In this work we consider the three-dimensional Lie group denoted by H2×R, equipped with left-invariant Riemannian metric. The existence of non-trivial (i.e., not Einstein) Ricci solitons on three-dimensional Lie group H2×R is proved. Moreover, we show that there are not gradient Ricci solitons.
Citation: Lakehal Belarbi. Ricci solitons of the H2×R Lie group[J]. Electronic Research Archive, 2020, 28(1): 157-163. doi: 10.3934/era.2020010
[1] |
Lakehal Belarbi .
Ricci solitons of the |
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In this work we consider the three-dimensional Lie group denoted by H2×R, equipped with left-invariant Riemannian metric. The existence of non-trivial (i.e., not Einstein) Ricci solitons on three-dimensional Lie group H2×R is proved. Moreover, we show that there are not gradient Ricci solitons.
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
LXg+Ric=λg, | (1) |
where
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
In the special case that
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
Let
(x,y,z)⋆(x′,y′,z′)=(x′y+x,yy′,z+z′) |
and the left invariant product metric
g=1y2(dx2+dy2)+dz2. | (2) |
With respect to the metric
E1=y∂∂x,E2=y∂∂y,E3=∂∂z. | (3) |
From this, the Lie brackets are given by
[E1,E2]=−E1,[E2,E3]=0,[E3,E1]=0. |
Throughout the paper, we shall endow the three-dimensional Lie group
We will denote by
R(X,Y)Z=∇X∇YZ−∇Y∇Xz−∇[X,Y]Z, |
and by
Ric(X,Y)=3∑k=1g(Ek,Ek)g(R(Ek,X)Y,Ek), |
where
The Levi-Civita connection
{∇E1E1=E2,∇E1E2=−E1,∇E1E3=0∇E2E1=0,∇E2E2=0,∇E2E3=0∇E3E1=0,∇E3E2=0,∇E3E3=0. | (4) |
The non-vanishing curvature tensor
R(E1,E2)E1=E2,R(E1,E2)E2=−E1. | (5) |
The Ricci curvature components
Ric11=Ric22=−1,Ric12=Ric13=Ric23=Ric33=0. | (6) |
The scalar curvature
τ=trRic=3∑i=1g(Ei,Ei)Ric(Ei,Ei)=−2. | (7) |
In this section we analyze the existence of Ricci solitons on three-dimensional Riemannian Lie group
Let
The Lie derivative of the metric (2) with respect to
{(LXg)(E1,E1)=−2(f2−∂xf1),(LXg)(E1,E2)=f1+∂yf1+∂xf2,(LXg)(E1,E3)=∂zf1+∂xf3,(LXg)(E2,E2)=2∂yf2,(LXg)(E2,E3)=∂zf2+∂yf3,(LXg)(E3,E3)=2∂zf3. | (8) |
Thus, by using (2), (6) and (8) in (1), a standard calculation gives that the three-dimensional Lie group
{−2(f2−∂xf1)−1=λ,f1+∂yf1+∂xf2=0,∂zf1+∂xf3=0,2∂yf2−1=λ,∂zf2+∂yf3=0,2∂zf3=λ. | (9) |
We deriving the third equation in (9) with respect to
∂2zf1=0. | (10) |
The equation (10) yields that
f1=φ(x,y)z+ψ(x,y), | (11) |
where
Deriving the first equation with respect to
∂x∂yf1=12(1+λ). | (12) |
Next, deriving the second equation with respect to
∂yf1+∂2yf1=0. | (13) |
Replacing
{(∂yφ+∂2yφ)z+∂yψ+∂2yψ=0,∂x∂yφz+∂x∂yψ=12(1+λ). | (14) |
By derivation of the equations (14) with respect to
{∂yφ+∂2yφ=0,∂yψ+∂2yψ=0,∂x∂yφ=0,∂x∂yψ=12(1+λ). | (15) |
We deriving the second equation with respect to
λ=−1 | (16) |
Integrating the first and second equations in (15), we find
{φ(x,y)=α1e−y+φ1(x),ψ(x,y)=α2e−y+ψ1(x), | (17) |
where
f1=(α1e−y+φ1(x))z+α2e−y+ψ1(x). | (18) |
Next, we replace
(φ1(x)+φ″1(x))z+ψ1(x)+ψ″1(x)=0. | (19) |
We derive equation (19) with respect to
{φ1(x)+φ″1(x)=0,ψ1(x)+ψ″1(x)=0. | (20) |
By integration of (20) with respect to
{φ1(x)=α3cos(x)+α4sin(x),ψ1(x)=α5cos(x)+α6sin(x), | (21) |
where
f1=(α1e−y+α3cos(x)+α4sin(x))z+α2e−y+α5cos(x)+α6sin(x). | (22) |
From the first equation in (9), gives
f2=(−α3sin(x)+α4cos(x))z−α5sin(x)+α6cos(x). | (23) |
The last equation in (9) gives
f3=−12z+ξ(x,y), | (24) |
where
{∂xξ=−α1e−y−α3cos(x)−α4sin(x),∂yξ=α3sin(x)−α4cos(x). | (25) |
Integrating the first equation in (25) with respect to
ξ(x,y)=−α1xe−y−α3sin(x)+α4cos(x)+α7, |
and replace
α1=α3=α4=0. |
finally for arbitrary reals constants
{f1=α2e−y+α5cos(x)+α6sin(x),f2=−α5sin(x)+α6cos(x),f3=−12z+α7, | (26) |
Thus, it easily follows that the vector field
Theorem 3.1. Let
Now, let
gradh=y2∂xh∂x+y2∂yh∂y+∂zh∂z. | (27) |
From (26) it follows that
{∂xh=α2ye−y+α5ycos(x)+α6ysin(x),∂yh=−α5ysin(x)+α6ycos(x),∂zh=−12z+α7. | (28) |
Hence, with direct integration we prove that
h(x,y,z)=ln(y)[α5sin(x)−α6cos(x)]−14z2+α7z+α8,αi∈R. | (29) |
But the function
Corollary 1. Let
Our study on the Ricci solitons of three-dimensional Lie group denoted by
We would like to thank the referee for valuable suggestions regarding both the contents and exposition of this article.
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