Lie Foliations Producing Harmonic Morphisms Sigmundur Gudmundsson - - PowerPoint PPT Presentation

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Lie Foliations Producing Harmonic Morphisms

Sigmundur Gudmundsson

Department of Mathematics Faculty of Science Lund University

Cagliari - 14 September 2018

www.matematik.lu.se/matematiklu/personal/sigma/2018-09-14-Cagliari.pdf

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Outline

1 Harmonic Morphisms

Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Outline

1 Harmonic Morphisms

Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

2 3-dimensional Lie groups

Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Outline

1 Harmonic Morphisms

Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

2 3-dimensional Lie groups

Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

3 4-dimensional Lie groups

Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Outline

1 Harmonic Morphisms

Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

2 3-dimensional Lie groups

Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

3 4-dimensional Lie groups

Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

4 5-dimensional Lie groups

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Definition 1.1 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Definition 1.1 (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 1.2 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality 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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality 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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Theorem 1.3 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Theorem 1.3 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Let φ : (M, g) → (N 2, h) be a submersive harmonic morphism from a Riemannian manifold to a surface. Then this induces a conformal foliation F on (M, g) with minimal leaves of codimension 2.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Let φ : (M, g) → (N 2, h) be a submersive harmonic morphism from a Riemannian manifold to a surface. Then this induces a conformal foliation F on (M, g) with minimal leaves of codimension 2. Let F be a conformal foliation on (M, g) with minimal leaves of codimension 2. Then F produces locally submersive harmonic morphisms from (M, g) to surfaces.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Let φ : (M, g) → (N 2, h) be a submersive harmonic morphism from a Riemannian manifold to a surface. Then this induces a conformal foliation F on (M, g) with minimal leaves of codimension 2. Let F be a conformal foliation on (M, g) with minimal leaves of codimension 2. Then F produces locally submersive harmonic morphisms from (M, g) to surfaces. Let V be the integrable subbundle of TM tangent to the fibres of F and H be its orthogonal complement. Then the second fundamental form for H is given by BH(X, Y ) = 1 2V(∇XY + ∇Y X) (X, Y ∈ H).

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Let φ : (M, g) → (N 2, h) be a submersive harmonic morphism from a Riemannian manifold to a surface. Then this induces a conformal foliation F on (M, g) with minimal leaves of codimension 2. Let F be a conformal foliation on (M, g) with minimal leaves of codimension 2. Then F produces locally submersive harmonic morphisms from (M, g) to surfaces. Let V be the integrable subbundle of TM tangent to the fibres of F and H be its orthogonal complement. Then the second fundamental form for H is given by BH(X, Y ) = 1 2V(∇XY + ∇Y X) (X, Y ∈ H). F is said to be conformal if there is a vector field V ∈ V such that BH = g ⊗ V. F is said to be Riemannian if V = 0.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Example 1.4 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Example 1.4 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Example 1.5 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Riemannian Geometry - Fuglede 1978, Ishihara 1979 Geometric Motivation - Baird-Eells 1981 Foliations - Minimality - Conformality Existence ?

Example 1.5 (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 Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

At the end of the 19th century, L. Bianchi classified the 3-dimensional real Lie algebras. They fall into nine disjoint types I-IX. Each contains a single isomorphy class except types VI and VII which are continuous families of different classes.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

At the end of the 19th century, L. Bianchi classified the 3-dimensional real Lie algebras. They fall into nine disjoint types I-IX. Each contains a single isomorphy class except types VI and VII which are continuous families of different classes. Example 2.1 (Type I) The Abelian Lie algebra R3; the corresponding simply connected Lie group is of course the Abelian group R3 which we equip with the standard flat metric.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

