Classification of harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups

Abstract

Purpose

The purpose of this study is to classify harmonic homomorphisms ϕ : (G, g) → (H, h), where G, H are connected and simply connected three-dimensional unimodular Lie groups and g, h are left-invariant Riemannian metrics.

Design/methodology/approach

This study aims the classification up to conjugation by automorphism of Lie groups of harmonic homomorphism, between twodifferent non-abelian connected and simply connected three-dimensional unimodular Lie groups (G, g) and (H, h), where g and h are two left-invariant Riemannian metrics on G and H, respectively.

Findings

This study managed to classify some homomorphisms between two different non-abelian connected and simply connected three-dimensional uni-modular Lie groups.

Originality/value

The theory of harmonic maps into Lie groups has been extensively studied related homomorphism in compact Lie groups by many mathematicians, harmonic maps into Lie group and harmonics inner automorphisms of compact connected semi-simple Lie groups and intensively study harmonic and biharmonic homomorphisms between Riemannian Lie groups equipped with a left-invariant Riemannian metric.

Citation

Abdelkader, Z., Nada, O. and Zegga, K. (2024), "Classification of harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups", Arab Journal of Mathematical Sciences, Vol. 30 No. 1, pp. 95-111. https://doi.org/10.1108/AJMS-01-2022-0010

1. Introduction

The theory of harmonic maps is old and rich and has gained a growing interest in the past decade (see Ref. [1] and others). The theory of harmonic maps into Lie groups has been extensively studied related homomorphism in compact Lie groups by many mathematicians (see for examples [2]), in particular, harmonic maps into Lie groups [3] and harmonic inner automorphisms of compact connected semi-simple Lie groups in Ref. [4] and intensively study harmonic and biharmonic homomorphisms between Riemannian Lie groups equipped with a left-invariant Riemannian metric in Ref. [5].

The investigations described here are motivated by the paper [6], the author studied the classification, up to conjugation by an automorphism of Lie groups, of harmonic and biharmonic maps f : (G, g₁) → (G, g₂), where G is non-abelian connected and simply connected three-dimensional unimodular Lie group, f is a homomorphism of Lie group and g₁, g₂ are two left-invariant Riemannian metrics. The Lie group is unimodular if every left Haar measure is a right Haar measure and vice versa. It is known that G is unimodular if and only if | det Ad_x| = 1 for all x ∈ G if and only if the tracead(X) = 0 for all X in its Lie algebra g if and only if g is unimodular.

There are five non-abelian connected and simply connected three-dimensional unimodular Lie groups, the nilpotent Lie group (or the Heisenberg group), the special unitary group SU(2), the universal covering group PSL̃(2,R) of the special linear group, the solvable Lie groups Sol and the universal covering group Ẽ0(2) of the connected component of the Euclidean group, for more detail, see Ref. [7].

In this paper, we aim the classification up to conjugation by an automorphism of Lie groups of harmonic homomorphism, between two different non-abelian connected, and simply connected three-dimensional unimodular Lie groups ϕ : (G, g) → (H, h), where g and h are two left-invariant Riemannian metrics on G and H, respectively.

2. Preliminaries

Let φ : (M, g) → (N, h) be a smooth map between two Riemannian manifolds with m = dim M and n = dim N. We denote by ∇^M and ∇^N the Levi-Civita connexions associated, respectively, to g and h and by T^φN the vector bundle over M pull-back of TN by φ. It is a Euclidean vector bundle and the tangent map of φ is a bundle homomorphism dφ : TM → T^φN. Moreover, T^φN carries a connexion ∇^φ pull-back of ∇^N by φ and there is a connexion on the vector bundle End(TM, T^φN) given by

(∇XA)(Y)=∇XϕA(Y)−A(∇XMY),X,Y∈Γ(TM),A∈Γ(End(TM,TφN)).

The map φ is called harmonic if it is a critical point of the energy

E(φ)=12∫M|dφ2|vg.

The corresponding Euler-Lagrange equation for the energy is given by the vanishing of the tension field

τ(φ)=tr∇dφ=∑i=1m(∇eidφ)ei,

where (ei)i=1m is a local frame of orthonormal vector fields.Let (G, g) be a Riemannian Lie group, i.e., a Lie group endowed with a left-invariant Riemannian metric. If g=TeG is its Lie algebra and <,>g=g(e), then there exists a unique bilinear map A:g×g→g called the Levi-Civita product associated with (g,<,>g) given by the formula:

2<Auv,w>g=<[u,v]g,w>g+<[w,u]g,v>g+<[w,v]g,u>g.

