Hypo contact Anna Fino SU(2)-structures in 5-dimensions Sasakian structures Sasaki-Einstein structures Hypo structures η-Einstein structures Hypo-contact structures Classification Consequences New metrics with holonomy SU(3) Sasakian structures on Lie groups 3-dimensional Lie groups General results 5-dimensional Lie groups SU(n)-structures in (2n + 1)-dimensions Generalized Killing spinors Contact SU(n)-structures Examples Contact reduction 1 Hypo contact and Sasakian structures on Lie groups “Workshop on CR and Sasakian Geometry”, Luxembourg – 24 - 26 March 2008 Anna Fino Dipartimento di Matematica Università di Torino
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An almost contact metric structure (η, ξ, ϕ,g) on N2n+1 is saidcontact metric if 2g(X , ϕY ) = dη(X ,Y ). On N5 the pair (η, ω3)defines a contact metric structure if dη = −2ω3.
A Sasakian structure on N2n+1 is a normal contact metricstructure.
Theorem (Boyer, Galicki)
A Riemannian manifold (N2n+1,g) has a compatible Sasakianstructure if and only if the cone N2n+1 × R+ equipped with theconic metric g = dr2 + r2g is Kähler.
An almost contact metric structure (η, ξ, ϕ,g) on N2n+1 is saidcontact metric if 2g(X , ϕY ) = dη(X ,Y ). On N5 the pair (η, ω3)defines a contact metric structure if dη = −2ω3.
A Sasakian structure on N2n+1 is a normal contact metricstructure.
Theorem (Boyer, Galicki)
A Riemannian manifold (N2n+1,g) has a compatible Sasakianstructure if and only if the cone N2n+1 × R+ equipped with theconic metric g = dr2 + r2g is Kähler.
An almost contact metric structure (η, ξ, ϕ,g) on N2n+1 is saidcontact metric if 2g(X , ϕY ) = dη(X ,Y ). On N5 the pair (η, ω3)defines a contact metric structure if dη = −2ω3.
A Sasakian structure on N2n+1 is a normal contact metricstructure.
Theorem (Boyer, Galicki)
A Riemannian manifold (N2n+1,g) has a compatible Sasakianstructure if and only if the cone N2n+1 × R+ equipped with theconic metric g = dr2 + r2g is Kähler.
An almost contact metric structure (η, ξ, ϕ,g) on N2n+1 is saidcontact metric if 2g(X , ϕY ) = dη(X ,Y ). On N5 the pair (η, ω3)defines a contact metric structure if dη = −2ω3.
A Sasakian structure on N2n+1 is a normal contact metricstructure.
Theorem (Boyer, Galicki)
A Riemannian manifold (N2n+1,g) has a compatible Sasakianstructure if and only if the cone N2n+1 × R+ equipped with theconic metric g = dr2 + r2g is Kähler.
An almost contact metric structure (η, ξ, ϕ,g) on N2n+1 is saidcontact metric if 2g(X , ϕY ) = dη(X ,Y ). On N5 the pair (η, ω3)defines a contact metric structure if dη = −2ω3.
A Sasakian structure on N2n+1 is a normal contact metricstructure.
Theorem (Boyer, Galicki)
A Riemannian manifold (N2n+1,g) has a compatible Sasakianstructure if and only if the cone N2n+1 × R+ equipped with theconic metric g = dr2 + r2g is Kähler.
On S2 × S3 there exist an infinite family of explicitSasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram,...].
Definition (Boyer, Galicki)
(N2n+1,g, η) is Sasaki-Einstein if the conic metric g = dr2 + r2gon the symplectic cone N2n+1 × R+ is Kähler and Ricci- flat(CY).
•N2n+1 × R+ has an integrable SU(n + 1)-structure, i.e. anHermitian structure (J, g), with F = d(r2η), and a(n + 1,0)-form Ψ = Ψ+ + iΨ− of lenght 1 such thatdF = dΨ = 0 ⇒ g has holonomy in SU(n + 1).• N2n+1 has a real Killing spinor, i.e. the restriction of a parallelspinor on the Riemannian cone [Friedrich, Kath].
