Harmonic Morphisms from Lie Groups and …...2018/05/22 · Symmetric Spaces G=Kof Non-Compact Type...

Harmonic Morphisms from Lie Groups and Symmetric Spaces

- Some Existence Theory -

Sigmundur Gudmundsson

Department of MathematicsFaculty of ScienceLund University

[email protected]

Copenhagen - 22 May 2018

Harmonic MorphismsThe Conjecture

Constructions by EigenfamiliesConstructions by Orthogonal Harmonic Families

Low-Dimensional ClassificationsReferences

Outline1 Harmonic Morphisms

The Origins - Jacobi 1848Riemannian Geometry - Fuglede 1978, Ishihara 1979Geometric Motivation - Baird-Eells 1981Existence ?

2 The ConjectureThe ConjectureRelevant History

3 Constructions by EigenfamiliesDefinitionUseful MachineryThe Classical Semisimple Lie Groups

4 Constructions by Orthogonal Harmonic FamiliesAnother Useful MachineSymmetric Spaces G/K of Non-Compact TypeNilpotent and Solvable Lie GroupsSymmetric Spaces U/K of Compact TypeExamplesHomogeneous Spaces of Positive Curvature

5 Low-Dimensional ClassificationsClassifications

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric Spaces

Definition 1.1 (Harmonic Morphisms (Jacobi 1848))

A map φ = u+ iv : U ⊂ R3 → C is said to be a harmonic morphism ifthe composition f φ with any holomorphic function f : W ⊂ C→ C isharmonic.

Theorem 1.2 (Jacobi 1848)

A map φ = u+ iv : U ⊂ R3 → C is a harmonic morphism if and only if it isharmonic and horizontally (weakly) conformal i.e.

∆u = ∆v = 0, 〈∇u,∇v〉 = 0 and |∇u|2 = |∇v|2.

Proof.

∆(f φ) =[∂f∂z

]·∆φ+

[∂2f

]· 〈∇φ,∇φ〉C = 0

∆u = ∆v = 0, 〈∇u,∇v〉 = 0 and |∇u|2 = |∇v|2.

Proof.

∆(f φ) =[∂f∂z

]·∆φ+

[∂2f

]· 〈∇φ,∇φ〉C = 0

∆u = ∆v = 0, 〈∇u,∇v〉 = 0 and |∇u|2 = |∇v|2.

Proof.

∆(f φ) =[∂f∂z

]·∆φ+

[∂2f

]· 〈∇φ,∇φ〉C = 0

Let f, g : W ⊂ C→ C be holomorphic functions, then every local solutionz : U ⊂ R3 → C to the equation

〈f(z(x))[1− g2(z(x)), i(1 + g2(z(x))), 2g(z(x))

], x〉C = 1

is a harmonic morphism.

Theorem 1.4 (Baird-Wood 1988)

Every harmonic morphism z : U → C defined locally on the Euclidean R3 isobtained this way.

Let f, g : W ⊂ C→ C be holomorphic functions, then every local solutionz : U ⊂ R3 → C to the equation

〈f(z(x))[1− g2(z(x)), i(1 + g2(z(x))), 2g(z(x))

], x〉C = 1

is a harmonic morphism.

Theorem 1.4 (Baird-Wood 1988)

Every harmonic morphism z : U → C defined locally on the Euclidean R3 isobtained this way.

Example 1.5 (The Outer Disc Example)

Let r ∈ R+ and choose g(z) = z, f(z) = −1/2irz then we yield

(x1 − ix2)z2 − 2(x3 + ir)z − (x1 + ix2) = 0

with the two solutions

z±r =−(x3 + ir)±

√x21 + x22 + x23 − r2 + 2irx3

x1 − ix2.

Definition 1.6 (Harmonic Morphisms (Fuglede 1978, Ishihara 1979))

A map φ : (Mm, g)→ (Nn, h) between Riemannian manifolds is called aharmonic morphism if, for any harmonic function f : U → R defined onan open subset U of N with φ−1(U) non-empty, f φ : φ−1(U)→ R is aharmonic function.