A is entirely determined by the following properties

1. for any u,v∈g,Auv−Avu=[u,v]g,

2. for any u,v,w∈g,<Auv,w>g+<v,Auw>g=0.

If we denote by u^ℓ the left-invariant vector field on G associated with u∈g then the Levi-Civita connection associated with (G, g) satisfies ∇uℓvℓ=(Auv)ℓ, the couple (g,<,>g) defines a vector denoted Ug by

<Ug,v>g=tr(adv),foranyv∈g.

One can deduce easily that, for any orthonormal basis (ei)i=1m of g,

Ug=∑i=1mAeiei.

Note that g is unimodular if and only if Ug=0.

Let φ : (G, g) → (H, h) be a Lie group homomorphism between two Riemannian Lie groups. The differential ξ:g→h of φ at e is a Lie algebra homomorphism. There is a left action of G on Γ(T^φH) given by

(a.X)(b)=Tφ(ab)Lφ(a−1)X(ab),a,b∈G,X∈Γ(TφH).

A section X of T^φH is called left-invariant if, for any a ∈ G, a.X = X. For any left-invariant section X of T^φH, we have for any a ∈ G, X(a) = (X(e))^ℓ(φ(a)). Thus the space of left-invariant sections is isomorphic to the Lie algebra h. Since φ is a homomorphism of Lie groups, g and h are leftinvariant, one can see easily that τ(φ) is left invariant and hence φ is harmonic if and only if τ(φ)(e) = 0. Now, one can see easily that

τ(ξ)≔τ(φ)(e)=Uξ−ξ(Ug),

where

Uξ=∑i=1mBξ(ei)ξ(ei),

where B is the Levi-Civita product associated with (h,<,>h) and (ei)i=1m is an orthonormal basis of g. So we get the following proposition.

Proposition 2.1.

Let ϕ : G → H be a homomorphism between two Riemannian Lie groups. Then ϕ is harmonic if only if τ(ξ) = 0, where ξ:g→h is the differential of ϕ at e. The classification of harmonic homomorphisms will be done up to conjugation.

Two homomorphisms between Euclidean Lie algebras:

ξ1:(g,<,>g)→(h,<,>h)andξ2:(g,<,>g)→(h,<,>h)

are conjugate if there exists two isometric automorphisms φ1:(g,<,>g)→(g,<,>g) and φ2:(h,<,>h)→(h,<,>h) such that

(1.1)φ2°ξ1=ξ2°φ1.

Proposition 2.2.

Let ξ:(g,<,>g)→(h,<,>h) be a homomorphism between unimodular Euclidean Lie algebras, the following formula was established in [5]

(1.2). <τ(ξ),X>h=trg(ξ*°adX°ξ)∀X∈h

where ξ*:h→g is given by

(1.3). <ξ*U,V>g=<U,ξV>h,forV∈gandU∈h.

3. Riemanian three-dimensional unimodular Lie groups G

Definition 3.1.

The Heisenberg group Nil

The nilpotent Lie group Nil known as Heisenberg group, whose Lie algebra will be denoted by n. We have

Nil=1ac01b001,witha,b,c∈R

and

n=0xz00y000,withx,y,z∈R.

The Lie algebra n has a basis {X, Y, Z}, where X=010000000, Y=000001000 and Z=001000000, where the non-vanishing Lie bracket is [X, Y] = Z.

Proposition 3.1.

[7]

Any left-invariant metric on Nil is equivalent up to automorphism to a metric whose associated matrix is of the form

(3.1)<,>n=ρ000ρ0001,whereρ>0.

Definition 3.2.

The solvable Lie group Sol

The solvable Lie group Sol whose Lie algebra will be denoted by sol. We have sol=R2⋊ιR where ι(t)=t00−t. We can choose a basis {X, Y, Z} of sol, where X=10,0, Y=01,0 and Z=00,1.

and the non-vanishing Lie brackets are [Z, X] = X and [Y, Z] = Y. The Lie group of the solvable Lie algebra sol=R2⋊ιR is the solvable Lie group Sol, which is the semi-direct product R2⋊ΘR, where t∈R acts on R2 by Θ(t)=et00e−t.

