Next Article in Journal
Multiscale Discrete Element Modeling
Previous Article in Journal
Contact Symmetries of a Model in Optimal Investment Theory

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups

by
Yong Wang
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Symmetry 2021, 13(2), 218; https://doi.org/10.3390/sym13020218
Submission received: 10 January 2021 / Revised: 23 January 2021 / Accepted: 27 January 2021 / Published: 28 January 2021

## Abstract

:
Riemann solitons are generalized fixed points of the Riemann flow. In this note, we study left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups. We completely classify left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.
MSC:
53C40; 53C42

## 1. Introduction

Riemann solitons are generalized fixed points of the Riemann flow. In the context of contact geometry, Hirica and Udriste proved [1] that if a Sasakian manifold admited a Riemann soliton with potential vector field pointwise collinear with the structure vector field, then it was a Sasakian space form. In [2], Blaga and Latcu studied almost Riemann solitons and almost Ricci solitons in an $( α , β )$-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton was pointwise collinear with the structure vector field. Geometric flows have many physical applications. Here we call attention to certain important applications of the Ricci flow theory in the study of nonlinear sigma models [3,4,5,6], research on geometric flow evolution of modified (non) holonomic commutative and noncommutative gravity theories [7,8,9,10], and exact solutions for (modified) gravity and geometric flows, Ricci solitons [11,12,13,14,15]. In [16], Calvaruso studied three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. He determined their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups. In [17], Batat and Onda studied algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. They got a complete classification of algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. In [18], Calvaruso completely classify three-dimensional homogeneous manifolds equipped with Einstein-like metrics. In [19], we classify affine Ricci solitons associated to canonical connections and Kobayashi–Nomizu connections and perturbed canonical connections and perturbed Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure. In this note, we completely classify the left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.