At the end of the 19th century, L. Bianchi classified the 3-dimensional real Lie algebras. They fall into nine disjoint types I-IX. Each contains a single isomorphy class except types VI and VII which are continuous families of different classes. Example 2.1 (Type I) The Abelian Lie algebra R3; the corresponding simply connected Lie group is of course the Abelian group R3 which we equip with the standard flat metric. Example 2.2 (Type II) The Lie algebra n3 with a basis X, Y, Z satisfying [X, Y ] = Z. The corresponding simply connected Lie group is the nilpotent Heisenberg group Nil3.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.3 (Type III) The Lie algebra h2 ⊕ R = span{X, Y, Z}, where h2 is the two-dimensional Lie algebra with basis X, Y satisfying [Y, X] = X. The corresponding simply connected Lie group is denoted by H2 × R. Here H2 is the standard hyperbolic plane.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.3 (Type III) The Lie algebra h2 ⊕ R = span{X, Y, Z}, where h2 is the two-dimensional Lie algebra with basis X, Y satisfying [Y, X] = X. The corresponding simply connected Lie group is denoted by H2 × R. Here H2 is the standard hyperbolic plane. Example 2.4 (Type IV) The Lie algebra g4 with a basis X, Y, Z satisfying [Z, X] = X, [Z, Y ] = X + Y. The corresponding simply connected Lie group is denoted by G4.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.5 (Type V) The Lie algebra h3 with a basis X, Y, Z satisfying [Z, X] = X, [Z, Y ] = Y. The corresponding simply connected Lie group H3 is the standard hyperbolic 3-space of constant sectional curvature −1

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.5 (Type V) The Lie algebra h3 with a basis X, Y, Z satisfying [Z, X] = X, [Z, Y ] = Y. The corresponding simply connected Lie group H3 is the standard hyperbolic 3-space of constant sectional curvature −1 Example 2.6 (Type VI) The Lie algebra sol3

α, where α > 0, is the Lie algebra with basis X, Y, Z

satisfying [Z, X] = αX, [Z, Y ] = −Y. The corresponding simply connected Lie group is denoted by Sol3

α. The

group Sol mentioned in the introduction is actually Sol3

1.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.7 (Type VII) The Lie algebra g7(α), where α ∈ R, is the the Lie algebra with basis X, Y, Z satisfying [Z, X] = αX − Y, [Z, Y ] = X + αY. The corresponding simply connected Lie group is denoted by G7(α).

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.7 (Type VII) The Lie algebra g7(α), where α ∈ R, is the the Lie algebra with basis X, Y, Z satisfying [Z, X] = αX − Y, [Z, Y ] = X + αY. The corresponding simply connected Lie group is denoted by G7(α). Example 2.8 (Type VIII) The Lie algebra sl2(R) with a basis X, Y, Z satisfying [X, Y ] = −2Z, [Z, X] = 2Y, [Y, Z] = 2X. The corresponding simply connected Lie group is denoted by

• SL2(R) as it

is the universal cover of the special linear group SL2(R).

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.9 (Type IX) The Lie algebra su(2) with a basis X, Y, Z satisfying [X, Y ] = 2Z, [Z, X] = 2Y, [Y, Z] = 2X. The corresponding simply connected Lie group is of course SU(2). This is isometric to the standard 3-sphere of constant curvature +1.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.9 (Type IX) The Lie algebra su(2) with a basis X, Y, Z satisfying [X, Y ] = 2Z, [Z, X] = 2Y, [Y, Z] = 2X. The corresponding simply connected Lie group is of course SU(2). This is isometric to the standard 3-sphere of constant curvature +1. Definition 2.10 (Lie Foliations) Let G be a Lie group equipped with a left-invariant Riemannian metric and K be a subgroup of G. Then the natural projection π : G → G/K induces a foliation F on G. The leaves of F are the fibres π−1(gK) of π.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.11 (G, Svensson (2009)) Let G be a connected 3-dimensional Lie group with a left-invariant metric

• f non-constant sectional curvature. Then any local conformal foliation by

geodesics of a connected open subset of G can be extended to a global conformal foliation F by geodesics of G. This is given by the left-translation

• f a 1-parameter subgroup of G i.e. F is a Lie foliation.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.11 (G, Svensson (2009)) Let G be a connected 3-dimensional Lie group with a left-invariant metric

• f non-constant sectional curvature. Then any local conformal foliation by

geodesics of a connected open subset of G can be extended to a global conformal foliation F by geodesics of G. This is given by the left-translation

• f a 1-parameter subgroup of G i.e. F is a Lie foliation.