On S2 × S3 there exist an infinite family of explicitSasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram,...].
Definition (Boyer, Galicki)
(N2n+1,g, η) is Sasaki-Einstein if the conic metric g = dr2 + r2gon the symplectic cone N2n+1 × R+ is Kähler and Ricci- flat(CY).
•N2n+1 × R+ has an integrable SU(n + 1)-structure, i.e. anHermitian structure (J, g), with F = d(r2η), and a(n + 1,0)-form Ψ = Ψ+ + iΨ− of lenght 1 such thatdF = dΨ = 0 ⇒ g has holonomy in SU(n + 1).• N2n+1 has a real Killing spinor, i.e. the restriction of a parallelspinor on the Riemannian cone [Friedrich, Kath].
On S2 × S3 there exist an infinite family of explicitSasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram,...].
Definition (Boyer, Galicki)
(N2n+1,g, η) is Sasaki-Einstein if the conic metric g = dr2 + r2gon the symplectic cone N2n+1 × R+ is Kähler and Ricci- flat(CY).
•N2n+1 × R+ has an integrable SU(n + 1)-structure, i.e. anHermitian structure (J, g), with F = d(r2η), and a(n + 1,0)-form Ψ = Ψ+ + iΨ− of lenght 1 such thatdF = dΨ = 0 ⇒ g has holonomy in SU(n + 1).• N2n+1 has a real Killing spinor, i.e. the restriction of a parallelspinor on the Riemannian cone [Friedrich, Kath].
On S2 × S3 there exist an infinite family of explicitSasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram,...].
Definition (Boyer, Galicki)
(N2n+1,g, η) is Sasaki-Einstein if the conic metric g = dr2 + r2gon the symplectic cone N2n+1 × R+ is Kähler and Ricci- flat(CY).
•N2n+1 × R+ has an integrable SU(n + 1)-structure, i.e. anHermitian structure (J, g), with F = d(r2η), and a(n + 1,0)-form Ψ = Ψ+ + iΨ− of lenght 1 such thatdF = dΨ = 0 ⇒ g has holonomy in SU(n + 1).• N2n+1 has a real Killing spinor, i.e. the restriction of a parallelspinor on the Riemannian cone [Friedrich, Kath].
On S2 × S3 there exist an infinite family of explicitSasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram,...].
Definition (Boyer, Galicki)
(N2n+1,g, η) is Sasaki-Einstein if the conic metric g = dr2 + r2gon the symplectic cone N2n+1 × R+ is Kähler and Ricci- flat(CY).
•N2n+1 × R+ has an integrable SU(n + 1)-structure, i.e. anHermitian structure (J, g), with F = d(r2η), and a(n + 1,0)-form Ψ = Ψ+ + iΨ− of lenght 1 such thatdF = dΨ = 0 ⇒ g has holonomy in SU(n + 1).• N2n+1 has a real Killing spinor, i.e. the restriction of a parallelspinor on the Riemannian cone [Friedrich, Kath].
An SU(2)-structure P on N5 induces a spin structure on N5
and P extends to PSpin(5) = P ×SU(2) Spin(5).
The spinor bundle is P ×SU(2) Σ, where Σ ∼= C4 and Spin(5)acts transitively on the sphere in Σ with stabilizer SU(2) in afixed unit spinor u0 ∈ Σ.
Then the SU(2)-structures are in one-to-one correspondencewith the pairs (PSpin(5), ψ), with ψ a unit spinor such thatψ = [u,u0] for any local section u of P, i.e.ψ ∈ P ×Spin(5) (Spin(5)u0).
An SU(2)-structure P on N5 induces a spin structure on N5
and P extends to PSpin(5) = P ×SU(2) Spin(5).
The spinor bundle is P ×SU(2) Σ, where Σ ∼= C4 and Spin(5)acts transitively on the sphere in Σ with stabilizer SU(2) in afixed unit spinor u0 ∈ Σ.
Then the SU(2)-structures are in one-to-one correspondencewith the pairs (PSpin(5), ψ), with ψ a unit spinor such thatψ = [u,u0] for any local section u of P, i.e.ψ ∈ P ×Spin(5) (Spin(5)u0).