## 2. Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups

Three-dimensional Lorentzian Lie groups have been classified in [20,21] (see Theorems 2.1 and 2.2 in [17]). Throughout this paper, we shall by ${ G i } i = 1 , ⋯ , 7$, denote the connected, simply connected three-dimensional Lie group equipped with a left-invariant Lorentzian metric g and having Lie algebra ${ g } i = 1 , ⋯ , 7$. Let be the Levi–Civita connection of $G i$ and R its curvature tensor, taken with the convention:
$R ( X , Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X , Y ] Z .$
Let $R ( X , Y , Z , W ) = − g ( R ( X , Y ) Z , W )$. Riemann solitons are defined by a smooth vector field and a real constant $λ$ which satisfy the following equation:
$R + 1 2 L V g ∧ g = λ 2 g ∧ g ,$
where $L V g$ denotes the Lie derivative of g and ∧ is the Kulkarni–Nomizu product. Let $T 1$ and $T 2$ be two arbitrary $( 0 , 2 )$-tensors, then their Kulkarni–Nomizu product is defined by:
$T 1 ∧ T 2 ( X , Y , Z , W ) : = T 1 ( X , W ) T 2 ( Y , Z ) + T 1 ( Y , Z ) T 2 ( X , W ) − T 1 ( X , Z ) T 2 ( Y , W ) − T 1 ( Y , W ) T 2 ( X , Z ) ,$
for any $X , Y , Z , W ∈ Γ ( T G i )$, where $Γ ( T G i )$ denotes the set of all vector fields on $G i$. By (2) and (3), we can express the Riemann soliton as follows:
$2 R ( X , Y , Z , W ) + g ( X , W ) ( L V g ) ( Y , Z ) + g ( Y , Z ) ( L V g ) ( X , W ) − g ( X , Z ) ( L V g ) ( Y , W ) − g ( Y , W ) ( L V g ) ( X , Z ) = 2 λ [ g ( X , W ) g ( Y , Z ) − g ( X , Z ) g ( Y , W ) ] .$
For $G i$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike. Let $V = λ 1 e 1 + λ 2 e 2 + λ 3 e 3$, where $λ 1 , λ 2 , λ 3$ are real numbers. Let $R i j k l = R ( e i , e j , e k , e l )$. Then $( G i , V , g )$ is a left-invariant Riemann soliton if and only if:
$2 R 1212 − ( L V g ) ( e 2 , e 2 ) − ( L V g ) ( e 1 , e 1 ) = − 2 λ , 2 R 1312 − ( L V g ) ( e 2 , e 3 ) = 0 , 2 R 2312 + ( L V g ) ( e 1 , e 3 ) = 0 , 2 R 1313 − ( L V g ) ( e 3 , e 3 ) + ( L V g ) ( e 1 , e 1 ) = 2 λ , 2 R 2313 + ( L V g ) ( e 1 , e 2 ) = 0 , 2 R 2323 − ( L V g ) ( e 3 , e 3 ) + ( L V g ) ( e 2 , e 2 ) = 2 λ .$
By Theorem 2.1 in [17], we have for $G 1$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 1$ satisfies:
$[ e 1 , e 2 ] = α e 1 − β e 3 , [ e 1 , e 3 ] = − α e 1 − β e 2 , [ e 2 , e 3 ] = β e 1 + α e 2 + α e 3 , α ≠ 0 .$
By (2.18) in [18], we have for $G 1$:
$R 1212 = − 2 α 2 − β 2 4 , R 1313 = β 2 4 − 2 α 2 , R 2323 = β 2 4 , R 1213 = 2 α 2 , R 1223 = − α β , R 1323 = α β .$
Let,
$L V g = ( L V g ) ( e 1 , e 1 ) ( L V g ) ( e 1 , e 2 ) ( L V g ) ( e 1 , e 3 ) ( L V g ) ( e 2 , e 1 ) ( L V g ) ( e 2 , e 2 ) ( L V g ) ( e 2 , e 3 ) ( L V g ) ( e 3 , e 1 ) ( L V g ) ( e 3 , e 2 ) ( L V g ) ( e 3 , e 3 ) .$
By page 7 in [16], we get for $G 1$,
$L V g = 2 α ( λ 2 − λ 3 ) − α λ 1 α λ 1 − α λ 1 2 α λ 3 − α ( λ 2 + λ 3 ) α λ 1 − α ( λ 2 + λ 3 ) 2 α λ 2 .$
By (5), (7), and (9) and $α ≠ 0$, we get that $( G 1 , V , g )$ is a left-invariant Riemann soliton if and only if:
$− 2 α 2 − β 2 4 − α λ 2 = − λ , 4 α + λ 2 + λ 3 = 0 , λ 1 = 2 β , β 2 4 − 2 α 2 − α λ 3 = λ , β 2 2 − 2 α λ 2 + 2 α λ 3 = 2 λ .$
The first equation plusing the fourth equation in (10), we get $λ 2 + λ 3 + 4 α = 0$. By the fourth equation and the fifth equation in (10), we have $λ 2 − 2 λ 3 − 2 α = 0$. Then $λ 2 = λ 3 = − 2 α$. By the first equation in (10), we get $λ = β 2 4$. So we have:
Theorem 1.
$( G 1 , V , g )$ is a left-invariant Riemann soliton if and only if $λ 1 = 2 β , λ 2 = − 2 α$, $λ 3 = − 2 α$, $λ = β 2 4$.
By Theorem 2.1 in [17], we have for $G 2$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 2$ satisfies:
$[ e 1 , e 2 ] = γ e 2 − β e 3 , [ e 1 , e 3 ] = − β e 2 − γ e 3 , [ e 2 , e 3 ] = α e 1 , γ ≠ 0 .$
By page 144 in [17], we have for $G 2$:
$R 1212 = − γ 2 − α 2 4 , R 1313 = α 2 4 + γ 2 , R 2323 = − γ 2 − 3 4 α 2 + α β , R 1213 = γ ( 2 β − α ) , R 1223 = 0 , R 1323 = 0 .$
By page 8 in [16], we get for $G 2$ (we correct a misprint in [16]),
$L V g = 0 γ λ 2 + ( α − β ) λ 3 ( − α + β ) λ 2 + γ λ 3 γ λ 2 + ( α − β ) λ 3 − 2 γ λ 1 0 ( − α + β ) λ 2 + γ λ 3 0 − 2 γ λ 1 .$
By (5), (12) and (13), we get that $( G 2 , V , g )$ is a left-invariant Riemann soliton if and only if:
$− γ 2 − α 2 4 + γ λ 1 = − λ , γ ( 2 β − α ) = 0 , ( − α + β ) λ 2 + γ λ 3 = 0 , α 2 4 + γ 2 + γ λ 1 = λ , γ λ 2 + ( α − β ) λ 3 = 0 , − γ 2 − 3 4 α 2 + α β = λ .$
By the first equation and the fourth equation and $γ ≠ 0$ in (14), we get $λ 1 = 0$ and $λ = α 2 4 + γ 2$. By the second equation and the sixth equation in (14), we get $λ = − α 2 4 − γ 2$. Then $γ = 0$ and this is a contradiction. So,
Theorem 2.
$( G 2 , V , g )$ is not a left-invariant Riemann soliton.
By Theorem 2.1 in [17], we have for $G 3$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 3$ satisfies:
$[ e 1 , e 2 ] = − γ e 3 , [ e 1 , e 3 ] = − β e 2 , [ e 2 , e 3 ] = α e 1 .$
By page 146 in [17], we have for $G 3$:
$R 1212 = − ( a 1 a 2 + γ a 3 ) , R 1313 = a 1 a 3 + β a 2 , R 2323 = − ( a 2 a 3 + α a 1 ) , R 1213 = 0 , R 1223 = 0 , R 1323 = 0 ,$
where
$a 1 = 1 2 ( α − β − γ ) , a 2 = 1 2 ( α − β + γ ) , a 3 = 1 2 ( α + β − γ ) .$
By page 9 in [16], we get for $G 3$,
$L V g = 0 ( α − β ) λ 3 ( γ − α ) λ 2 ( α − β ) λ 3 0 ( β − γ ) λ 1 ( γ − α ) λ 2 ( β − γ ) λ 1 0 .$
By (5), (16) and (18), we get that $( G 3 , V , g )$ is a left-invariant Riemann soliton if and only if:
$a 1 a 2 + γ a 3 = λ , ( β − γ ) λ 1 = 0 , ( α − γ ) λ 2 = 0 , ( α − β ) λ 3 = 0 , a 1 a 3 + β a 2 = λ , a 2 a 3 + α a 1 = − λ .$
Theorem 3.
$( G 3 , V , g )$ is a left-invariant Riemann soliton if and only if:
(i)
$β = γ$, $α ≠ γ$, $λ 2 = λ 3 = 0$, $α = 0$, $λ = 0$,
(ii)
$α = β = γ$, $λ = 1 4 α 2$,
(iii)
$β ≠ γ$, $α = β$, $λ 1 = λ 2 = 0$, $γ = 0$, $λ = 0$,
(iv)
$β ≠ γ$, $α = γ$, $λ 1 = λ 3 = 0$, $β = 0$, $λ = 0$.
Proof.
By the first equation and the fifth equation in (19), we get $a 1 ( a 2 − a 3 ) + γ a 3 − β a 2 = 0$. By (17), then we get $( α − β − γ ) ( β − γ ) = 0$. By the fifth equation and the sixth equation in (19), we get $( α + β − γ ) ( α − β ) = 0$ and:
$( β − γ ) λ 1 = 0 , ( α − γ ) λ 2 = 0 , ( α − β ) λ 3 = 0 , ( α − β − γ ) ( β − γ ) = 0 , ( α + β − γ ) ( α − β ) = 0 , λ = a 1 a 2 + γ a 3 .$
Case (1) $β ≠ γ , α ≠ γ , α ≠ β$. Then by the fourth equation and the fifth equation in (20), we get $α = γ$. This is a contradiction and there are no solutions.
Case (2) $β = γ$, $α ≠ γ$. Solving (20), we get the case (i).
Case (3) $α = β = γ$. Solving (20), we get the case (ii).
Case (4) $β ≠ γ$, $α = β$. Solving (20), we get the case (iii).
Case (5) $β ≠ γ$, $α = γ$. Solving (20), we get the case (iv). □
By Theorem 2.1 in [17], we have for $G 4$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 4$ satisfies:
$[ e 1 , e 2 ] = − e 2 + ( 2 η − β ) e 3 , η = 1 or − 1 , [ e 1 , e 3 ] = − β e 2 + e 3 , [ e 2 , e 3 ] = α e 1 .$
By (2.32) in [18], we have for $G 4$:
$R 1212 = ( 2 η − β ) b 3 − b 1 b 2 − 1 , R 1313 = b 1 b 3 + β b 2 + 1 , R 2323 = − ( b 2 b 3 + α b 1 + 1 ) , R 1213 = 2 η − β + b 1 + b 2 , R 1223 = 0 , R 1323 = 0 ,$
where
$b 1 = α 2 + η − β , b 2 = α 2 − η , b 3 = α 2 + η .$
By page 11 in [16], we get for $G 4$,
$L V g = 0 − λ 2 + ( α − β ) λ 3 ( β − α − 2 η ) λ 2 − λ 3 − λ 2 + ( α − β ) λ 3 2 λ 1 2 η λ 1 ( β − α − 2 η ) λ 2 − λ 3 2 η λ 1 2 λ 1 .$
By (5), (22) and (24), we get that $( G 4 , V , g )$ is a left-invariant Riemann soliton if and only if:
$( 2 η − β ) b 3 − b 1 b 2 − 1 − λ 1 = − λ , 2 η − β + b 1 + b 2 − η λ 1 = 0 , ( β − α − 2 η ) λ 2 − λ 3 = 0 , b 1 b 3 + β b 2 + 1 − λ 1 = λ , − λ 2 + ( α − β ) λ 3 = 0 , − ( b 2 b 3 + α b 1 + 1 ) = λ .$
Theorem 4.
$( G 4 , V , g )$ is a left-invariant Riemann soliton if and only if:
(i)
$β ≠ η$, $α = 0$, $λ 1 = 2 − 2 η β$, $λ 2 = λ 3 = 0$, $λ = 0$,
(ii)
$α − β + η = 0$, $λ 2 = − η λ 3$, $λ 1 = 1 − η β$, $λ = α 2 4$.
Proof.
The fourth equation minusing the first equation in (25), we get $b 1 b 3 + β b 2 + 1 − ( 2 η − β ) b 3 + b 1 b 2 + 1 = 2 λ$. By the sixth equation in (25), we get $α ( α − β + η ) = 0$.
Case (1) $α − β + η ≠ 0$. Then $α = 0$, solving (25), we get case (i).
Case (2) $α − β + η = 0$. Solving (25), we get case (ii). □