Theorem 1.7 (Fuglede 1978, Ishihara 1979)

A map φ : (M, g)→ (N,h) between Riemannian manifolds is a harmonicmorphism if and only if it is harmonic and horizontally (weakly)conformal.

Definition 1.6 (Harmonic Morphisms (Fuglede 1978, Ishihara 1979))

A map φ : (Mm, g)→ (Nn, h) between Riemannian manifolds is called aharmonic morphism if, for any harmonic function f : U → R defined onan open subset U of N with φ−1(U) non-empty, f φ : φ−1(U)→ R is aharmonic function.

Theorem 1.7 (Fuglede 1978, Ishihara 1979)

A map φ : (M, g)→ (N,h) between Riemannian manifolds is a harmonicmorphism if and only if it is harmonic and horizontally (weakly)conformal.

(Harmonicity)

For local coordinates x on (M, g) and y on (N,h), we have the non-linearsystem

τ(φ) =m∑

∂2φγ

∂xi∂xj−

m∑k=1

Γkij∂φγ

∂xk+

n∑α,β=1

Γγαβ φ∂φα

∂φβ

where φα = yα φ and Γ,Γ are the Christoffel symbols on M,N , resp.

(Horizontal (weak) Conformality)

There exists a continuous function λ : M → R+0 such that for all

α, β = 1, 2, . . . , n

m∑i,j=1

gij(x)∂φα

∂xi(x)

∂φβ

∂xj(x) = λ2(x)hαβ(φ(x)).

This is a first order non-linear system of [(n+12

)− 1] equations.

(Harmonicity)

For local coordinates x on (M, g) and y on (N,h), we have the non-linearsystem

τ(φ) =m∑

∂2φγ

∂xi∂xj−

m∑k=1

Γkij∂φγ

∂xk+

n∑α,β=1

Γγαβ φ∂φα

∂φβ

where φα = yα φ and Γ,Γ are the Christoffel symbols on M,N , resp.

(Horizontal (weak) Conformality)

There exists a continuous function λ : M → R+0 such that for all

α, β = 1, 2, . . . , n

m∑i,j=1

gij(x)∂φα

∂xi(x)

∂φβ

∂xj(x) = λ2(x)hαβ(φ(x)).

This is a first order non-linear system of [(n+12

)− 1] equations.

Theorem 1.8 (Baird, Eells 1981)

Let φ : (M, g)→ (N2, h) be a horizontally conformal map from aRiemannian manifold to a surface. Then φ is harmonic if and only if itsfibres are minimal at regular points φ.

The problem is invariant under isometries on (M, g). If the codomain(N,h) is a surface (n = 2) then it is also invariant under conformalchanges σ2h of the metric on N2. This means, at least for local studies,that (N2, h) can be chosen to be the standard complex plane C.

Theorem 1.8 (Baird, Eells 1981)

Let φ : (M, g)→ (N2, h) be a horizontally conformal map from aRiemannian manifold to a surface. Then φ is harmonic if and only if itsfibres are minimal at regular points φ.

The problem is invariant under isometries on (M, g). If the codomain(N,h) is a surface (n = 2) then it is also invariant under conformalchanges σ2h of the metric on N2. This means, at least for local studies,that (N2, h) can be chosen to be the standard complex plane C.

Example 1.9 (The Nilpotent Lie Group Nil3)

(x, y, z) ∈ R3 7→

1 x z0 1 y0 0 1

∈ SL3(R).

The left-invariant metric, with orthonormal basis

X = ∂/∂x, Y = ∂/∂y, Z = ∂/∂z

at the neutral element e = (0, 0, 0), is given by

ds2 = dx2 + dy2 + (dz − xdy)2.

(Baird, Wood 1990): Every local solution is a restriction of the globallydefined harmonic morphism φ : Nil3 → C with

1 x z0 1 y0 0 1

7→ x+ iy.