Proposition 3.2.

[7]

Any left-invariant metric on Sol=R2⋊ΘR is equivalent up to automorphism to a metric whose associated matrix is of the form

(3.2)<,>sol=10001000ν,whereν>0,

Or

(3.3)<,>sol=1101μ000ν,whereν>0andμ>1.

Definition 3.3.

The solvable Lie group Ẽ0(2)

The solvable Lie group Ẽ0 whose Lie algebra will be denoted by e0(2), where e0(2)=R2⋊so(2). We can choose a basis {X, Y, Z} of e0(2) where X=10,0000, Y=01,0000, Z=00,0−110 and the non-vanishing Lie brackets are [Z, X] = Y, [Y, Z] = X.

The Lie algebra e0(2)=R2⋊so(2) is Lie algebra of the Lie group E0(2)=R2⋊SO(2).

The group E₀(2) is not simply connected. The unique simply connected Lie group corresponding to the Lie algebra e0=R2⋊so(2) is universal covering group Ẽ0(2) of E₀(2).

The group Ẽ0(2) is the semi-direct product C⋊R, where (z, t).(z′, t′) = (z + z′e^2iπt, t + t′) has a faithful matrix representation in GL(3,C) by

(z,t)↦e2iπtz001000et,

where z∈C and t∈R.

Proposition 3.3.

[7]

Any left-invariant metric on Ẽ0(2) is equivalent up to automorphism to a metric whose associated matrix is of the form

(3.4)1000ϱ000σ,whereσ>0and0<ϱ≤1.

4. Harmonic homomorphisms between Sol and Nil

The following result gives a complete classification of harmonic homomorphisms between sol equipped with the left-invariant metric defined in (3.2) or (3.3) and n equipped with the left-invariant metric defined in (3.1).

Theorem 4.1.

A homomorphism from sol to n is conjugate to ξ:sol→n, where

ξ=00a00b00cwitha,b,c∈R.

Proof.

The basis of sol is {X, Y, Z} where [Z, X] = X, and [Y, Z] = Y.

The basis of n is {E, F, H} with [E, F] = H. we put

ξ:X↦a1E+b1F+c1HY↦a2E+b2F+c2HZ↦a3E+b3F+c3H.

Thus, we obtain

[ξX,ξY]=ξ[X,Y]=0[ξX,ξZ]=ξ[X,Z]=−ξX[ξY,ξZ]=ξ[Y,Z]=ξY⇔a1=b1=c1=a2=b2=c2=0.

Theorem 4.2.

Let ξ:sol→n a homomorphism, where

(4.1)ξ=00a00b00c,

the Lie algebra sol equipped with the left-invariant metric defined in (3.2) or (3.3) and n equipped with the left-invariant metric defined in (3.1). Then

(4.2)τ(ξ)=bcνE−acνF.

Proof.

We have

adE=000000010,adF=000000−100andadH=000000000.

Using formula (1.3) where U∈n and V∈sol, we obtain

ξ*=000000ρaρbc.

Using formula (1.2), a simple calculation gives us

<τ(ξ),E>n=tr(ξ*°adE°ξ)=bcν,

<τ(ξ),F>n=tr(ξ*°adF°ξ)=−acν

and

<τ(ξ),H>n=tr(ξ*°adH°ξ)=0

Corollary 4.1.

ξ:(sol,<,>sol),→(n<,>sol) is harmonic if and only if (a = b = 0 or c = 0).

Theorem 4.3.

A homomorphism from n to sol is conjugate to ξi=1,2:n→sol, where

(4.3)ξ1=00a00b000witha,b∈R,

Or

(4.4)ξ2=a1a20b1b20000withai,bi∈Rfori=1,2.

Proof.

The basis of sol is {X, Y, Z}, where [Z, X] = X and [Y, Z] = Y, the basis of n is {E, F, H} with [E, F] = H, then we can suppose

E↦a1X+b1Y+c1Z,

F↦a2X+b2Y+c2Z

and

H↦a3X+b3Y+c3Z.

Thus we obtain

ξ[E,F]=[ξE,ξF]=ξHξ[E,H]=[ξE,ξH]=0ξ[F,H]=[ξE,ξH]=0⇔(c3=a1=b1=c1=a2=b2=c2=0ora3=b3=c3=c1=c2=0).