By Theorem 2.2 in [17], we have for $G 5$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 5$ satisfies:
$[ e 1 , e 2 ] = 0 , [ e 1 , e 3 ] = α e 1 + β e 2 , [ e 2 , e 3 ] = γ e 1 + δ e 2 , α + δ ≠ 0 , α γ + β δ = 0 .$
By (2.36) in [18], we have for $G 5$:
$R 1212 = α δ − ( β + γ ) 2 4 , R 1313 = − α 2 − β ( β + γ ) 2 − β 2 − γ 2 4 , R 2323 = − δ 2 − γ ( β + γ ) 2 + β 2 − γ 2 4 , R 1213 = 0 , R 1223 = 0 , R 1323 = 0 .$
By page 13 in [16], we get for $G 5$,
$L V g = 2 α λ 3 ( β + γ ) λ 3 − α λ 1 − γ λ 2 ( β + γ ) λ 3 2 δ λ 3 − β λ 1 − δ λ 2 − α λ 1 − γ λ 2 − β λ 1 − δ λ 2 0 .$
By (5), (27), and (28), we get that $( G 5 , V , g )$ is a left-invariant Riemann soliton if and only if:
$α δ − ( β + γ ) 2 4 − δ λ 3 − α λ 3 = − λ , β λ 1 + δ λ 2 = 0 , α λ 1 + γ λ 2 = 0 , − α 2 − β ( β + γ ) 2 − β 2 − γ 2 4 + α λ 3 = λ , ( β + γ ) λ 3 = 0 , − δ 2 − γ ( β + γ ) 2 + β 2 − γ 2 4 + δ λ 3 = λ .$
Theorem 5.
$( G 5 , V , g )$ is a left-invariant Riemann soliton if and only if:
(i)
$β + γ = 0$, $β ≠ 0$, $α = δ$, $α ≠ 0$, $λ 1 = λ 2 = λ 3 = 0$, $λ = − α 2$,
(ii)
$β = γ = 0$, $α = δ$, $α ≠ 0$, $λ 1 = λ 2 = λ 3 = 0$, $λ = − α 2$.
Proof.
Case (1) $β + γ ≠ 0$. Then $λ 3 = 0$. By the fourth equation and the sixth equation in (29), we get $α 2 − δ 2 + β 2 − γ 2 = 0$. By the first equation and the fourth equation in (29), we get $α 2 + β 2 + β γ − α δ = 0$.
Case (1-a) $β γ − α δ = 0$. We get $α = β = γ = δ = 0$. This is a contradiction.
Case (1-b) $β γ − α δ ≠ 0$. By the second equation and the third equation in (29), we get $λ 1 = λ 2 = 0$.
Case (1-b-1) $α = 0$. Then $δ ≠ 0$ and $β = 0$, then $δ = γ = 0$ by $α 2 − δ 2 + β 2 − γ 2 = 0$. This is a contradiction.
Case (1-b-2) $α ≠ 0$. Then $γ = − β δ α$. Then $β ≠ 0$ and $α ≠ δ$ by $β + γ ≠ 0$. By $α + δ ≠ 0$, then $α 2 ≠ δ 2$. By $α 2 − δ 2 + β 2 − γ 2 = 0$, we get $1 + β 2 α 2 = 0$. This is a contradiction.
Case (2) $β + γ = 0$. By (29), we have:
$α δ − δ λ 3 − α λ 3 = − λ , β λ 1 + δ λ 2 = 0 , α λ 1 + γ λ 2 = 0 , − α 2 + α λ 3 = λ , − δ 2 + δ λ 3 = λ .$
By $α γ + β δ = 0$, we have $β ( α − δ ) = 0$.
Case (2-a) $β ≠ 0$. Then $α = δ$. Solving (30), we get the case (i).
Case (2-b) $β = 0$. Then $γ = 0$. So $δ λ 2 = 0$, $α λ 1 = 0$.
Case (2-b-1) $α ≠ 0$, $δ ≠ 0$. Then $λ 1 = λ 2 = 0$. Solving (30), we get the case (ii).
Case (2-b-2) $α = 0$, $δ ≠ 0$. Solving (30), we get $δ = 0$. This is a contradiction.
Case (2-b-3) $α ≠ 0$, $δ = 0$. Solving (30), we get $α = 0$. This is a contradiction. □
By Theorem 2.2 in [17], we have for $G 6$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 6$ satisfies:
$[ e 1 , e 2 ] = α e 2 + β e 3 , [ e 1 , e 3 ] = γ e 2 + δ e 3 , [ e 2 , e 3 ] = 0 , α + δ ≠ 0 , α γ − β δ = 0 .$
By (2.40) in [18], we have for $G 6$:
$R 1212 = − α 2 + β ( β − γ ) 2 + β 2 − γ 2 4 , R 1313 = δ 2 + γ ( β − γ ) 2 + β 2 − γ 2 4 , R 2323 = α δ + ( β − γ ) 2 4 , R 1213 = 0 , R 1223 = 0 , R 1323 = 0 .$
By page 14 in [16], we get for $G 6$,
$L V g = 0 α λ 2 + γ λ 3 − β λ 2 − δ λ 3 α λ 2 + γ λ 3 − 2 α λ 1 ( β − γ ) λ 1 − β λ 2 − δ λ 3 ( β − γ ) λ 1 2 δ λ 1 .$
By (5), (32), and (33), we get that $( G 6 , V , g )$ is a left-invariant Riemann soliton if and only if:
$− α 2 + β ( β − γ ) 2 + β 2 − γ 2 4 + α λ 1 = − λ , ( β − γ ) λ 1 = 0 , β λ 2 + δ λ 3 = 0 , δ 2 + γ ( β − γ ) 2 + β 2 − γ 2 4 − δ λ 1 = λ , α λ 2 + γ λ 3 = 0 , α δ + ( β − γ ) 2 4 − δ λ 1 − α λ 1 = λ .$
Theorem 6.
$( G 6 , V , g )$ is a left-invariant Riemann soliton if and only if:
(i)
$β ≠ γ$, $λ 1 = 0$, $α = β = 0$, $λ = γ 2 4$, $λ 3 = 0$, $δ 2 = γ 2$,
(ii)
$β ≠ γ$, $λ 1 = 0$, $α ≠ 0$, $α 2 = β 2$, $δ = β γ α$, $λ = ( β + γ ) 2 4$, $λ 2 = − γ α λ 3$,
(iii)
$β = γ$, $β ≠ 0$, $α = δ$, $α ≠ 0$, $λ 1 = λ 2 = λ 3 = 0$, $λ = α 2$,
(iv)
$λ 3 ≠ 0$, $λ 2 = − δ β λ 3$, $α ≠ 0$, $β ≠ 0$, $β = γ$, $α = δ$, $α 2 = β 2$, $λ 1 = 0$, $λ = α 2$,
(v)
$β = γ = 0$, $α ≠ 0$, $δ ≠ 0$, $λ 1 = λ 2 = λ 3 = 0$, $α = δ$, $λ = α 2$.
Proof.
Case (1) $β − γ ≠ 0$. Then $λ 1 = 0$. So by the first, the fourth, and sixth equations in (34), we get:
$δ 2 − α 2 + β 2 − γ 2 = 0 , α 2 − β 2 + β γ − α δ = 0 .$
Case (1-a) $β γ − α δ = 0$. So $α 2 = β 2$ and $δ 2 = γ 2$ by (35).
Case (1-a-1) $α = 0$. Solving (34), we get the case (i).
Case (1-a-2) $α ≠ 0$. Then $δ = β γ α$. Solving (34), we get the case (ii).
Case (1-b) $β γ − α δ ≠ 0$. So $λ 2 = λ 3 = 0$.
Case (1-b-1) $α = 0$. So $δ ≠ 0$ and $β = 0$. This is a contradiction with $β γ − α δ ≠ 0$.
Case (1-b-2) $α ≠ 0$. We get $γ = β δ α$ and $α 2 = β 2$ by (35). Then $β γ − α δ = 0$. This is a contradiction.
Case (2) $β − γ = 0$. Then $β ( α − δ ) = 0$. By (34), we have:
$− α 2 + α λ 1 = − λ , β λ 2 + δ λ 3 = 0 , δ 2 − δ λ 1 = λ , α λ 2 + γ λ 3 = 0 , α δ − δ λ 1 − α λ 1 = λ .$
Case (2-a) $β ≠ 0$. Then $α = δ$ and $λ 2 = − δ β λ 3 = − γ α λ 3$.
Case (2-a-1) $λ 3 = 0$. Then we get the case (iii).
Case (2-a-2) $λ 3 ≠ 0$. Then we get the case (iv).
Case (2-b) $β = 0$. Then $γ = 0$ and $δ λ 3 = 0$, $α λ 2 = 0$.
Case (2-b-1) $α ≠ 0$, $δ ≠ 0$. Then $λ 2 = λ 3 = 0$. Solving (36), we get the case (v).
Case (2-b-2) $α = 0$, $δ ≠ 0$. Solving (36), we get $δ = 0$. This is a contradiction.
Case (2-b-3) $α ≠ 0$, $δ = 0$. Solving (36), we get $α = 0$. This is a contradiction. □
By Theorem 4.2 in [17], we have for $G 7$, there exists a pseudo-orthonormal basis ${ e 1 , e 2 , e 3 }$ with $e 3$ timelike such that the Lie algebra of $G 7$ satisfies:
$[ e 1 , e 2 ] = − α e 1 − β e 2 − β e 3 , [ e 1 , e 3 ] = α e 1 + β e 2 + β e 3 , [ e 2 , e 3 ] = γ e 1 + δ e 2 + δ e 3 , , α + δ ≠ 0 , α γ = 0 .$