Example 1.9 (The Nilpotent Lie Group Nil3)

(x, y, z) ∈ R3 7→

1 x z0 1 y0 0 1

∈ SL3(R).

X = ∂/∂x, Y = ∂/∂y, Z = ∂/∂z

ds2 = dx2 + dy2 + (dz − xdy)2.

(Baird, Wood 1990): Every local solution is a restriction of the globallydefined harmonic morphism φ : Nil3 → C with

1 x z0 1 y0 0 1

7→ x+ iy.

Example 1.10 (The Solvable Lie Group Sol3)

(x, y, z) ∈ R3 7→

ez 0 x0 e−z y0 0 1

∈ SL3(R).

X = ∂/∂x, Y = ∂/∂y, Z = ∂/∂z

ds2 = e2zdx2 + e−2zdy2 + dz2.

(Baird, Wood 1990): No solutions exist, not even locally.

e−2z ∂2φ

∂x2+ e2z

∂2φ

∂y2+∂2φ

∂z2= 0,

e−2z

(∂φ

+ e2z(∂φ

(∂φ

Example 1.10 (The Solvable Lie Group Sol3)

(x, y, z) ∈ R3 7→

ez 0 x0 e−z y0 0 1

∈ SL3(R).

X = ∂/∂x, Y = ∂/∂y, Z = ∂/∂z

ds2 = e2zdx2 + e−2zdy2 + dz2.

(Baird, Wood 1990): No solutions exist, not even locally.

e−2z ∂2φ

∂x2+ e2z

∂2φ

∂y2+∂2φ

∂z2= 0,

e−2z

(∂φ

+ e2z(∂φ

(∂φ

The ConjectureRelevant History

Conjecture 1 (SG 1995)

Let (M, g) be an irreducible Riemannian symmetric space of dimensionm ≥ 2. For each point p ∈M there exists a non-constant complex-valuedharmonic morphism φ : U ⊂M → C defined on an open neighbourhood Uof p. If M is of non-compact type then the domain U can be chosen to bethe whole of M .

Definition 2.1 (Symmetric Space)

A Riemannian manifold (M, g) is said to be a symmetric space if for eachpoint p ∈M there exists a global geodesic reflective isometryσ : (M, g)→ (M, g) i.e. such that its differential dσp : TpM → TpM at psatisfies

dσp = −idTpM .

The ConjectureRelevant History

Baird-Eells (1981): S3 = SO(1 + 3)/SO(1)× SO(3). The Hopf mapφ : S3 → S2 ∼= C with

φ : (x1, x2, x3, x4) 7→ (x1 + ix2)/(x3 + ix4).

Baird-Wood (1989): H3 = SOo(1, 3)/SO(1)× SO(3)

Wood (1991): S4 = SO(1 + 4)/SO(1)× SO(4)

Baird (1992): H4 = SOo(1, 4)/SO(1)× SO(4)

SG (1994): CP q = U(1 + q)/U(1)×U(q)

SG (1994): HP q = Sp(1 + q)/Sp(1)× Sp(q)

SG (1995): H2n+1 = SOo(1, 2n+ 1)/SO(1)× SO(2n+ 1). The ”dual”Hopf map φ : H3 → C with

φ : (x1, x2, x3, x4) 7→ (x1 + ix2)/(x3 − x4).

DefinitionUseful MachineryThe Classical Semisimple Lie Groups

Definition 3.1 (The Laplacian - The Conformality Operator)

For complex-valued functions φ, ψ : (M, g)→ C on a Riemannian manifoldwe have the complex-valued Laplacian τ(φ) and the symmetric bilinearconformality operator κ given by

κ(φ, ψ) = g(∇φ,∇ψ).

The harmonicity and the horizontal conformality of φ : (M, g)→ Care then given by the following relations

τ(φ) = 0 and κ(φ, φ) = 0.

Definition 3.2 (Eigenfamilies)

A set E = φα : (M, g)→ C | α ∈ I of complex-valued functions is calledan eigenfamily on (M, g) if there exist complex numbers λ, µ ∈ C suchthat for all φ, ψ ∈ E

τ(φ) = λφ and κ(φ, ψ) = µφψ.

