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 + β 2 � = 0) Case (B) - ( λ � = 0 and ( λ − α ) 2 + β 2 = 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 + β 2 � = 0) Case (B) - ( λ � = 0 and ( λ − α ) 2 + β 2 = 0) Case (C-F) - ( λ = 0) 4 5-dimensional Lie groups Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms
Harmonic Morphisms Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References (Harmonicity) For local coordinates x on ( M, g ) and y on ( N, h ), we have the non-linear system m m n ∂ 2 φ γ ∂φ γ αβ ◦ φ∂φ α ∂φ β � g ij � Γ k ˆ � Γ γ = 0 , τ ( φ ) = ∂x i ∂x j − ∂x k + ij ∂x i ∂x j i,j =1 k =1 α,β =1 where φ α = y α ◦ φ and ˆ Γ , Γ are the Christoffel symbols on M, N , resp. Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms
Harmonic Morphisms Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References (Harmonicity) For local coordinates x on ( M, g ) and y on ( N, h ), we have the non-linear system m m n ∂ 2 φ γ ∂φ γ αβ ◦ φ∂φ α ∂φ β � g ij � Γ k ˆ � Γ γ = 0 , τ ( φ ) = ∂x i ∂x j − ∂x k + ij ∂x i ∂x j i,j =1 k =1 α,β =1 where φ α = y α ◦ 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 g ij ( x ) ∂φ α ∂x i ( x ) ∂φ β � ∂x j ( x ) = λ 2 ( x ) h αβ ( φ ( x )) . i,j =1 � n +1 � This is a first order non-linear system of [ − 1] equations. 2 Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms
Harmonic Morphisms Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 σ 2 h 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 Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 Riemannian Geometry - Fuglede 1978, Ishihara 1979 3-dimensional Lie groups Geometric Motivation - Baird-Eells 1981 4-dimensional Lie groups Foliations - Minimality - Conformality 5-dimensional Lie groups Existence ? References 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 B H ( X, Y ) = 1 2 V ( ∇ X Y + ∇ Y X ) ( X, Y ∈ H ) . Sigmundur Gudmundsson Lie Foliations Producing Harmonic Morphisms
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