By (2.44) in [18], we have for $G 7$:
$R 1212 = α δ − α 2 − β γ − γ 2 4 , R 1313 = α δ − α 2 − β γ + γ 2 4 , R 2323 = − 3 4 γ 2 , R 1213 = α 2 − α δ + β γ , R 1223 = 0 , R 1323 = 0 .$
By page 16 in [16], we get for $G 7$,
$L V g = − 2 α ( λ 2 − λ 3 ) α λ 1 − β λ 2 + ( β + γ ) λ 3 − α λ 1 + ( β − γ ) λ 2 − β λ 3 α λ 1 − β λ 2 + ( β + γ ) λ 3 2 β λ 1 + 2 δ λ 3 − 2 β λ 1 − δ λ 2 − δ λ 3 − α λ 1 + ( β − γ ) λ 2 − β λ 3 − 2 β λ 1 − δ λ 2 − δ λ 3 2 β λ 1 + 2 δ λ 2 .$
By (5), (38), and (39), we get that $( G 7 , V , g )$ is a left-invariant Riemann soliton if and only if:
$α δ − α 2 − β γ − γ 2 4 − ( β λ 1 + δ λ 3 ) + α ( λ 2 − λ 3 ) = − λ , 2 ( α 2 − α δ + β γ ) + 2 β λ 1 + δ λ 2 + δ λ 3 = 0 , − α λ 1 + ( β − γ ) λ 2 − β λ 3 = 0 , α δ − α 2 − β γ + γ 2 4 − β λ 1 − δ λ 2 − α ( λ 2 − λ 3 ) = λ , α λ 1 − β λ 2 + ( β + γ ) λ 3 = 0 , − 3 4 γ 2 − δ λ 2 + δ λ 3 = λ .$
Theorem 7.
$( G 7 , V , g )$ is a left-invariant Riemann soliton if and only if:
(i)
$α = 0$, $δ ≠ 0$, $β = γ = 0$, $λ 2 = λ 3 = λ = 0$,
(ii)
$α = 0$, $δ ≠ 0$, $γ = 0$, $β ≠ 0$, $λ 2 = λ 3$, $λ = 0$, $λ 1 = − δ β λ 2$,
(iii)
$α ≠ 0$, $γ = 0$, $α = δ$, $λ 1 = λ 2 = λ 3 = λ = 0$.
Proof.
Case (1) $α = 0$. Then $δ ≠ 0$. By (40), we have:
$− β γ − γ 2 4 − ( β λ 1 + δ λ 3 ) = − λ , 2 β γ + 2 β λ 1 + δ λ 2 + δ λ 3 = 0 , ( β − γ ) λ 2 − β λ 3 = 0 , − β γ + γ 2 4 − β λ 1 − δ λ 2 = λ , − β λ 2 + ( β + γ ) λ 3 = 0 , − 3 4 γ 2 − δ λ 2 + δ λ 3 = λ .$
Case (1-a) $γ ≠ 0$. Then $λ 2 = λ 3 = 0$ by the third equation and the fifth equation in (41). By (41), we have:
$− β γ − γ 2 4 − β λ 1 = − λ , β γ + β λ 1 = 0 , − β γ + γ 2 4 − β λ 1 = λ , − 3 4 γ 2 = λ .$
Case (1-a-1) $β = 0$. By (42), we get $γ = 0$. This is a contradiction.
Case (1-a-2) $β ≠ 0$. By (42), we get $λ 1 = − γ$ and $γ = 0$. This is a contradiction.
Case (1-b) $γ = 0$. By (41), we have:
$β λ 1 + δ λ 3 = λ , 2 β λ 1 + δ λ 2 + δ λ 3 = 0 , β ( λ 2 − λ 3 ) = 0 , − β λ 1 − δ λ 2 = λ , − δ λ 2 + δ λ 3 = λ .$
Case (1-b-1) $β = 0$. Solving (43), we get the case (i).
Case (1-b-2) $β ≠ 0$. Solving (43), we get the case (ii).
Case (2) $α ≠ 0$. Then $γ = 0$. By (40), we get:
$α δ − α 2 − ( β λ 1 + δ λ 3 ) + α ( λ 2 − λ 3 ) = − λ , 2 ( α 2 − α δ ) + 2 β λ 1 + δ λ 2 + δ λ 3 = 0 , − α λ 1 + β λ 2 − β λ 3 = 0 , α δ − α 2 − β λ 1 − δ λ 2 − α ( λ 2 − λ 3 ) = λ , − δ λ 2 + δ λ 3 = λ .$
By the third equation in (44), we have $λ 1 = β α ( λ 2 − λ 3 )$. By the first, the second, and the fourth equations in (44), we get $α = δ$ and $2 β λ 1 + δ λ 2 + δ λ 3 = 0$. By the fourth and the fifth equations in (44), we get $β λ 1 + δ λ 2 = 0$ and $λ 2 = λ 3$. Then by the fifth equation in (44), we get $λ = 0$. So $λ 1 = λ 2 = λ 3 = 0$. This is the case (iii). □

