Harmonic Morphisms from Lie Groups and Symmetric Spaces - Some - - PowerPoint PPT Presentation

Harmonic Morphisms from Lie Groups and Symmetric Spaces

• Some Existence Theory -

Sigmundur Gudmundsson

Department of Mathematics Faculty of Science Lund University Sigmundur.Gudmundsson@math.lu.se

Copenhagen - 22 May 2018

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References

Outline

1 Harmonic Morphisms

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

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References

Outline

1 Harmonic Morphisms

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

2 The Conjecture

The Conjecture Relevant History

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References

Outline

1 Harmonic Morphisms

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

2 The Conjecture

The Conjecture Relevant History

3 Constructions by Eigenfamilies

Definition Useful Machinery The Classical Semisimple Lie Groups

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References

Outline

1 Harmonic Morphisms

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

2 The Conjecture

The Conjecture Relevant History

3 Constructions by Eigenfamilies

Definition Useful Machinery The Classical Semisimple Lie Groups

4 Constructions by Orthogonal Harmonic Families

Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References

Outline

1 Harmonic Morphisms

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

2 The Conjecture

The Conjecture Relevant History

3 Constructions by Eigenfamilies

Definition Useful Machinery The Classical Semisimple Lie Groups

4 Constructions by Orthogonal Harmonic Families

Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

5 Low-Dimensional Classifications

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Definition 1.1 (Harmonic Morphisms (Jacobi 1848)) A map φ = u + iv : U ⊂ R3 → C is said to be a harmonic morphism if the composition f ◦ φ with any holomorphic function f : W ⊂ C → C is harmonic.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Definition 1.1 (Harmonic Morphisms (Jacobi 1848)) A map φ = u + iv : U ⊂ R3 → C is said to be a harmonic morphism if the composition f ◦ φ with any holomorphic function f : W ⊂ C → C is harmonic. Theorem 1.2 (Jacobi 1848) A map φ = u + iv : U ⊂ R3 → C is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal i.e. ∆u = ∆v = 0, ∇u, ∇v = 0 and |∇u|2 = |∇v|2.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Definition 1.1 (Harmonic Morphisms (Jacobi 1848)) A map φ = u + iv : U ⊂ R3 → C is said to be a harmonic morphism if the composition f ◦ φ with any holomorphic function f : W ⊂ C → C is harmonic. Theorem 1.2 (Jacobi 1848) A map φ = u + iv : U ⊂ R3 → C is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal i.e. ∆u = ∆v = 0, ∇u, ∇v = 0 and |∇u|2 = |∇v|2. Proof. ∆(f ◦ φ) = ∂f ∂z

• · ∆φ +

∂2f ∂z2

• · ∇φ, ∇φC = 0

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Theorem 1.3 (Jacobi 1848) Let f, g : W ⊂ C → C be holomorphic functions, then every local solution z : U ⊂ R3 → C to the equation f(z(x))

• 1 − g2(z(x)), i(1 + g2(z(x))), 2g(z(x))
• , xC = 1

is a harmonic morphism.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Theorem 1.3 (Jacobi 1848) Let f, g : W ⊂ C → C be holomorphic functions, then every local solution z : U ⊂ R3 → C to the equation f(z(x))

• 1 − g2(z(x)), i(1 + g2(z(x))), 2g(z(x))
• , xC = 1

is a harmonic morphism. Theorem 1.4 (Baird-Wood 1988) Every harmonic morphism z : U → C defined locally on the Euclidean R3 is

• btained this way.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

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) ±

• x2

1 + x2 2 + x2 3 − r2 + 2irx3

x1 − ix2 .

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Definition 1.6 (Harmonic Morphisms (Fuglede 1978, Ishihara 1979)) A map φ : (M m, g) → (N n, h) between Riemannian manifolds is called a harmonic morphism if, for any harmonic function f : U → R defined on an open subset U of N with φ−1(U) non-empty, f ◦ φ : φ−1(U) → R is a harmonic function.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Definition 1.6 (Harmonic Morphisms (Fuglede 1978, Ishihara 1979)) A map φ : (M m, g) → (N n, h) between Riemannian manifolds is called a harmonic morphism if, for any harmonic function f : U → R defined on an open subset U of N with φ−1(U) non-empty, f ◦ φ : φ−1(U) → R is a harmonic function. Theorem 1.7 (Fuglede 1978, Ishihara 1979) A map φ : (M, g) → (N, h) between Riemannian manifolds is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

