}{}^{n-1} = -\rho^2 \epsilon_i, R_{inn}{}^i = \rho^2, i \le n-2.$$

and consequently the fundamental non-zero $R_{ijkl}$ are

$$R_{inni} = \rho^2 \epsilon_i, i \le n-2.$$

In particular, the Ricci tensor is

$$(\mathrm{Ric}(e_i, e_j)) = \begin{pmatrix} 0 & \cdots & 0 & 0 \\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 0 & 0 \\ 0 & \cdots & 0 & (n-2)\rho^2 \end{pmatrix},$$

and the scalar curvature vanishes. Then by (11), we know that the Schouten tensor $P = \frac{2}{n-2} \mathrm{Ric}$, and the fundamental nonzero $H_{ijkl} = (P \circ B)(e_i, e_j, e_k, e_l)$ are

$$H_{inni} = \rho^2 \epsilon_i, i \le n-2.$$

Since the Weyl conformal curvature $W_{ijkl} = R_{ijkl} - H_{ijkl}$, by Theorem 12, we know that $(G, \langle \cdot, \cdot \rangle)$ is conformally flat.

## 5. Acknowledgement

The first author would like to thank Prof. L.B. Huang for some useful discussions. Also we would like to express our thanks to the referee for meaningful suggestions and corrections, especially for pointing out Lichnérowicz's Conjecture to us. This paper is partially supported by NSFC (11571182 and 11931009) and NSF of Tianjin (19JCYBJC30600).
References

[1] D. Alekseevskii, "Self-similar Lorentzian manifolds", *Ann. Global Anal. Geom.* **3** (1985), no. 1, p. 59-84.

[2] A. Araujo, Z. Chen, B. Leandro, "Conformal vector fields on Lie groups", https://arxiv.org/abs/1608.05943v2, 2016.

[3] E. Barbosa, E. j. Ribeiro, "On conformal solutions of the Yamabe flow", *Arch. Math.* **101** (2013), no. 1, p. 79-89.

[4] E. Calviño-Louzao, J. Seoane-Bascay, M. E. Vázquez-Abal, R. Vázquez-Lorenzo, "Three-dimensional homogeneous Lorentzian Yamabe solitons", *Abh. Math. Semin. Univ. Hamb.* **82** (2012), no. 2, p. 193-203.

[5] A. Caminha, "The geometry of closed conformal vector fields on Riemannian spaces", *Bull. Braz. Math. Soc. (N.S.)* **42** (2011), no. 2, p. 277-300.

[6] H.-D. Cao, X. Sun, Y. Zhang, "On the structure of gradient Yamabe solitons", *Math. Res. Lett.* **19** (2012), no. 4, p. 767-774.

[7] J. Ferrand, "The action of conformal transformations on a Riemannian manifold", *Math. Ann.* **304** (1996), no. 2, p. 277-291.

[8] C. Frances, "About pseudo-Riemannian Lichnerowicz conjecture", *Transform. Groups* **20** (2015), no. 4, p. 1015-1022.

[9] S.-Y. Hsu, "A note on compact gradient Yamabe solitons", *J. Math. Anal. Appl.* **388** (2012), no. 2, p. 725-726.

[10] N. Jacobson, "A note on automorphisms and derivations of Lie algebras", *Proc. Am. Math. Soc.* **6** (1955), p. 281-283.

[11] W. Kühnel, H.-B. Rademacher, "Essential conformal fields in pseudo-Riemannian geometry. II", *J. Math. Sci., Tokyo* **4** (1997), no. 3, p. 649-662.

[12] M. Obata, "Certain conditions for a Riemannian manifold to be isometric with a sphere", *J. Math. Soc. Japan* **14** (1962), p. 152-164.

[13] B. O'Neill, *Semi-Riemannian geometry with applications to relativity*, Pure and Applied Mathematics, vol. 103, Academic Press Inc., 1983.

[14] M.-N. Podoksënov, "Conformally homogeneous Lorentzian manifolds. II", *Sib. Mat. Zh.* **33** (1992), no. 6, p. 154-161, translation in *Siberian Math. J.* **33** (1992), p. 1087-1093.

[15] J. Tan, Z. Chen, N. Xu, "Conformal vector fields on Lorentzian Lie groups of dimension 4", *J. Lie Theory* **28** (2018), no. 3, p. 761-769.

[16] H. Weyl, "Reine Infinitesimalgeometrie", *Math. Z.* **26** (1918), p. 384-411.

[17] H. Zhang, Z. Chen, S. Zhang, "Conformal vector fields on Lorentzian Lie groups of dimension 5", to appear in *J. Lie Theory*, 2020.