Let G be a 3-dimensional Riemannian Lie group with Lie algebra g generated by the elements of the orthonormal basis {X, Y, Z}. Let F be a 1-dimensional conformal and minimal foliation of G generated by the left-invariant vector field Z. Then the Lie brackets of g must satisfy [Z, X] = aX + bY, [Z, Y ] = −bX + aY, [X, Y ] = xX + yY + zZ, where a, b, x, y, z ∈ R. The Jacobi identity shows that here we have a Lie algebra if and only if the following system of 3 quadratic equations in 5 variables are satisfied az = 0, ax + by = 0, bx − ay = 0.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.12 (G, Svensson (2011)) The following THREE families of 3-dimensional Lie algebras give a complete classification.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.12 (G, Svensson (2011)) The following THREE families of 3-dimensional Lie algebras give a complete classification. Example 2.13 When x = y = a = 0 we obtain a 2-dimensional family with the bracket relations [Z, X] = bY, [Z, Y ] = −bX, [X, Y ] = zZ, When bz < 0 the Lie algebra is of type VIII and of type IX if bz > 0. The case when z = 0 and b = 0 is of type VII (α = 0), and the case when b = 0 and z = 0 is of type II. The case when b = z = 0 is of type I.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.14 With a = b = 0 we yield a 3-dimensional family of Lie groups with bracket relation [X, Y ] = xX + yY + zZ. If x = y = z = 0 then the type is I. If x or y non-zero then we have type III. If z = 0 and x = y = 0, then the type is II.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Example 2.14 With a = b = 0 we yield a 3-dimensional family of Lie groups with bracket relation [X, Y ] = xX + yY + zZ. If x = y = z = 0 then the type is I. If x or y non-zero then we have type III. If z = 0 and x = y = 0, then the type is II. Example 2.15 In the case of x = y = z = 0 we get semi-direct products R2 ⋊ R with bracket relations [Z, X] = aX + bY, [Z, Y ] = −bX + aY. If b = 0 then the Lie algebra is of type VII. If b = 0 then the Lie algebra is

• f type V or of type I if also a = 0.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.16 Let G be a 3-dimensional Lie group with Lie algebra g. Then there exists a left-invariant Riemannian metric g on G and a left-invariant horizontally conformal foliation on (G, g) by geodesics if and only if the Lie algebra g is neither of type IV nor of type VI.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.16 Let G be a 3-dimensional Lie group with Lie algebra g. Then there exists a left-invariant Riemannian metric g on G and a left-invariant horizontally conformal foliation on (G, g) by geodesics if and only if the Lie algebra g is neither of type IV nor of type VI. Type IV: The single case of G4.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Bianchi’s Classification Lie Foliations Producing Harmonic Morphisms

Theorem 2.16 Let G be a 3-dimensional Lie group with Lie algebra g. Then there exists a left-invariant Riemannian metric g on G and a left-invariant horizontally conformal foliation on (G, g) by geodesics if and only if the Lie algebra g is neither of type IV nor of type VI. Type IV: The single case of G4. Type VI: The continuous family of Sol3

α, where α > 0.

Note that in the cases of type I, II, III, V, VII, VIII and IX the possible left-invariant Riemannian metrics are completely determined via isomorphisms to the standard examples.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

Let G be a 4-dimensional Riemannian Lie group with Lie algebra g generated by the elements of the orthonormal basis {X, Y, Z, W}. Let F be a 2-dimensional conformal and minimal Lie foliation of G generated by the left-invariant vector fields Z and W. Then the Lie bracket of g must satisfy the following SIX relations [W, Z] = λW, [Z, X] = αX + βY + z1Z + w1W, [Z, Y ] = −βX + αY + z2Z + w2W, [W, X] = aX + bY + z3Z − z1W, [W, Y ] = −bX + aY + z4Z − z2W, [Y, X] = rX + θ1Z + θ2W

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

Let G be a 4-dimensional Riemannian Lie group with Lie algebra g generated by the elements of the orthonormal basis {X, Y, Z, W}. Let F be a 2-dimensional conformal and minimal Lie foliation of G generated by the left-invariant vector fields Z and W. Then the Lie bracket of g must satisfy the following SIX relations [W, Z] = λW, [Z, X] = αX + βY + z1Z + w1W, [Z, Y ] = −βX + αY + z2Z + w2W, [W, X] = aX + bY + z3Z − z1W, [W, Y ] = −bX + aY + z4Z − z2W, [Y, X] = rX + θ1Z + θ2W Proposition 1 Let G be a 4-dimensional Lie group and {X, Y, Z, W} be an orthonormal basis for its Lie algebra as above. Then (i) F is totally geodesic if and only if z1 = z2 = z3 + w1 = z4 + w2 = 0, (ii) F is Riemannian if and only if α = a = 0, and (iii) H is integrable if and only if θ1 = θ2 = 0.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