(Harmonicity) For local coordinates x on (M, g) and y on (N, h), we have the non-linear system τ(φ) =

m

• i,j=1

gij   ∂2φγ ∂xi∂xj −

m

• k=1

ˆ Γk

ij

∂φγ ∂xk +

n

• α,β=1

Γγ

αβ ◦ φ∂φα

∂xi ∂φβ ∂xj   = 0, where φα = yα ◦ φ and ˆ Γ, Γ are the Christoffel symbols on M, N, resp.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

(Harmonicity) For local coordinates x on (M, g) and y on (N, h), we have the non-linear system τ(φ) =

m

• i,j=1

gij   ∂2φγ ∂xi∂xj −

m

• k=1

ˆ Γk

ij

∂φγ ∂xk +

n

• α,β=1

Γγ

αβ ◦ φ∂φα

∂xi ∂φβ ∂xj   = 0, where φα = yα ◦ φ and ˆ 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+1

2

• − 1] equations.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Theorem 1.8 (Baird, Eells 1981) Let φ : (M, g) → (N 2, h) be a horizontally conformal map from a Riemannian manifold to a surface. Then φ is harmonic if and only if its fibres are minimal at regular points φ.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Theorem 1.8 (Baird, Eells 1981) Let φ : (M, g) → (N 2, h) be a horizontally conformal map from a Riemannian manifold to a surface. Then φ is harmonic if and only if its fibres 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 conformal changes σ2h of the metric on N 2. This means, at least for local studies, that (N 2, h) can be chosen to be the standard complex plane C.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Example 1.9 (The Nilpotent Lie Group Nil3) (x, y, z) ∈ R3 →   1 x z 1 y 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.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Example 1.9 (The Nilpotent Lie Group Nil3) (x, y, z) ∈ R3 →   1 x z 1 y 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 globally defined harmonic morphism φ : Nil3 → C with φ :   1 x z 1 y 1   → x + iy.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Example 1.10 (The Solvable Lie Group Sol3) (x, y, z) ∈ R3 →   ez x e−z y 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 = e2zdx2 + e−2zdy2 + dz2.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Origins - Jacobi 1848 Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Existence ?

Example 1.10 (The Solvable Lie Group Sol3) (x, y, z) ∈ R3 →   ez x e−z y 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 = 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 ∂x 2 + e2z ∂φ ∂y 2 + ∂φ ∂z 2 = 0.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Conjecture Relevant History

Conjecture 1 (SG 1995) Let (M, g) be an irreducible Riemannian symmetric space of dimension m ≥ 2. For each point p ∈ M there exists a non-constant complex-valued harmonic morphism φ : U ⊂ M → C defined on an open neighbourhood U

• f p. If M is of non-compact type then the domain U can be chosen to be

the whole of M. Definition 2.1 (Symmetric Space) A Riemannian manifold (M, g) is said to be a symmetric space if for each point p ∈ M there exists a global geodesic reflective isometry σ : (M, g) → (M, g) i.e. such that its differential dσp : TpM → TpM at p satisfies dσp = −idTpM.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References The Conjecture Relevant History