κ(φ, ψ) = g(∇φ,∇ψ).

τ(φ) = 0 and κ(φ, φ) = 0.

κ(φ, ψ) = g(∇φ,∇ψ).

τ(φ) = 0 and κ(φ, φ) = 0.

Theorem 3.3 (SG, Sakovich 2008)

Let (M, g) be a Riemannian manifold and E = φ1, . . . , φn be a finiteeigenfamily of complex-valued functions on M . If P,Q : Cn → C arelinearily independent homogeneous polynomials of the same positive degreethen the quotient

P (φ1, . . . , φn)/Q(φ1, . . . , φn)

is a non-constant harmonic morphism on the open and dense subset

p ∈M | Q(φ1(p), . . . , φn(p)) 6= 0.

The authors apply this machinery to construct solutions on the classicalsemisimple Lie groups SO(n), SU(n), Sp(n), SLn(R), SU∗(2n) andSp(n,R) equipped with their standard Riemannian metrics.

They also develop a duality principle and use this to construct solutionsfrom the semisimple Lie groups SO(n), SU(n), Sp(n), SLn(R), SU∗(2n),Sp(n,R), SO∗(2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with theirstandard dual semi-Riemannian metrics.

Let (M, g) be a Riemannian manifold and E = φ1, . . . , φn be a finiteeigenfamily of complex-valued functions on M . If P,Q : Cn → C arelinearily independent homogeneous polynomials of the same positive degreethen the quotient

P (φ1, . . . , φn)/Q(φ1, . . . , φn)

is a non-constant harmonic morphism on the open and dense subset

p ∈M | Q(φ1(p), . . . , φn(p)) 6= 0.

The authors apply this machinery to construct solutions on the classicalsemisimple Lie groups SO(n), SU(n), Sp(n), SLn(R), SU∗(2n) andSp(n,R) equipped with their standard Riemannian metrics.

They also develop a duality principle and use this to construct solutionsfrom the semisimple Lie groups SO(n), SU(n), Sp(n), SLn(R), SU∗(2n),Sp(n,R), SO∗(2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with theirstandard dual semi-Riemannian metrics.

Equip the special orthogonal group

SO(n) = x ∈ GLn(R) | xt · x = In, detx = 1

with the standard Riemannian metric g induced by the Euclidean scalarproduct g(X,Y ) = trace(Xt · Y ) on the Lie algebra

so(n) = X ∈ gln(R)| Xt +X = 0.

Lemma 3.4 (SG, Sakovich 2008)

For 1 ≤ i, j ≤ n, let xij : SO(n)→ R be the real valued coordinate functionsgiven by xij : x 7→ 〈ei, x · ej〉 where e1, . . . , en is the canonical basis forRn. Then the following relations hold

τ(xij) = − (n− 1)

2xij , κ(xij , xkl) = −1

2(xilxkj − δjlδik).

Equip the special orthogonal group

SO(n) = x ∈ GLn(R) | xt · x = In, detx = 1

with the standard Riemannian metric g induced by the Euclidean scalarproduct g(X,Y ) = trace(Xt · Y ) on the Lie algebra

so(n) = X ∈ gln(R)| Xt +X = 0.

Lemma 3.4 (SG, Sakovich 2008)

For 1 ≤ i, j ≤ n, let xij : SO(n)→ R be the real valued coordinate functionsgiven by xij : x 7→ 〈ei, x · ej〉 where e1, . . . , en is the canonical basis forRn. Then the following relations hold

τ(xij) = − (n− 1)

2xij , κ(xij , xkl) = −1

2(xilxkj − δjlδik).

Let p ∈ Cn be a non-zero isotropic element i.e. 〈p, p〉C = 0. Then thefollowing is an eigenfamily on SO(n)

Ep = φa : SO(n)→ C | φa(x) = 〈p, x · a〉C, a ∈ Cn.

Example 3.6 (Eigenfamilies on SO(4))

For z, w ∈ C, let p be the isotropic element of C4 given by

p(z, w) = (1 + zw, i(1− zw), i(z + w), z − w).

This gives us the complex 2-dimensional deformation of eigenfamilies Epeach consisting of functions φa : SO(4)→ C with

φa(x) = (1 + zw)(x11a1 + x21a2 + x31a3 + x41a4)

+i(1− zw)(x12a1 + x22a2 + x32a3 + x42a4)

+i(z + w)(x13a1 + x23a2 + x33a3 + x43a4)

+(z − w)(x14a1 + x24a2 + x34a3 + x44a4).

Let p ∈ Cn be a non-zero isotropic element i.e. 〈p, p〉C = 0. Then thefollowing is an eigenfamily on SO(n)

Ep = φa : SO(n)→ C | φa(x) = 〈p, x · a〉C, a ∈ Cn.

Example 3.6 (Eigenfamilies on SO(4))

For z, w ∈ C, let p be the isotropic element of C4 given by

p(z, w) = (1 + zw, i(1− zw), i(z + w), z − w).

This gives us the complex 2-dimensional deformation of eigenfamilies Epeach consisting of functions φa : SO(4)→ C with

φa(x) = (1 + zw)(x11a1 + x21a2 + x31a3 + x41a4)

+i(1− zw)(x12a1 + x22a2 + x32a3 + x42a4)

+i(z + w)(x13a1 + x23a2 + x33a3 + x43a4)

+(z − w)(x14a1 + x24a2 + x34a3 + x44a4).

Another Useful MachineSymmetric Spaces G/K of Non-Compact TypeNilpotent and Solvable Lie GroupsSymmetric Spaces U/K of Compact TypeExamplesHomogeneous Spaces of Positive Curvature

Definition 4.1 (Orthogonal Harmonic Family)

A set Ω = φα : (M, g)→ C | α ∈ I of complex-valued functions is calledan orthogonal harmonic family on (M, g) if for all φ, ψ ∈ Ω

τ(φ) = 0 and κ(φ, ψ) = 0.

Example 4.2

Let Ω = φα : (M, g, J)→ C | α ∈ I be a collection of holomorphicfunctions on a Kahler manifold. Then Ω is an orthogonal harmonicfamily.

Definition 4.1 (Orthogonal Harmonic Family)

A set Ω = φα : (M, g)→ C | α ∈ I of complex-valued functions is calledan orthogonal harmonic family on (M, g) if for all φ, ψ ∈ Ω

τ(φ) = 0 and κ(φ, ψ) = 0.

Example 4.2

Let Ω = φα : (M, g, J)→ C | α ∈ I be a collection of holomorphicfunctions on a Kahler manifold. Then Ω is an orthogonal harmonicfamily.

Theorem 4.3 (SG 1997)

Let (M, g) be a Riemannian manifold and U be an open subset of Cncontaining the image of Φ = (φ1, . . . , φn) : M → Cn. Further let

H = Fα : U → C | α ∈ I

be a collection of holomorphic functions defined on U . If the finite set

Ω = φk : (M, g)→ C | k = 1, . . . , n

is an orthogonal harmonic family on (M, g) then

ΩH = ψ : M → C | ψ = F (φ1, . . . , φn), F ∈ H

is again an orthogonal harmonic family.

Let (M, g) be an irreducible Riemannian symmetric space ofnon-compact type presented as the quotient G/K where G a connectedsemisimple Lie group and K its maximal compact subgroup.

Let G = NAK be the Iwasawa decomposition of G, where N isnilpotent and A is abelian.

Fact 4.4 (solvable Lie group - rank)

The non-compact symmetric space (M, g) can be identified with thesolvable subgroup S = NA of G and its rank r is the dimension ofabelian subgroup A.

Let s, n, a be the Lie algebras of S,N,A, respectively. For this situation wehave s = a + n = a + [s, s], hence

a = s/[s, s].

Let G be a connected and simply connected Lie group with Lie algebra g.Then the natural projection π : g→ a ∼= g/[g, g] to the abelian algebra a isa Lie algebra homomorphism inducing a natural group epimorphismΦ : G→ Rd with d = dim a.

Fact 4.5 (semisimple - solvable - nilpotent)

If the group G is semisimple then d = 0, if G is solvable then d ≥ 1 andif G is nilpotent then d ≥ 2.

Equip Rd with its standard Euclidean metric and the Lie group G with aleft-invariant Riemannian metric g such that the natural groupepimorphism Φ : G→ Rd is a Riemannian submersion. Then the kernel[g, g] of the linear map π : g→ g/[g, g] generates a left-invariant Riemannianfoliation V on (G, g) with orthogonal distribution H = [g, g]⊥.

Theorem 4.6 (SG, Svensson 2009)

Let B = X1, . . . , Xd be an ONB for the horizontal subspace [g, g]⊥ of gand ξ ∈ Cd be given by ξ = (trace adX1 , . . . , trace adXd). For a maximalisotropic subspace V of Cd put

Vξ = v ∈ V | 〈ξ, v〉C = 0.

If the real dimension of the isotropic subspace Vξ is at least 2 then

Ω = φv(x) = 〈Φ(x), v〉C | v ∈ Vξ

is an orthogonal harmonic family on (G, g).

Proof.

The tension field of natural group epimorphism Φ : G→ Rd satisfies

τ(Φ)(p) =

d∑k=1

(trace adXk )dΦe(Xk).

Let B = X1, . . . , Xd be an ONB for the horizontal subspace [g, g]⊥ of gand ξ ∈ Cd be given by ξ = (trace adX1 , . . . , trace adXd). For a maximalisotropic subspace V of Cd put

Vξ = v ∈ V | 〈ξ, v〉C = 0.

If the real dimension of the isotropic subspace Vξ is at least 2 then

Ω = φv(x) = 〈Φ(x), v〉C | v ∈ Vξ

is an orthogonal harmonic family on (G, g).

Proof.

The tension field of natural group epimorphism Φ : G→ Rd satisfies

τ(Φ)(p) =d∑k=1

(trace adXk )dΦe(Xk).

Example 4.7

For the nilpotent Riemannian Lie group

1 x12 · · · x1,n−1 x1n

0 1. . .

......

. . .. . .

. . ....

.... . . 1 xn−1,n

0 · · · · · · 0 1

∈ SLn(R) | xij ∈ R

the natural group epimorphism Φ : Nn → Rn−1 is given by

Φ(x) = (x12, . . . , xn−1,n)

and the vector ξ ∈ Cn satisfies ξ = 0.

Example 4.8

For the solvable Riemannian Lie group

et1 x12 · · · x1,n−1 x1n0 et2 · · · x2,n−1 x2n...

. . .. . .

......

0 · · · 0 etn−1 xn−1,n

0 · · · 0 0 etn

∈ GLn(R) | xij , ti ∈ R

the natural group epimorphism Φ : Sn → Rn is given by

Φ(x) = (t1, t2, . . . , tn)

and the vector ξ ∈ Cn satisfies

ξ = ((n+ 1)− 2, (n+ 1)− 4, . . . , (n+ 1)− 2n).

As an immediate consequence of Theorem 4.6 we now have the existence ofglobally defined harmonic morphisms from any simply connected symmetricspace G/K of non-compact type and rank r ≥ 3.

With a series of other additional methods we have the following result.

Let (M, g) be an irreducible Riemannian symmetric space of non-cmpacttype other than G∗2/SO(4). Then there exists a non-constant globallydefined complex-valued harmonic morphism φ : M → C.

This can be extended to the following result.

Let (M, g) be an irreducible Riemannian symmetric space other thanG∗2/SO(4) or its compact dual G2/SO(4). Then for each point p ∈Mthere exists a non-constant complex-valued harmonic morphismφ : U → C defined on an open neighbourhood U of p. If the space (M, g) isof non-compact type then the domain U can be chosen to be the whole of M .

An essential tool is the following Duality Principle:

Let F be a family of local maps φ : W ⊂ G/K → C and F∗ be the dualfamily consisting of the local maps φ∗ : W ∗ ⊂ U/K → C. Then F∗ is a localorthogonal harmonic family on U/K if and only if F is a local orthogonalharmonic family on G/K.

This can be extended to the following result.

Let (M, g) be an irreducible Riemannian symmetric space other thanG∗2/SO(4) or its compact dual G2/SO(4). Then for each point p ∈Mthere exists a non-constant complex-valued harmonic morphismφ : U → C defined on an open neighbourhood U of p. If the space (M, g) isof non-compact type then the domain U can be chosen to be the whole of M .

An essential tool is the following Duality Principle:

Let F be a family of local maps φ : W ⊂ G/K → C and F∗ be the dualfamily consisting of the local maps φ∗ : W ∗ ⊂ U/K → C. Then F∗ is a localorthogonal harmonic family on U/K if and only if F is a local orthogonalharmonic family on G/K.

The Duality Principle explains the following.

Example 4.12 (Baird, Eells 1981)

The map φ : U ⊂ S3 ⊂ R4 = C2 → C given by

φ : (x1, x2, x3, x4) 7→ x1 + ix2x3 + ix4

is a locally defined harmonic morphism.

Example 4.13 (SG 1996)

The map φ : H3 ⊂ R41 → C given by

φ : (x1, x2, x3, x4) 7→ x1 + ix2x3 − x4

is a globally defined harmonic morphism.

The Duality Principle explains the following.

Example 4.12 (Baird, Eells 1981)

The map φ : U ⊂ S3 ⊂ R4 = C2 → C given by

φ : (x1, x2, x3, x4) 7→ x1 + ix2x3 + ix4

is a locally defined harmonic morphism.

Example 4.13 (SG 1996)

The map φ : H3 ⊂ R41 → C given by

φ : (x1, x2, x3, x4) 7→ x1 + ix2x3 − x4

is a globally defined harmonic morphism.

Our existence result for symmetric spaces has the following interestingconsequence:

Let (M, g) be a Riemannian homogeneous space of positive curvatureother than the Berger space Sp(2)/SU(2). Then M admits localcomplex-valued harmonic morphisms.

Classifications

Fact 5.1

Every Riemannian homogeneous space (M, g) of dimension 3 or 4 iseither symmetric or a Lie group with a left-invariant metric.

(SG, Svensson 2011): Give a classification for 3-dimensionalRiemannian Lie groups admitting solutions. Find a continuous family ofgroups, containing Sol3, not carrying any left-invariant metric admittingcomplex-valued harmonic morphisms.

(SG, Svensson 2013): Give a classification for 4-dimensionalRiemannian Lie groups admitting left-invariant solutions. Most of thesolutions constructed are NOT holomorphic with respect to any(integrable) Hermitian structure.

(SG 2016): Gives a large collection of 5-dimensional Riemannian Liegroups admitting left-invariant solutions.

Classifications

Fact 5.1

Classifications

Fact 5.1

Classifications

Fact 5.1

[1] S. Gudmundsson, On the existence of harmonic morphisms from symmetricspaces of rank one, Manuscripta Math. 93 (1997), 421-433.

[2] S. Gudmundsson and M. Svensson, Harmonic morphisms from theGrassmannians and their non-compact duals, Ann. Global Anal. Geom. 30(2006), 313-333.

[3] S. Gudmundsson, A. Sakovich, Harmonic morphisms from the classical compactsemisimple Lie groups, Ann. Global Anal. Geom. 33 (2008), 343-356.

[4] S. Gudmundsson and A. Sakovich, Harmonic morphisms from the classicalnon-compact semisimple Lie groups, Differential Geom. Appl. 27 (2009), 47-63.

[5] S. Gudmundsson, M. Svensson, Harmonic morphisms from solvable Lie groups,Math. Proc. Cambridge Philos. Soc. 147 (2009), 389-408.

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