= λa, = λb, = −w2z3 + w1z4 − 2αθ1 + rz1, = −2z4z1 + 2z3z2 − 2aθ1 + rz3, = −λz3 − z2b + z4β − z1a + z3α, = −λz4 − z2a + z4α + z1b − z3β, = λθ1 − w1z4 + w2z3 − 2aθ2 − rz1, = −λθ2 + 2z1w2 − 2z2w1 − 2αθ2 + rw1, = −w2a − w1b − z2α − z1β − αr, = −w2b + w1a − z2β + z1α + rβ, = λz1 − w2b − z2β − w1a − z1α, = z2a + z1b − z4α − z3β − ar, = z2b − z1a − z4β + z3α + rb, = λz2 − w2a − z2α + w1b + z1β. Maple 17: Time 0.89 seconds - Memory: 38.18 Mb

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

Theorem 3.1 (G, Svensson (2014)) Let G be a 4-dimensional Riemannian Lie group. Let F be a conformal Lie foliation on G with minimal leaves of codimension 2. Then the Lie algebra g of G belongs to one of 20 multi-dimensional families g1, . . . , g20 of Lie algebras and F is the corresponding foliation generated by g.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

An elementary calculation shows that the two Jacobi equations [[W, Z], X] + [[X, W], Z] + [[Z, X], W] = 0, (1) [[W, Z], Y ] + [[Y, W], Z] + [[Z, Y ], W] = 0 (2) are equivalent to the following relations for the real structure constants

• β

λ − α λ − α −β z1 z4 z2 −z3

• =
• .

(3)

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

An elementary calculation shows that the two Jacobi equations [[W, Z], X] + [[X, W], Z] + [[Z, X], W] = 0, (1) [[W, Z], Y ] + [[Y, W], Z] + [[Z, Y ], W] = 0 (2) are equivalent to the following relations for the real structure constants

• β

λ − α λ − α −β z1 z4 z2 −z3

• =
• .

(3) Applying equation (3) we see that z = 0 so the Lie brackets satisfy [W, Z] = λW, [Z, X] = αX + βY + w1W, [Z, Y ] = −βX + αY + w2W, [Y, X] = rX + θ1Z + θ2W.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

In this situation it is easily seen that the two remaining Jacobi equations are equivalent to θ1 = rα = rβ = 0 and θ2(λ + 2α) = rw1.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

In this situation it is easily seen that the two remaining Jacobi equations are equivalent to θ1 = rα = rβ = 0 and θ2(λ + 2α) = rw1. Example 3.2 (g1(λ, r, w1, w2)) If r = 0 then clearly α = β = 0 and rw1 = λθ2. This gives a 4-dimensional family of solutions satisfying the following Lie bracket relations [W, Z] = λW, [Z, X] = w1W, [Z, Y ] = w2W, λ[Y, X] = λrX + rw1W.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

On the other hand, if r = 0 then clearly θ1 = θ2(λ + 2α) = 0 providing us with the following two examples. Example 3.3 (g2(λ, α, β, w1, w2)) For r = θ1 = θ2 = 0 we have the family g = g2(λ, α, β, w1, w2) given by [W, Z] = λW, [Z, X] = αX + βY + w1W, [Z, Y ] = −βX + αY + w2W.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

On the other hand, if r = 0 then clearly θ1 = θ2(λ + 2α) = 0 providing us with the following two examples. Example 3.3 (g2(λ, α, β, w1, w2)) For r = θ1 = θ2 = 0 we have the family g = g2(λ, α, β, w1, w2) given by [W, Z] = λW, [Z, X] = αX + βY + w1W, [Z, Y ] = −βX + αY + w2W. Example 3.4 (g3(α, β, w1, w2, θ2)) If r = θ1 = 0 and θ2 = 0 then λ = −2α provides us with the family g = g3(α, β, w1, w2, θ2) of solutions satisfying [W, Z] = −2αW, [Z, X] = αX + βY + w1W, [Z, Y ] = −βX + αY + w2W, [Y, X] = θ2W.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