Baird-Eells (1981): S3 = SO(1 + 3)/SO(1) × SO(3). The Hopf map φ : S3 → S2 ∼ = ˆ C with φ : (x1, x2, x3, x4) → (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) → (x1 + ix2)/(x3 − x4).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Definition 3.1 (The Laplacian - The Conformality Operator) For complex-valued functions φ, ψ : (M, g) → C on a Riemannian manifold we have the complex-valued Laplacian τ(φ) and the symmetric bilinear conformality operator κ given by κ(φ, ψ) = g(∇φ, ∇ψ).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Definition 3.1 (The Laplacian - The Conformality Operator) For complex-valued functions φ, ψ : (M, g) → C on a Riemannian manifold we have the complex-valued Laplacian τ(φ) and the symmetric bilinear conformality operator κ given by κ(φ, ψ) = g(∇φ, ∇ψ). The harmonicity and the horizontal conformality of φ : (M, g) → C are then given by the following relations τ(φ) = 0 and κ(φ, φ) = 0.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Definition 3.1 (The Laplacian - The Conformality Operator) For complex-valued functions φ, ψ : (M, g) → C on a Riemannian manifold we have the complex-valued Laplacian τ(φ) and the symmetric bilinear conformality operator κ given by κ(φ, ψ) = g(∇φ, ∇ψ). The harmonicity and the horizontal conformality of φ : (M, g) → C are 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 called an eigenfamily on (M, g) if there exist complex numbers λ, µ ∈ C such that for all φ, ψ ∈ E τ(φ) = λφ and κ(φ, ψ) = µφψ.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Theorem 3.3 (SG, Sakovich 2008) Let (M, g) be a Riemannian manifold and E = {φ1, . . . , φn} be a finite eigenfamily of complex-valued functions on M. If P, Q : Cn → C are linearily independent homogeneous polynomials of the same positive degree then 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)) = 0}.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Theorem 3.3 (SG, Sakovich 2008) Let (M, g) be a Riemannian manifold and E = {φ1, . . . , φn} be a finite eigenfamily of complex-valued functions on M. If P, Q : Cn → C are linearily independent homogeneous polynomials of the same positive degree then 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)) = 0}. The authors apply this machinery to construct solutions on the classical semisimple Lie groups SO(n), SU(n), Sp(n), SLn(R), SU∗(2n) and Sp(n, R) equipped with their standard Riemannian metrics. They also develop a duality principle and use this to construct solutions from 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 their standard dual semi-Riemannian metrics.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Equip the special orthogonal group SO(n) = {x ∈ GLn(R) | xt · x = In, det x = 1} with the standard Riemannian metric g induced by the Euclidean scalar product g(X, Y ) = trace(Xt · Y ) on the Lie algebra so(n) = {X ∈ gln(R)| Xt + X = 0}.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Equip the special orthogonal group SO(n) = {x ∈ GLn(R) | xt · x = In, det x = 1} with the standard Riemannian metric g induced by the Euclidean scalar product 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 functions given by xij : x → ei, x · ej where {e1, . . . , en} is the canonical basis for

• Rn. Then the following relations hold

τ(xij) = −(n − 1) 2 xij, κ(xij, xkl) = −1 2(xilxkj − δjlδik).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Theorem 3.5 (SG, Sakovich 2008) Let p ∈ Cn be a non-zero isotropic element i.e. p, pC = 0. Then the following is an eigenfamily on SO(n) Ep = {φa : SO(n) → C | φa(x) = p, x · aC, a ∈ Cn}.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Definition Useful Machinery The Classical Semisimple Lie Groups

Theorem 3.5 (SG, Sakovich 2008) Let p ∈ Cn be a non-zero isotropic element i.e. p, pC = 0. Then the following is an eigenfamily on SO(n) Ep = {φa : SO(n) → C | φa(x) = p, x · aC, 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 Ep each 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).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

Definition 4.1 (Orthogonal Harmonic Family) A set Ω = {φα : (M, g) → C | α ∈ I} of complex-valued functions is called an orthogonal harmonic family on (M, g) if for all φ, ψ ∈ Ω τ(φ) = 0 and κ(φ, ψ) = 0.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

Definition 4.1 (Orthogonal Harmonic Family) A set Ω = {φα : (M, g) → C | α ∈ I} of complex-valued functions is called an 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 holomorphic functions on a K¨ ahler manifold. Then Ω is an orthogonal harmonic family.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

Theorem 4.3 (SG 1997) Let (M, g) be a Riemannian manifold and U be an open subset of Cn containing 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.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

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

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

Let (M, g) be an irreducible Riemannian symmetric space of non-compact type presented as the quotient G/K where G a connected semisimple Lie group and K its maximal compact subgroup. Let G = NAK be the Iwasawa decomposition of G, where N is nilpotent and A is abelian.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Another Useful Machine Symmetric Spaces G/K of Non-Compact Type Nilpotent and Solvable Lie Groups Symmetric Spaces U/K of Compact Type Examples Homogeneous Spaces of Positive Curvature