An SU(2)-structure P on N5 induces a spin structure on N5
and P extends to PSpin(5) = P ×SU(2) Spin(5).
The spinor bundle is P ×SU(2) Σ, where Σ ∼= C4 and Spin(5)acts transitively on the sphere in Σ with stabilizer SU(2) in afixed unit spinor u0 ∈ Σ.
Then the SU(2)-structures are in one-to-one correspondencewith the pairs (PSpin(5), ψ), with ψ a unit spinor such thatψ = [u,u0] for any local section u of P, i.e.ψ ∈ P ×Spin(5) (Spin(5)u0).
• Any oriented hypersurface N5 of (M6,F ,Ψ) with an integrableSU(3)-structure (F ,Ψ) has in a natural way a hypo structure.The generalized Killing spinor ψ on N5 is the restriction of theparallel spinor on M6 and O is just given by the Weingartenoperator. If ψ is the restriction of a parallel spinor over theRiemannian cone then O is a constant multiple of the identity.
• Any oriented hypersurface N5 of (M6,F ,Ψ) with an integrableSU(3)-structure (F ,Ψ) has in a natural way a hypo structure.The generalized Killing spinor ψ on N5 is the restriction of theparallel spinor on M6 and O is just given by the Weingartenoperator. If ψ is the restriction of a parallel spinor over theRiemannian cone then O is a constant multiple of the identity.
• Any oriented hypersurface N5 of (M6,F ,Ψ) with an integrableSU(3)-structure (F ,Ψ) has in a natural way a hypo structure.The generalized Killing spinor ψ on N5 is the restriction of theparallel spinor on M6 and O is just given by the Weingartenoperator. If ψ is the restriction of a parallel spinor over theRiemannian cone then O is a constant multiple of the identity.
A real analytic hypo structure (η, ωi) on N5 determines anintegrable SU(3)-structure on N5 × I, with I some openinterval, if (η, ωi) belongs to a one-parameter family of hypostructures (η(t), ωi(t)) which satisfy the evolution equations
A real analytic hypo structure (η, ωi) on N5 determines anintegrable SU(3)-structure on N5 × I, with I some openinterval, if (η, ωi) belongs to a one-parameter family of hypostructures (η(t), ωi(t)) which satisfy the evolution equations
Studying the Conti-Salamon evolution equations for the left-invarianthypo-contact structures on the simply-connected solvable Lie groupsGi (1 ≤ i ≤ 5) with Lie algebra gi :
Theorem (De Andres, Fernandez, –, Ugarte)
Any left-invariant hypo-contact structure on any Gi (1 ≤ i ≤ 5)determines a Riemannian metric with holonomy SU(3) onGi × I, for some open interval I.
For the nilpotent Lie group G1 we get the metric found byGibbons, Lü, Pope and Stelle.
Studying the Conti-Salamon evolution equations for the left-invarianthypo-contact structures on the simply-connected solvable Lie groupsGi (1 ≤ i ≤ 5) with Lie algebra gi :
Theorem (De Andres, Fernandez, –, Ugarte)
Any left-invariant hypo-contact structure on any Gi (1 ≤ i ≤ 5)determines a Riemannian metric with holonomy SU(3) onGi × I, for some open interval I.
For the nilpotent Lie group G1 we get the metric found byGibbons, Lü, Pope and Stelle.
Studying the Conti-Salamon evolution equations for the left-invarianthypo-contact structures on the simply-connected solvable Lie groupsGi (1 ≤ i ≤ 5) with Lie algebra gi :
Theorem (De Andres, Fernandez, –, Ugarte)
Any left-invariant hypo-contact structure on any Gi (1 ≤ i ≤ 5)determines a Riemannian metric with holonomy SU(3) onGi × I, for some open interval I.
For the nilpotent Lie group G1 we get the metric found byGibbons, Lü, Pope and Stelle.
Studying the Conti-Salamon evolution equations for the left-invarianthypo-contact structures on the simply-connected solvable Lie groupsGi (1 ≤ i ≤ 5) with Lie algebra gi :
Theorem (De Andres, Fernandez, –, Ugarte)
Any left-invariant hypo-contact structure on any Gi (1 ≤ i ≤ 5)determines a Riemannian metric with holonomy SU(3) onGi × I, for some open interval I.
For the nilpotent Lie group G1 we get the metric found byGibbons, Lü, Pope and Stelle.
A Sasakian manifold (N2n+1, η, ξ, ϕ,g) is called homogeneousSasakian if (η, ξ, ϕ,g) are invariant under the group ofisometries acting transitively on the manifold.
Theorem (Perrone)
A homogeneous 3-dimensional Sasakian manifold has to be aLie group endowed with a left-invariant Sasakian structure.
Theorem (Geiges, Cho-Chung)
Any 3-dimensional Sasakian Lie algebra is isomorphic to oneof the following: su(2), sl(2,R), aff(R)× R, h3, where aff(R) isthe Lie algebra of the Lie group of affine motions of R.
Problem
Classify 5-dimensional Lie groups with a left-invariant Sasakianstructure.
A Sasakian manifold (N2n+1, η, ξ, ϕ,g) is called homogeneousSasakian if (η, ξ, ϕ,g) are invariant under the group ofisometries acting transitively on the manifold.
Theorem (Perrone)
A homogeneous 3-dimensional Sasakian manifold has to be aLie group endowed with a left-invariant Sasakian structure.
Theorem (Geiges, Cho-Chung)
Any 3-dimensional Sasakian Lie algebra is isomorphic to oneof the following: su(2), sl(2,R), aff(R)× R, h3, where aff(R) isthe Lie algebra of the Lie group of affine motions of R.
Problem
Classify 5-dimensional Lie groups with a left-invariant Sasakianstructure.
A Sasakian manifold (N2n+1, η, ξ, ϕ,g) is called homogeneousSasakian if (η, ξ, ϕ,g) are invariant under the group ofisometries acting transitively on the manifold.
Theorem (Perrone)
A homogeneous 3-dimensional Sasakian manifold has to be aLie group endowed with a left-invariant Sasakian structure.
Theorem (Geiges, Cho-Chung)
Any 3-dimensional Sasakian Lie algebra is isomorphic to oneof the following: su(2), sl(2,R), aff(R)× R, h3, where aff(R) isthe Lie algebra of the Lie group of affine motions of R.
Problem
Classify 5-dimensional Lie groups with a left-invariant Sasakianstructure.
A Sasakian manifold (N2n+1, η, ξ, ϕ,g) is called homogeneousSasakian if (η, ξ, ϕ,g) are invariant under the group ofisometries acting transitively on the manifold.
Theorem (Perrone)
A homogeneous 3-dimensional Sasakian manifold has to be aLie group endowed with a left-invariant Sasakian structure.
Theorem (Geiges, Cho-Chung)
Any 3-dimensional Sasakian Lie algebra is isomorphic to oneof the following: su(2), sl(2,R), aff(R)× R, h3, where aff(R) isthe Lie algebra of the Lie group of affine motions of R.
Problem
Classify 5-dimensional Lie groups with a left-invariant Sasakianstructure.
A Sasakian manifold (N2n+1, η, ξ, ϕ,g) is called homogeneousSasakian if (η, ξ, ϕ,g) are invariant under the group ofisometries acting transitively on the manifold.
Theorem (Perrone)
A homogeneous 3-dimensional Sasakian manifold has to be aLie group endowed with a left-invariant Sasakian structure.
Theorem (Geiges, Cho-Chung)
Any 3-dimensional Sasakian Lie algebra is isomorphic to oneof the following: su(2), sl(2,R), aff(R)× R, h3, where aff(R) isthe Lie algebra of the Lie group of affine motions of R.
Problem
Classify 5-dimensional Lie groups with a left-invariant Sasakianstructure.
Let (g, η, ξ) be a contact Lie algebra. Then dim z(g) ≤ 1.
Proposition (Andrada,–, Vezzoni)
Let (g, η, ξ, ϕ,g) be a Sasakian Lie algebra.• If dim z(g) = 1, then z(g) = R ξ and (ker η, θ, ϕ,g) is a KählerLie algebra, where θ is the component of the Lie bracket of gon ker η.• If z(g) = 0, then adξϕ = ϕ adξ, and one has the orthogonaldecomposition
g = ker adξ ⊕ (Im adξ).
If g is a (2n + 1)-dimensional Sasakian nilpotent Lie algebra,then g ∼= h2n+1.
Let (g, η, ξ) be a contact Lie algebra. Then dim z(g) ≤ 1.
Proposition (Andrada,–, Vezzoni)
Let (g, η, ξ, ϕ,g) be a Sasakian Lie algebra.• If dim z(g) = 1, then z(g) = R ξ and (ker η, θ, ϕ,g) is a KählerLie algebra, where θ is the component of the Lie bracket of gon ker η.• If z(g) = 0, then adξϕ = ϕ adξ, and one has the orthogonaldecomposition
g = ker adξ ⊕ (Im adξ).
If g is a (2n + 1)-dimensional Sasakian nilpotent Lie algebra,then g ∼= h2n+1.
Let (g, η, ξ) be a contact Lie algebra. Then dim z(g) ≤ 1.
Proposition (Andrada,–, Vezzoni)
Let (g, η, ξ, ϕ,g) be a Sasakian Lie algebra.• If dim z(g) = 1, then z(g) = R ξ and (ker η, θ, ϕ,g) is a KählerLie algebra, where θ is the component of the Lie bracket of gon ker η.• If z(g) = 0, then adξϕ = ϕ adξ, and one has the orthogonaldecomposition
g = ker adξ ⊕ (Im adξ).
If g is a (2n + 1)-dimensional Sasakian nilpotent Lie algebra,then g ∼= h2n+1.
Let (g, η, ξ) be a contact Lie algebra. Then dim z(g) ≤ 1.
Proposition (Andrada,–, Vezzoni)
Let (g, η, ξ, ϕ,g) be a Sasakian Lie algebra.• If dim z(g) = 1, then z(g) = R ξ and (ker η, θ, ϕ,g) is a KählerLie algebra, where θ is the component of the Lie bracket of gon ker η.• If z(g) = 0, then adξϕ = ϕ adξ, and one has the orthogonaldecomposition
g = ker adξ ⊕ (Im adξ).
If g is a (2n + 1)-dimensional Sasakian nilpotent Lie algebra,then g ∼= h2n+1.
Let g be a 5-dimensional Sasakian Lie algebra. Then1 if z(g) 6= 0, g is solvable with dim z(g) = 1 and the
quotient g/z(g) carries an induced Kähler structure;2 if z(g) = 0, g is isomorphic to sl(2,R)× aff(R), or
su(2)× aff(R), or g3 ∼= R2 n h3.
• g is either solvable or a direct sum.• A 5-dimensional Sasakian solvmanifold is either a compactquotient of H5 or of R n (H3 × R) with structure equations
(0,−13,12,0,14 + 23).
• A 5-dimensional Sasakian η-Einstein Lie algebra isisomorphic either to g1 ∼= h5, or g3 or to sl(2,R)× aff(R).
Let g be a 5-dimensional Sasakian Lie algebra. Then1 if z(g) 6= 0, g is solvable with dim z(g) = 1 and the
quotient g/z(g) carries an induced Kähler structure;2 if z(g) = 0, g is isomorphic to sl(2,R)× aff(R), or
su(2)× aff(R), or g3 ∼= R2 n h3.
• g is either solvable or a direct sum.• A 5-dimensional Sasakian solvmanifold is either a compactquotient of H5 or of R n (H3 × R) with structure equations
(0,−13,12,0,14 + 23).
• A 5-dimensional Sasakian η-Einstein Lie algebra isisomorphic either to g1 ∼= h5, or g3 or to sl(2,R)× aff(R).
Let g be a 5-dimensional Sasakian Lie algebra. Then1 if z(g) 6= 0, g is solvable with dim z(g) = 1 and the
quotient g/z(g) carries an induced Kähler structure;2 if z(g) = 0, g is isomorphic to sl(2,R)× aff(R), or
su(2)× aff(R), or g3 ∼= R2 n h3.
• g is either solvable or a direct sum.• A 5-dimensional Sasakian solvmanifold is either a compactquotient of H5 or of R n (H3 × R) with structure equations
(0,−13,12,0,14 + 23).
• A 5-dimensional Sasakian η-Einstein Lie algebra isisomorphic either to g1 ∼= h5, or g3 or to sl(2,R)× aff(R).
Let g be a 5-dimensional Sasakian Lie algebra. Then1 if z(g) 6= 0, g is solvable with dim z(g) = 1 and the
quotient g/z(g) carries an induced Kähler structure;2 if z(g) = 0, g is isomorphic to sl(2,R)× aff(R), or
su(2)× aff(R), or g3 ∼= R2 n h3.
• g is either solvable or a direct sum.• A 5-dimensional Sasakian solvmanifold is either a compactquotient of H5 or of R n (H3 × R) with structure equations
(0,−13,12,0,14 + 23).
• A 5-dimensional Sasakian η-Einstein Lie algebra isisomorphic either to g1 ∼= h5, or g3 or to sl(2,R)× aff(R).
As for the case of SU(2)-structures in dimensions 5 we havethat an SU(n)-structure PSU on N2n+1 induces a spin structurePSpin and if we fix a unit element u0 ∈ Σ = (C2)⊗2n we have that
PSU = u ∈ PSpin | [u,u0] = ψ.
The pair (η, φ) defines a U(n)-structure or an almost contactmetric structure on N2n+1.The U(n)-structure is a contact metric structure if dη = −2φ.
As for the case of SU(2)-structures in dimensions 5 we havethat an SU(n)-structure PSU on N2n+1 induces a spin structurePSpin and if we fix a unit element u0 ∈ Σ = (C2)⊗2n we have that
PSU = u ∈ PSpin | [u,u0] = ψ.
The pair (η, φ) defines a U(n)-structure or an almost contactmetric structure on N2n+1.The U(n)-structure is a contact metric structure if dη = −2φ.
As for the case of SU(2)-structures in dimensions 5 we havethat an SU(n)-structure PSU on N2n+1 induces a spin structurePSpin and if we fix a unit element u0 ∈ Σ = (C2)⊗2n we have that
PSU = u ∈ PSpin | [u,u0] = ψ.
The pair (η, φ) defines a U(n)-structure or an almost contactmetric structure on N2n+1.The U(n)-structure is a contact metric structure if dη = −2φ.
As for the case of SU(2)-structures in dimensions 5 we havethat an SU(n)-structure PSU on N2n+1 induces a spin structurePSpin and if we fix a unit element u0 ∈ Σ = (C2)⊗2n we have that
PSU = u ∈ PSpin | [u,u0] = ψ.
The pair (η, φ) defines a U(n)-structure or an almost contactmetric structure on N2n+1.The U(n)-structure is a contact metric structure if dη = −2φ.
N2n+1 → M2n+2 (with holonomy in SU(n + 1)).Then the restriction of the parallel spinor defines anSU(n)-structure (η, φ,Ω) where the forms φ and Ω ∧ η are thepull-back of the Kähler form and the complex volume form onthe CY manifold M2n+2.
Proposition (Conti, –)
Let N2n+1 be a real analytic manifold with a real analytcSU(n)-structure PSU defined by (η, φ,Ω). The following areequivalent:
1 The spinor ψ associated to PSU is a generalized Killingspinor, i.e. ∇Xψ = 1
2 O(X ) · ψ.2 dφ = 0 and d(η ∧ Ω) = 0.3 A neighbourhood of M × 0 in M × R has a CY structure
N2n+1 → M2n+2 (with holonomy in SU(n + 1)).Then the restriction of the parallel spinor defines anSU(n)-structure (η, φ,Ω) where the forms φ and Ω ∧ η are thepull-back of the Kähler form and the complex volume form onthe CY manifold M2n+2.