Theorem 4.4.

Let ξ1,ξ2:n→sol be homomorphisms, where ξ₁ and ξ₂ are defined in formulas (4.3) and (4.4), the Lie algebra sol is equipped with the left-invariant metric defined in formula (3.2). Then

(4.5)τ(ξ1)=(a2−b2)Z

and

(4.6)τ(ξ2)=(a12−b12)+(a22−b22)ρZ.

Proof.

We have

adX=00−1000000,adY=000001000andadZ=1000−10000.

For the homomorphism ξ₁, using formula (1.3), where V∈n and U∈sol, we obtain

ξ1*=000000ab0.

Using formula (1.2), a simple calculation gives us

<τ(ξ1),X>sol=tr(ξ1*°adX°ξ1)=0,

<τ(ξ1),Y>sol=tr(ξ1*°adY°ξ1)=0

and

<τ(ξ1),Z>sol=tr(ξ1*°adZ°ξ1)=a2−b2.

For the homomorphism ξ₂, we have

ξ2*=a1b10a2b20000.

By using formula (1.2), we obtain

<τ(ξ2),X>sol=tr(ξ2*°adX°ξ2)=0,

<τ(ξ2),Y>sol=tr(ξ2*°adY°ξ2)=0

and

<τ(ξ2),Z>sol=tr(ξ2*°adZ°ξ2)=(a12−b12)+(a22−b22)ρ.

Corollary 4.2.

ξ1:(n,<,>n)→(sol,<,>sol) is harmonic if and only if a = ±b.

ξ2:(n,<,>n)→(sol,<,>sol) is harmonic if and only if a12+a22=b22+b12.

Theorem 4.5.

Let ξ1,ξ2:n→sol be homomorphisms, where ξ₁ and ξ₂ are defined in formulas (4.3), (4.4) and the Lie algebra sol is equipped with the left-invariant metric defined in formula (3.3). Then

(4.7)τ(ξ1)=(a2−μb2)Z,

(4.8)τ(ξ2)=(a12−μb12)+(a22−μb22)ρZ.

Proof.

By using formula (1.3), where V∈n and U∈sol, we obtain:

For ξ₁

ξ1*=000000a+ba+μb0.

Using formula (1.2), we get

<τ(ξ1),X>sol=tr(ξ1*°adX°ξ1)=0,

<τ(ξ1),Y>sol=tr(ξ1*°adY°ξ1)=0

and

<τ(ξ1),Z>sol=tr(ξ1*°adZ°ξ1)=a2−μb2.

For ξ₂, we have

ξ2*=a1+b1a1+μb10a2+b2a2+μb20000,

furthermore

<τ(ξ2),X>sol=tr(ξ2*°adX°ξ2)=0,

<τ(ξ2),Y>sol=tr(ξ2*°adY°ξ2)=0

and

<τ(ξ2),Z>sol=tr(ξ2*°adZ°ξ2)=(a12−μb12)+(a22−μb22)ρ.

Corollary 4.3.

ξ1:(n<,>n)→(sol,<,>sol) is harmonic if and only if a=±μb.

ξ2:(n<,>n)→(sol,<,>sol) is harmonic if and only if a12+a22=μ(b22+b12).

5. Harmonic homomorphisms between Sol and Ẽ0(2)

The following result gives a complete classification of harmonic homomorphisms between sol equipped with the left-invariant metric defined in (3.2), (3.3) and e0(2) equipped with the left-invariant metric defined in (3.4).

Theorem 5.1.

Any homomorphism from sol to e0(2) is conjugate to ξ:sol→e0(2), where

ξ=00a00b00c,suchthata,b,c∈R.

Proof.

The basis of sol is {X, Y, Z} where [Z, X] = X, [Y, Z] = Y and the basis of e0(2) is {A, B, C} with [A, B] = 0, [C, A] = B and [B, C] = A, we suppose

ξ(X)=a1A+b1B+c1C,

ξ(Y)=a2A+b2B+c2C

and

ξ(Z)=a3A+b3B+c3C.

Thus we obtain

[ξX,ξY]=ξ[X,Y]=0[ξX,ξZ]=ξ[X,Z]=−ξX[ξY,ξZ]=ξ[Y,Z]=ξY⇔(a1=b1=c1=a2=b2=c2=0).

Theorem 5.2.

Let ξ:sol→e0(2) be a homomorphism, where

(5.1)ξ=00a00b00c,

and sol equipped with the left-invariant metric defined in (3.2) or in (3.3), then

(5.2)τ(ξ)=1ν(−ϱbcA+acB+(ϱ−1)abC).

Proof.

We have

adA=00000−1000,adB=001000000,adC=0−10100000.

By using formula (1.3), where U∈e0(2) and V∈sol, we obtain

ξ*=000000aϱbσc.

Use formula (1.2), we get

<τ(ξ),A>n=tr(ξ*°adA°ξ)=−ϱbcν,

<τ(ξ),B>n=tr(ξ*°adB°ξ)=acν

and

<τ(ξ),C>n=tr(ξ*°adC°ξ)=(ϱ−1)abν.

Corollary 5.1.

ξ:(sol,<,>sol),→(e0(2)<,>e0(2)) is harmonic if and only if (ϱ = 1 and c = 0) or (a = b = 0).

Theorem 5.3.

A homomorphism from e0(2) to sol is conjugate to ξ:e0(2)→sol, where

ξ=00a00b00c,suchthata,b,c∈R.

Proof.

ξ:e0(2)→sol, we have

A↦a1X+b1Y+c1Z,

B↦a2X+b2Y+c2Z

and

C↦a3X+b3Y+c3Z.

Thus we obtain

[ξA,ξB]=ξ[A,B]=0[ξA,ξC]=ξ[A,C]=−ξB[ξB,ξC]=ξ[B,C]=ξA⇔(a1=b1=c1=a2=b2=c2=0).

Theorem 5.4.

Let ξ:e0(2)→sol be a homomorphism, where

(5.3)ξ=00a00b00c,

and sol equipped with the left-invariant metric defined in (3.2). Then

(5.4)τ(ξ)=1σ−acX+bcY+(a2−b2)Z.

Proof.

By using formula (1.3) where V∈e0(2) and U∈sol, we obtain

ξ*=000000abνc.

By direct calculation and we use formula (1.2), we obtain

<τ(ξ),X>n=tr(ξ*°adX°ξ)=−acσ,

<τ(ξ),Y>n=tr(ξ*°adY°ξ)=bcσ

and

<τ(ξ),Z>n=tr(ξ*°adZ°ξ)=(a2−b2)σ.

Corollary 5.2.

ξ:(e0(2)<,>e0(2))→(sol,<,>sol) is harmonic if and only if (c = 0 and a = ±b or a = b = 0).

Theorem 5.5.

Let ξ:e0(2)→sol be a homomorphism, where

(5.5)ξ=00a00b00c.

Where sol equipped with left-invariant metric define in (3.3).

Then

(5.6)τ(ξ)=1σ(−(a+b)cX+μbcY+(a2−μb2+ab)Z).

Proof.

by a similar calculation, we get ξ*=000000a+bμbνc.Using formula (1.2), a direct calculation gives us

<τ(ξ),X>n=tr(ξ*°adX°ξ)=−(a+b)cσ,

<τ(ξ),Y>n=tr(ξ*°adY°ξ)=μbcσ

and

<τ(ξ),Z>n=tr(ξ*°adZ°ξ)=a2−μb2+abσ.

Corollary 5.3.

ξ:(e0(2)<,>e0(2))→(sol,<,>sol) is harmonic if and only if (a = b = 0) or (b = c = 0).

6. Harmonic homomorphisms between Nil and Ẽ0(2)

The following result gives a complete classification of harmonic homomorphisms between n equipped with the left-invariant metric defined in (3.1) and e0(2) equipped with the left-invariant metric defined in (3.4).

Theorem 6.1.

A homomorphism from e0(2) to n is conjugate to ξ:e0(2)→n, where

ξ=00a00b00c,suchthata,b,c∈R.

Proof.

The basis of e0(2) is {A, B, C} with [A, B] = 0, [C, A] = B, [B, C] = A and the basis of n is {E, F, H} with [E, F] = H. Suppose that

A↦a1E+b1F+c1H,

B↦a2E+b2F+c2H,

and

C↦a3E+b3F+c3H.

Thus, we obtain

[ξA,ξB]=ξ[A,B]=0[ξA,ξC]=ξ[A,C]=−ξB[ξB,ξC]=ξ[B,C]=ξA⇔(a1=b1=c1=a2=b2=c2=0).

Theorem 6.2.

Let ξ:e0(2)→n a homomorphism, where

(6.1)ξ=00a00b00c.

Then

(6.2)τ(ξ)=bcνE−acνF.

Proof.

We have adE=000000010,adF=000000−100, and adH=000000000.

using formula (1.3), where U∈n and V∈e0(2), we get

ξ*=000000ρaρbc.

Using formula (1.2), a simple calculation gives us

<τ(ξ),E>n=tr(ξ*°adE°ξ)=bcσ

and

<τ(ξ),F>n=tr(ξ*°adF°ξ)=−acσ.

Corollary 6.1.

ξ:(e0(2),<,>e0(2))→(n<,>n) is harmonic if and only if (a = b = 0 or c = 0).

Theorem 6.3.

A homomorphism from n to e0(2) is conjugate to ξi:n→e0(2), with i = 1, 2, 3 where

ξ1=ab0cd0000,wherea,b,c,d∈R,

ξ2=0a00b00c0wherea,b,c∈R

and

ξ3=ad0bcda0cbda0withb,c,d∈Randa∈R*.

Proof.

The basis of e0(2) is {A, B, C} such that [A, B] = 0, [C, A] = B, [B, C] = A and the basis of n is {E, F, H} with [E, F] = H. We put

E↦a1A+b1B+c1C,

F↦a2A+b2B+c2C

and

H↦a3A+b3B+c3C.

Thus, we obtain

[ξE,ξF]=ξ[E,F]=ξH[ξE,ξH]=0[ξF,ξH]=ξ[F,H]=0⇔a3=b3=c3=0c1=c2=0ora3=b3=c3=0a1=b1=c1=0ora3=b3=c3=0c1×a2=b2×a1b1×a2=c2×a1

Theorem 6.4.

Let ξi:n→e0(2) be homomorphisms, where (ξi)i=1,2,3 are defined in (Theorem 5.3.), then

τ(ξ1)=(ϱ−1)ac+bdρC.

τ(ξ2)=1ρ−ϱbcA+acB+ab(ϱ−1)C.

τ(ξ3)=−ϱbcρ(1+d2a2)A+1ρ(ac+b2da)B+1ρcda(ϱd−1)+ab(ϱ−1)C.

Proof.

We have adA=00000−1000,adB=001000000, and adC=0−10100000.

By using formula (1.3), where U∈e0(2) and V∈n, we obtain

ξ1*=aϱc0bϱd0000.

We use formula (1.2), we obtain

<τ(ξ1),A>e0(2)=tr(ξ*°adA°ξ1)=0,

<τ(ξ1),B>e0(2)=tr(ξ*°adB°ξ)=0,

and

<τ(ξ1),C>e0(2)=tr(ξ*°adC°ξ)=(ϱ−1)ad+bdρ.

For ξ = ξ₂, we have ξ2*=000aϱbσc000.

We use formula (1.2), we obtain

<τ(ξ2),A>e0(2)=tr(ξ2*°adA°ξ2)=−1ρϱbc,

<τ(ξ2),B>e0(2)=tr(ξ2*°adB°ξ2)=1ρac

and

<τ(ξ2),C>e0(2)=tr(ξ2*°adC°ξ2)=1ρab(ϱ−1).

For ξ = ξ₃, we have

(6.3)ξ3*=abϱcσdϱcdaσbda000.

We use formula (1.2), we obtain

<τ(ξ3),A>e0(2)=tr(ξ3*°adA°ξ3)=−ϱbcρ(1+d2a2),

<τ(ξ3),B>e0(2)=tr(ξ3*°adB°ξ3)=1ρ(ac+bd2a)

and

<τ(ξ3),C>e0(2)=tr(ξ3*°adC°ξ3)=ϱ−1ρcd2a+ab.

Corollary 6.2.

ξ1:(n<,>n)→(e0(2),<,>e0(2)), is harmonic if and only if (ϱ = 1 or ac + bd = 0).

ξ2:(n<,>n)→(e0(2),<,>e0(2)), is harmonic if and only if (b = c = 0 or ϱ = 1, and c = 0).

ξ3:(n<,>n)→(e0(2),<,>e0(2)), is harmonic if and only if (b = c = 0 or c = d = 0 and ϱ = 1).