## 3. Conclusions

During the last years, geometric evolution equations have been used to study geometric questions like isoperimetric inequalities, the Poincare conjecture, and Thurston’s geometrization conjecture. In particular, the geometric flow enjoys rapid growth. The Riemann flow is an important geometric flow. Riemann solitons are generalized fix points of the Riemann flow. Thus it is interesting to study Riemann solitons. In this note, a classification of Riemann solitons on three dimensional Lorentzian Lie group was given. In particular, $( G 2 , V , g )$ was not a left-invariant Riemann soliton, while $( G i , V , g )$ for $i = 1 , 3 , 4 , 5 , 6 ,$ and 7, were left invariant Riemann solitons if and only if the parameters satisfied particular conditions.
Our classified theorems are proven by some algebraic calculations. In fact, by (5), we needed to compute the geometric objects $R i j k l$ and $L V ( e i , e j )$. Moreover our classified theorems will have some geometric applications.

## Funding

This research was funded by National Natural Science Foundation of China: No.11771070.

Not applicable.

Not applicable.

## Data Availability Statement

Data sharing not applicable.

## Acknowledgments

The author was supported in part by NSFC No. 11771070. The author thanks the referees for their careful reading and helpful comments.

## Conflicts of Interest

The author declares no conflict of interest.

## References

1. Hirica, I.; Udriste, C. Ricci and Riemann Solitons. Balkan J. Geom. Appl. 2016, 21, 35–44. [Google Scholar]
2. Blaga, A.; Latcu, D. Remarks on Riemann and Ricci solitons in (α,β)-contact metric manifolds. arXiv 2021, arXiv:2009.02506. [Google Scholar]
3. Carfora, M. The Wasserstein geometry of nonlinear models and the Hamilton-Perelman Ricci flow. Rev. Math. Phys. 2017, 29, 1750001. [Google Scholar] [CrossRef]
4. Nitta, M. Conformal sigma models with anomalous dimensions and Ricci solitons. Mod. Phys. Lett. A 2005, 20, 577–584. [Google Scholar] [CrossRef] [Green Version]
5. Oliynyk, T.; Suneeta, V.; Woolgar, E. A gradient flow for worldsheet nonlinear sigma models. Nucl. Phys. B 2006, 739, 441–458. [Google Scholar] [CrossRef] [Green Version]
6. Tseytlin, A. Sigma model renormalization group flow, central charge action, and Perelman’s entropy. Phys. Rev. D 2007, 75, 064024. [Google Scholar] [CrossRef] [Green Version]
7. Bakas, I.; Bourliot, F.; Lust, D.; Petropoulos, J. Geometric flows in Horava-Lifshitz gravity. J. High Energy Phys. 2010, 4, 131. [Google Scholar] [CrossRef] [Green Version]
8. Streets, J. Ricci Yang-Mills flow on surfaces. Adv. Math. 2010, 223, 454–475. [Google Scholar] [CrossRef] [Green Version]
9. Vacaru, S. Nonholonomic Ricci flows. II. Evolution equations and dynamics. J. Math. Phys. 2008, 49, 043504. [Google Scholar] [CrossRef] [Green Version]
10. Vacaru, S. Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows. J. Math. Phys. 2009, 50, 073503. [Google Scholar] [CrossRef] [Green Version]
11. Carfora, M. Ricci-flow-conjugated initial data sets for Einstein equations. Adv. Theor. Math. Phys. 2011, 15, 1411–1484. [Google Scholar] [CrossRef] [Green Version]
12. Carstea, S.; Visinescu, M. Special solutions for Ricci flow equation in 2D using the linearization approach. Mod. Phys. Lett. A 2005, 20, 2993–3002. [Google Scholar] [CrossRef] [Green Version]
13. Gheorghiu, T.; Ruchin, V.; Vacaru, O.; Vacaru, S. Geometric flows and Perelman’s thermodynamics for black ellipsoids in R2 and Einstein gravity theories. Ann. Phys. 2016, 369, 1–35. [Google Scholar]
14. Ruchin, V.; Vacaru, O.; Vacaru, S. On relativistic generalization of Perelman’s W-entropy and thermodynamic description of gravitational fields and cosmology. Eur. Phys. J. C 2017, 77, 1–27. [Google Scholar] [CrossRef] [Green Version]
15. Vacaru, S. Finsler and Lagrange geometries in Einstein and string gravity. Int. J. Geom. Methods Mod. Phys. 2008, 5, 473–511. [Google Scholar] [CrossRef] [Green Version]
16. Calvaruso, G. Three-dimensional homogeneous generalized Ricci solitons. Mediterr. J. Math. 2017, 14, 216. [Google Scholar] [CrossRef] [Green Version]
17. Batat, W.; Onda, K. Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. J. Geom. Phys. 2017, 114, 138–152. [Google Scholar] [CrossRef] [Green Version]
18. Calvaruso, G. Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds. Geom. Dedicata 2007, 127, 99–119. [Google Scholar] [CrossRef]
19. Wang, Y. Affine Ricci solitons of three-dimensional Lorentzian Lie groups. arXiv 2020, arXiv:2012.11421. [Google Scholar]
20. Calvaruso, G. Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 2007, 57, 1279–1291. [Google Scholar]
21. Cordero, L.A.; Parker, P.E. Left-invariant Lorentzian metrics on 3-dimensional Lie groups. Rend. Mat. Appl. 1997, 17, 129–155. [Google Scholar]
 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Share and Cite

MDPI and ACS Style

Wang, Y. Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups. Symmetry 2021, 13, 218. https://doi.org/10.3390/sym13020218

AMA Style

Wang Y. Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups. Symmetry. 2021; 13(2):218. https://doi.org/10.3390/sym13020218

Chicago/Turabian Style

Wang, Yong. 2021. "Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups" Symmetry 13, no. 2: 218. https://doi.org/10.3390/sym13020218

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.