Let (M, g) be an irreducible Riemannian symmetric space of non-compact type presented as the quotient G/K where G a connected semisimple Lie group and K its maximal compact subgroup. Let G = NAK be the Iwasawa decomposition of G, where N is nilpotent and A is abelian. Fact 4.4 (solvable Lie group - rank) The non-compact symmetric space (M, g) can be identified with the solvable subgroup S = NA of G and its rank r is the dimension of abelian subgroup A.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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Let (M, g) be an irreducible Riemannian symmetric space of non-compact type presented as the quotient G/K where G a connected semisimple Lie group and K its maximal compact subgroup. Let G = NAK be the Iwasawa decomposition of G, where N is nilpotent and A is abelian. Fact 4.4 (solvable Lie group - rank) The non-compact symmetric space (M, g) can be identified with the solvable subgroup S = NA of G and its rank r is the dimension of abelian subgroup A. Let s, n, a be the Lie algebras of S, N, A, respectively. For this situation we have s = a + n = a + [s, s], hence a = s/[s, s].

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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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 is a Lie algebra homomorphism inducing a natural group epimorphism Φ : G → Rd with d = dim a.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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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 is a Lie algebra homomorphism inducing a natural group epimorphism Φ : 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 and if G is nilpotent then d ≥ 2.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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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 is a Lie algebra homomorphism inducing a natural group epimorphism Φ : 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 and if G is nilpotent then d ≥ 2. Equip Rd with its standard Euclidean metric and the Lie group G with a left-invariant Riemannian metric g such that the natural group epimorphism 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 Riemannian foliation V on (G, g) with orthogonal distribution H = [g, g]⊥.

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Theorem 4.6 (SG, Svensson 2009) Let B = {X1, . . . , Xd} be an ONB for the horizontal subspace [g, g]⊥ of g and ξ ∈ Cd be given by ξ = (trace adX1, . . . , trace adXd). For a maximal isotropic subspace V of Cd put Vξ = {v ∈ V | ξ, vC = 0}. If the real dimension of the isotropic subspace Vξ is at least 2 then Ω = {φv(x) = Φ(x), vC | v ∈ Vξ} is an orthogonal harmonic family on (G, g).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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Theorem 4.6 (SG, Svensson 2009) Let B = {X1, . . . , Xd} be an ONB for the horizontal subspace [g, g]⊥ of g and ξ ∈ Cd be given by ξ = (trace adX1, . . . , trace adXd). For a maximal isotropic subspace V of Cd put Vξ = {v ∈ V | ξ, vC = 0}. If the real dimension of the isotropic subspace Vξ is at least 2 then Ω = {φv(x) = Φ(x), vC | 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).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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Example 4.7 For the nilpotent Riemannian Lie group Nn =

• 

         1 x12 · · · x1,n−1 x1n 1 ... . . . . . . ... ... ... . . . . . . ... 1 xn−1,n · · · · · · 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.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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Example 4.8 For the solvable Riemannian Lie group Sn =

• 

      et1 x12 · · · x1,n−1 x1n et2 · · · x2,n−1 x2n . . . ... ... . . . . . . · · · etn−1 xn−1,n · · · 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).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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As an immediate consequence of Theorem 4.6 we now have the existence of globally defined harmonic morphisms from any simply connected symmetric space G/K of non-compact type and rank r ≥ 3. With a series of other additional methods we have the following result. Theorem 4.9 (SG, Svensson 2009) Let (M, g) be an irreducible Riemannian symmetric space of non-cmpact type other than G∗

2/SO(4). Then there exists a non-constant globally

defined complex-valued harmonic morphism φ : M → C.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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This can be extended to the following result. Theorem 4.10 (SG, Svensson 2009) Let (M, g) be an irreducible Riemannian symmetric space other than G∗

2/SO(4) or its compact dual G2/SO(4). Then for each point p ∈ M

there exists a non-constant complex-valued harmonic morphism φ : U → C defined on an open neighbourhood U of p. If the space (M, g) is

• f non-compact type then the domain U can be chosen to be the whole of M.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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This can be extended to the following result. Theorem 4.10 (SG, Svensson 2009) Let (M, g) be an irreducible Riemannian symmetric space other than G∗

2/SO(4) or its compact dual G2/SO(4). Then for each point p ∈ M

there exists a non-constant complex-valued harmonic morphism φ : U → C defined on an open neighbourhood U of p. If the space (M, g) is

• f non-compact type then the domain U can be chosen to be the whole of M.

An essential tool is the following Duality Principle: Theorem 4.11 (SG, Svensson 2006) Let F be a family of local maps φ : W ⊂ G/K → C and F ∗ be the dual family consisting of the local maps φ∗ : W ∗ ⊂ U/K → C. Then F ∗ is a local

• rthogonal harmonic family on U/K if and only if F is a local orthogonal

harmonic family on G/K.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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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) → x1 + ix2 x3 + ix4 is a locally defined harmonic morphism.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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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) → x1 + ix2 x3 + ix4 is a locally defined harmonic morphism. Example 4.13 (SG 1996) The map φ : H3 ⊂ R4

1 → C given by

φ : (x1, x2, x3, x4) → x1 + ix2 x3 − x4 is a globally defined harmonic morphism.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

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Our existence result for symmetric spaces has the following interesting consequence: Theorem 4.14 (SG, Svensson 2013) Let (M, g) be a Riemannian homogeneous space of positive curvature

• ther than the Berger space Sp(2)/SU(2). Then M admits local

complex-valued harmonic morphisms.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Classifications

Fact 5.1 Every Riemannian homogeneous space (M, g) of dimension 3 or 4 is either symmetric or a Lie group with a left-invariant metric.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Classifications

Fact 5.1 Every Riemannian homogeneous space (M, g) of dimension 3 or 4 is either symmetric or a Lie group with a left-invariant metric. (SG, Svensson 2011): Give a classification for 3-dimensional Riemannian Lie groups admitting solutions. Find a continuous family of groups, containing Sol3, not carrying any left-invariant metric admitting complex-valued harmonic morphisms.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Classifications

Fact 5.1 Every Riemannian homogeneous space (M, g) of dimension 3 or 4 is either symmetric or a Lie group with a left-invariant metric. (SG, Svensson 2011): Give a classification for 3-dimensional Riemannian Lie groups admitting solutions. Find a continuous family of groups, containing Sol3, not carrying any left-invariant metric admitting complex-valued harmonic morphisms. (SG, Svensson 2013): Give a classification for 4-dimensional Riemannian Lie groups admitting left-invariant solutions. Most of the solutions constructed are NOT holomorphic with respect to any (integrable) Hermitian structure.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References Classifications

Fact 5.1 Every Riemannian homogeneous space (M, g) of dimension 3 or 4 is either symmetric or a Lie group with a left-invariant metric. (SG, Svensson 2011): Give a classification for 3-dimensional Riemannian Lie groups admitting solutions. Find a continuous family of groups, containing Sol3, not carrying any left-invariant metric admitting complex-valued harmonic morphisms. (SG, Svensson 2013): Give a classification for 4-dimensional Riemannian Lie groups admitting left-invariant solutions. Most of the solutions constructed are NOT holomorphic with respect to any (integrable) Hermitian structure. (SG 2016): Gives a large collection of 5-dimensional Riemannian Lie groups admitting left-invariant solutions.

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric

Harmonic Morphisms The Conjecture Constructions by Eigenfamilies Constructions by Orthogonal Harmonic Families Low-Dimensional Classifications References

[1] S. Gudmundsson, On the existence of harmonic morphisms from symmetric spaces of rank one, Manuscripta Math. 93 (1997), 421-433. [2] S. Gudmundsson and M. Svensson, Harmonic morphisms from the Grassmannians and their non-compact duals, Ann. Global Anal. Geom. 30 (2006), 313-333. [3] S. Gudmundsson, A. Sakovich, Harmonic morphisms from the classical compact semisimple Lie groups, Ann. Global Anal. Geom. 33 (2008), 343-356. [4] S. Gudmundsson and A. Sakovich, Harmonic morphisms from the classical non-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.

[6] S. Gudmundsson, M. Svensson, On the existence of harmonic morphisms from three-dimensional Lie groups, Contemp. Math. 542 (2011), 279-284. [7] S. Gudmundsson, M. Svensson, Harmonic morphisms from four-dimensional Lie groups, J. Geom. Phys. 83 (2014), 1-11. [8] S. Gudmundsson, Harmonic morphisms from five-dimensional Lie groups, preprint (2016). [9] S. Gudmundsson, M. Svensson, Harmonic morphisms from homogeneous spaces

• f positive curvature, Math. Proc. Cambridge Philos. Soc. (to appear).

Sigmundur Gudmundsson Harmonic Morphisms from Lie Groups and Symmetric