Under the assumptions that α = λ and β = 0 we have [W, Z] = λW, [Z, X] = λX + z1Z + w1W, [Z, Y ] = λY + z2Z + w2W, [W, X] = z3Z − z1W, [W, Y ] = z4Z − z2W, [Y, X] = rX + θ1Z + θ2W. The last two Jacobi equations are easily seen to be equivalent to z1 = z3 = z4 = θ1 = 0, z2 = −r and λθ2 = −z2w1. Example 3.5 (g4(λ, z2, w1, w2)) In this case we have the family of solutions given by [W, Z] = λW, [Z, X] = λX + w1W, [Z, Y ] = λY + z2Z + w2W, [W, Y ] = −z2W, λ[Y, X] = −z2λX − z2w1W.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References Lie Foliations Producing Harmonic Morphisms Case (A) - (λ = 0 and (λ − α)2 + β2 = 0) Case (B) - (λ = 0 and (λ − α)2 + β2 = 0) Case (C-F) - (λ = 0)

That story is far too long for this talk :-)

We get the following collection of 4-dimensional Lie foliations producing harmonic morphisms. g5(α, a, β, b, r), g6(z1, z2, z3, r, θ1, θ2), g7(z2, w1, w2, θ1, θ2), g8(z2, z4, w2, r, θ1, θ2), g9(z2, z3, z4, θ1, θ2), g10(α, a, β, b), g11(z1, z2, z3, w1, θ1, θ2), g12(z3, w1, w2, θ1, θ2), g13(z3, z4, θ1, θ2), g14(z2, z4, w2, θ1, θ2), g15(α, w1, w2), g16(β, w1, w2, θ1, θ2), g17(α, a, w1, w2), g18(β, b, z3, z4, θ1, θ2), g19(α, β, w1, w2), g20(α, a, β, w1, w2).

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Theorem 4.1 (k = su(2)) Let G be a 5-dimensional Riemannian Lie group carrying a conformal and minimal Lie foliation F, generated by the subalgebra su(2) of g as above. Then the Lie bracket relations are of the form [X, Y ] = 2λZ, [Z, X] = 2λY, [Y, Z] = 2X/λ. [X, A] = −λ2x3Y − λ2x5Z, [X, B] = −λ2x4Y − λ2x6Z, [Y, A] = x3X + z3Z, [Y, B] = x4X + z4, [Z, A] = x5X − z3Y, [Z, B] = x6X − z4Y, [A, B] = rA + θ1X + θ2Y + θ3Z, where θ1, θ2, θ3 are given by   θ1 θ2 θ3   = 1 2   rz3/λ + λ(x3x6 − x4x5) λ(rx5 − x3z4 + x4z3) −λ(rx3 + z4x5 − z3x6)   . The foliation F is Riemannian. It is totally geodesic if and only if λ = 1 i.e. the leaves are 3-dimensional round spheres.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

Theorem 4.2 (k = 2) Let G be a 5-dimensional Riemannian Lie group carrying a left-invariant, minimal and conformal distribution V, generated by the Riemannian subalgebra sl2(R) of g as above. Then the Lie bracket relations are of the form [X, Y ] = 2λZ, [Z, X] = 2λY, [Y, Z] = −2X/λ. [X, A] = λ2x3Y + λ2x5Z, [X, B] = λ2x4Y + λ2x6Z, [Y, A] = x3X + z3Z, [Y, B] = x4X + z4Z, [Z, A] = x5X − z3Y, [Z, B] = x6X − z4Y, [A, B] = rA + θ1X + θ2Y + θ3Z, where θ1, θ2, θ3 are given by   θ1 θ2 θ3   = 1 2   rz3/λ − λ(x3x6 − x4x5) −λ(rx5 − x3z4 + x4z3) λ(rx3 + z4x5 − z3x6)   . The corresponding foliation F is Riemannian but not totally geodesic for any values of λ ∈ R+.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms

Harmonic Morphisms 3-dimensional Lie groups 4-dimensional Lie groups 5-dimensional Lie groups References

[1] S. Gudmundsson, M. Svensson, On the existence of harmonic morphisms from three-dimensional Lie groups, Contemp. Math. 542 (2011), 279-284. [2] S. Gudmundsson, M. Svensson, Harmonic morphisms from four-dimensional Lie groups, J. Geom. Phys. 83 (2014), 1-11. [3] S. Gudmundsson, Harmonic morphisms from five-dimensional Lie groups, Geom. Dedicata 184 (2016), 143-157.

Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms