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Ricci solitons of the H2×R Lie group

  • Received: 01 October 2019 Revised: 01 February 2020
  • 53C50, 53B30

  • 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

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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 g on a smooth manifold M such that there exists a smooth vector field X on M satisfying the following equation:

    LXg+Ric=λg, (1)

    where LX denotes the Lie derivative in the direction of X, Ric denotes the Ricci tensor and λ is a real number. A Ricci soliton is said to be a shrinking, steady or expanding, respectively, if λ>0, λ=0 or λ<0. Moreover, we say that a Ricci soliton (M,g) is a gradient Ricci soliton if it admits a vector field X satisfying X=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 M is a Lie group and g is a left-invariant metric, we say that g is a left-invariant Ricci soliton on 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 H2×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 H2×R are characterized via a system of partial differential equations. In particular, we show that three-dimensional Lie group H2×R admit a vector fields in expanding Ricci solitons. Finally, we show that there are not gradient Ricci solitons.

    Let H2 be represented by the upper half-plane model {(x,y)R2|y>0} equipped with the metric gH2=1y2(dx2+dy2). The space H2, with the group structure derived by the composition of proper affine map, is a Lie group and the metric gH2 is left invariant. Therefore the Riemannian product space H2×R is a Lie group with respect to the operation

    (x,y,z)(x,y,z)=(xy+x,yy,z+z)

    and the left invariant product metric

    g=1y2(dx2+dy2)+dz2. (2)

    With respect to the metric g an orthonormal basis of left invariant vector fields on H2×R is

    E1=yx,E2=yy,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 H2×R with left-invariant Riemannian g.

    We will denote by the Levi-Civita connection of (H2×R,g), by R its curvature tensor, taken with the sign convention:

    R(X,Y)Z=XYZYXz[X,Y]Z,

    and by Ric the Ricci tensor of (H2×R,g), which is defined by

    Ric(X,Y)=3k=1g(Ek,Ek)g(R(Ek,X)Y,Ek),

    where {Ek}k=1,..,3 is an orthonormal basis.

    The Levi-Civita connection of the H2×R Lie group with respect to this frame (3) is

    {E1E1=E2,E1E2=E1,E1E3=0E2E1=0,E2E2=0,E2E3=0E3E1=0,E3E2=0,E3E3=0. (4)

    The non-vanishing curvature tensor R components are computed as

    R(E1,E2)E1=E2,R(E1,E2)E2=E1. (5)

    The Ricci curvature components {Ricij} are computed as

    Ric11=Ric22=1,Ric12=Ric13=Ric23=Ric33=0. (6)

    The scalar curvature τ of the H2×R Lie group is constant and we have

    τ=trRic=3i=1g(Ei,Ei)Ric(Ei,Ei)=2. (7)

    In this section we analyze the existence of Ricci solitons on three-dimensional Riemannian Lie group (H2×R,g) equipped with the left-invariant Riemannian metric (2).

    Let X=f1E1+f2E2+f3E3 be an arbitrary vector field on (H2×R,g), where f1,..,f3 are smooth functions of the variables x,y,z. We will denote the coordinate basis {x,y,z} by {x,y,z}.

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

    {(LXg)(E1,E1)=2(f2xf1),(LXg)(E1,E2)=f1+yf1+xf2,(LXg)(E1,E3)=zf1+xf3,(LXg)(E2,E2)=2yf2,(LXg)(E2,E3)=zf2+yf3,(LXg)(E3,E3)=2zf3. (8)

    Thus, by using (2), (6) and (8) in (1), a standard calculation gives that the three-dimensional Lie group (H2×R,g) is a Ricci soliton if and only if the following system holds,

    {2(f2xf1)1=λ,f1+yf1+xf2=0,zf1+xf3=0,2yf21=λ,zf2+yf3=0,2zf3=λ. (9)

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

    2zf1=0. (10)

    The equation (10) yields that

    f1=φ(x,y)z+ψ(x,y), (11)

    where φ and ψ are smooth functions.

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

    xyf1=12(1+λ). (12)

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

    yf1+2yf1=0. (13)

    Replacing f1 in equations (12) and (13), we get

    {(yφ+2yφ)z+yψ+2yψ=0,xyφz+xyψ=12(1+λ). (14)

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

    {yφ+2yφ=0,yψ+2yψ=0,xyφ=0,xyψ=12(1+λ). (15)

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

    λ=1 (16)

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

    {φ(x,y)=α1ey+φ1(x),ψ(x,y)=α2ey+ψ1(x), (17)

    where αiR, and φ1,ψ1 are smooth functions with only one variable x. Thus

    f1=(α1ey+φ1(x))z+α2ey+ψ1(x). (18)

    Next, we replace f1 in the second equation in (9), we get

    (φ1(x)+φ1(x))z+ψ1(x)+ψ1(x)=0. (19)

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

    {φ1(x)+φ1(x)=0,ψ1(x)+ψ1(x)=0. (20)

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

    {φ1(x)=α3cos(x)+α4sin(x),ψ1(x)=α5cos(x)+α6sin(x), (21)

    where αiR. Thus

    f1=(α1ey+α3cos(x)+α4sin(x))z+α2ey+α5cos(x)+α6sin(x). (22)

    From the first equation in (9), gives

    f2=(α3sin(x)+α4cos(x))zα5sin(x)+α6cos(x). (23)

    The last equation in (9) gives

    f3=12z+ξ(x,y), (24)

    where ξ is a smooth function depending to x and y. We replacing f1,f2, and f3 in the third and fifth equations in (9), we get

    {xξ=α1eyα3cos(x)α4sin(x),yξ=α3sin(x)α4cos(x). (25)

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

    ξ(x,y)=α1xeyα3sin(x)+α4cos(x)+α7,

    and replace ξ in the second equation in (25), we find that

    α1=α3=α4=0.

    finally for arbitrary reals constants αi we have

    {f1=α2ey+α5cos(x)+α6sin(x),f2=α5sin(x)+α6cos(x),f3=12z+α7, (26)

    Thus, it easily follows that the vector field X= f1E1+f2E2+f3E3 where f1,..,f3 are given by (26) satisfies (9). Note that λ=1. Summarizing, we proved that the three-dimensional Lie group H2×R admits appropriate vector fields for which (1) holds, obtaining the following result.

    Theorem 3.1. Let (H2×R,g) be the three-dimensional Lie group equipped with the left-invariant Riemannian metric g given by (2). Then, (H2×R,g) is an Expanding Ricci soliton.

    Now, let X=grad h be an arbitrary gradient vector field on (H2×R,g) with potential function h, X is then given by

    gradh=y2xhx+y2yhy+zhz. (27)

    From (26) it follows that (H2×R,g) is gradient soliton if and only if the potential function h satisfy the following systems

    {xh=α2yey+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,αiR. (29)

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

    Corollary 1. Let (H2×R,g) be the three-dimensional Lie group equipped with the left-invariant Riemannian metric g given by (2). Then, (H2×R,g) is not gradient Ricci soliton.

    Our study on the Ricci solitons of 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. More precisely, we proved that there are not gradient Ricci solitons.

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



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