harmonic morphisms from homogeneous spaces · harmonic morphisms from homogeneous spaces - some...

Harmonic Morphisms from Homogeneous Spaces

- Some Existence Theory -

Sigmundur Gudmundsson

Department of MathematicsFaculty of ScienceLund University

[email protected]

February 8, 2016

Harmonic MorphismsEigenfamilies - Constructions

Orthogonal Harmonic Families - ConstructionsLow-Dimensional Classifications

Outline

1 Harmonic MorphismsBasicsGeometric MotivationHomogeneous Spaces

2 Eigenfamilies - ConstructionsDefinitionA Useful MachineThe Classical Semisimple Lie Groups

3 Orthogonal Harmonic Families - ConstructionsAnother Useful MachineNilpotent and Solvable Lie GroupsSymmetric Spaces G/K of Non-Compact TypeSymmetric Spaces U/K of Compact TypeExamplesHomogeneous Spaces of Positive Curvature

4 Low-Dimensional ClassificationsClassifications

Sigmundur Gudmundsson Harmonic Morphisms from Homogeneous Spaces

Outline

BasicsGeometric MotivationHomogeneous Spaces

Definition 1.1 (harmonic morphism)

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 1.2 (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 1.1 (harmonic morphism)

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 1.2 (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 and y on N , we have the non-linear system

τ(φ) =

m∑i,j=1

∂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 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 and y on N , we have the non-linear system

τ(φ) =

m∑i,j=1

∂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 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 1.3 (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 isa surface (n = 2) then it is also invariant under conformal changes σ2h ofthe metric on (N2, h). This means, at least for local studies, that (N2, h)can be chosen to be the standard complex plane C.

Theorem 1.3 (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 isa surface (n = 2) then it is also invariant under conformal changes σ2h ofthe metric on (N2, h). This means, at least for local studies, that (N2, h)can be chosen to be the standard complex plane C.

Definition 1.4 (Riemannian homogeneous space)

A Riemannian manifold (M, g) is said to be homogeneous if it possesses atransitive group G of isometries i.e. if for all p, q ∈M there exists anisometry φqp : M →M such that φqp(p) = q.

Example 1.5 (Riemannian Lie group)

Every Lie group (G, g) equipped with a left-invariant Riemannian metricacts transitively on itself.

Example 1.6 (Riemannian symmetric space)

Every Riemannian symmetric space M = (G/K, g) is homogeneous.

Example 1.7 (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 solution is a restriction of the globallydefined harmonic morphism φ : Nil3 → C with

1 x z0 1 y0 0 1

7→ x+ iy.

Example 1.7 (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 solution is a restriction of the globallydefined harmonic morphism φ : Nil3 → C with

1 x z0 1 y0 0 1

7→ x+ iy.

Example 1.8 (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 1.8 (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

DefinitionA Useful MachineThe Classical Semisimple Lie Groups

Definition 2.1 (Laplacian, conformality operator)

For functions φ, ψ : (M, g)→ C the metric g induces the complex-valuedLaplacian τ(φ) and the symmetric bilinear conformality operator κ 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 2.2 (eigenfamily)

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 2.3 (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 machine 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 2.4 (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 2.4 (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〉 = 0. Then

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

is an eigenfamily on SO(n)

Example 2.6 (eigenfamilies on SO(n))

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〉 = 0. Then

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

is an eigenfamily on SO(n)

Example 2.6 (eigenfamilies on SO(n))

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 MachineNilpotent and Solvable Lie GroupsSymmetric Spaces G/K of Non-Compact TypeSymmetric Spaces U/K of Compact TypeExamplesHomogeneous Spaces of Positive Curvature

Definition 3.1 (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 3.2

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 3.1 (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 3.2

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 3.3 (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 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 3.4 (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 3.5 (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〉 = 0.

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

Ω = φv(x) = 〈Φ(x), v〉 | v ∈ Vξ

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

Proof.

The tension field of Φ 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〉 = 0.

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

Ω = φv(x) = 〈Φ(x), v〉 | v ∈ Vξ

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

Proof.

The tension field of Φ satisfies

τ(Φ)(p) =d∑k=1

(trace adXk )dΦe(Xk).

Example 3.6

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).

Example 3.7

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).

Let (M, g) be an irreducible Riemannian symmetric space ofnon-compact type and write M = G/K with G a semisimple, connectedand simply connected Lie group and K a maximal compact subgroup of G.

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

Fact 3.8 (solvable Lie group - rank)

The symmetric space (M, g) can be identified with the solvablesubgroup S = NA of G and its rank r is the dimension of abeliansubgroup 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].

With a series of different methods we have obtained 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.

With a series of different methods we have obtained 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 3.11 (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 3.12 (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 3.11 (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 3.12 (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 4.1

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 4.1

Classifications

Fact 4.1

Classifications

Fact 4.1

Classifications

S. Gudmundsson, On the existence of harmonic morphisms fromsymmetric spaces of rank one, Manuscripta Math. 93 (1997), 421-433.

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

S. Gudmundsson, A. Sakovich, Harmonic morphisms from the classicalcompact semisimple Lie groups, Ann. Global Anal. Geom. 33 (2008),343-356.

S. Gudmundsson and A. Sakovich, Harmonic morphisms from theclassical non-compact semisimple Lie groups, Differential Geom. Appl.27 (2009), 47-63.

S. Gudmundsson, M. Svensson, Harmonic morphisms from solvable Liegroups, Math. Proc. Cambridge Philos. Soc. 147 (2009), 389-408.

S. Gudmundsson, M. Svensson, On the existence of harmonicmorphisms from three-dimensional Lie groups, Contemp. Math. 542(2011), 279-284.

S. Gudmundsson, M. Svensson, Harmonic morphisms fromfour-dimensional Lie groups, J. Geom. Phys. 83 (2014), 1-11.

S. Gudmundsson, Harmonic morphisms from five-dimensional Liegroups, preprint (2016).

S. Gudmundsson, M. Svensson, Harmonic morphisms fromhomogeneous spaces of positive curvature, Math. Proc. CambridgePhilos. Soc. (to appear).

harmonic morphisms from homogeneous spaces · harmonic morphisms from homogeneous spaces - some...

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