Proposition (Conti, –)
Let N2n+1 be a real analytic manifold with a real analytcSU(n)-structure PSU defined by (η, φ,Ω). The following areequivalent:
1 The spinor ψ associated to PSU is a generalized Killingspinor, i.e. ∇Xψ = 1
2 O(X ) · ψ.2 dφ = 0 and d(η ∧ Ω) = 0.3 A neighbourhood of M × 0 in M × R has a CY structure
N2n+1 → M2n+2 (with holonomy in SU(n + 1)).Then the restriction of the parallel spinor defines anSU(n)-structure (η, φ,Ω) where the forms φ and Ω ∧ η are thepull-back of the Kähler form and the complex volume form onthe CY manifold M2n+2.
Proposition (Conti, –)
Let N2n+1 be a real analytic manifold with a real analytcSU(n)-structure PSU defined by (η, φ,Ω). The following areequivalent:
1 The spinor ψ associated to PSU is a generalized Killingspinor, i.e. ∇Xψ = 1
2 O(X ) · ψ.2 dφ = 0 and d(η ∧ Ω) = 0.3 A neighbourhood of M × 0 in M × R has a CY structure
The assumption of real analycity is certainly necessary toprove that (1) or (2) implies (3), but the fact that (1) implies (2)does not require this hypothesis.(2) ⇒ (3) can be described in terms of evolution equations inthe sense of Hitchin. Indeed, suppose that there is a family(η(t), φ(t),Ω(t)) of SU(n)-structures on N2n+1, with t in someinterval I, then the forms
η(t) ∧ dt + φ(t), (η(t) + idt) ∧ Ω(t)
define a CY structure on N2n+1 × I if and only if (2) holds fort = 0 and the evolution equations
The assumption of real analycity is certainly necessary toprove that (1) or (2) implies (3), but the fact that (1) implies (2)does not require this hypothesis.(2) ⇒ (3) can be described in terms of evolution equations inthe sense of Hitchin. Indeed, suppose that there is a family(η(t), φ(t),Ω(t)) of SU(n)-structures on N2n+1, with t in someinterval I, then the forms
η(t) ∧ dt + φ(t), (η(t) + idt) ∧ Ω(t)
define a CY structure on N2n+1 × I if and only if (2) holds fort = 0 and the evolution equations
The assumption of real analycity is certainly necessary toprove that (1) or (2) implies (3), but the fact that (1) implies (2)does not require this hypothesis.(2) ⇒ (3) can be described in terms of evolution equations inthe sense of Hitchin. Indeed, suppose that there is a family(η(t), φ(t),Ω(t)) of SU(n)-structures on N2n+1, with t in someinterval I, then the forms
η(t) ∧ dt + φ(t), (η(t) + idt) ∧ Ω(t)
define a CY structure on N2n+1 × I if and only if (2) holds fort = 0 and the evolution equations
An SU(n)-structure (η, φ,Ω) on N2n+1 is contact if dη = −2φ.
In this case N2n+1 is contact metric with contact form η and wemay consider the symplectic cone over (N2n+1, η) as thesymplectic manifold (N2n+1 × R+,− 1
2 d(r2η)).
If N2n+1 is Sasaki-Einstein, we know that the symplectic coneis CY with the cone metric r2g + dr2 and the Kähler form equalto the conical symplectic form.
Problem
If one thinks the form φ as the pullback to N2n+1 ∼= N2n+1 × 1of the conical symplectic form, which types of contactSU(n)-structures give rise to a CY symplectic cone but notnecessarily with respect to the cone metric?
An SU(n)-structure (η, φ,Ω) on N2n+1 is contact if dη = −2φ.
In this case N2n+1 is contact metric with contact form η and wemay consider the symplectic cone over (N2n+1, η) as thesymplectic manifold (N2n+1 × R+,− 1
2 d(r2η)).
If N2n+1 is Sasaki-Einstein, we know that the symplectic coneis CY with the cone metric r2g + dr2 and the Kähler form equalto the conical symplectic form.
Problem
If one thinks the form φ as the pullback to N2n+1 ∼= N2n+1 × 1of the conical symplectic form, which types of contactSU(n)-structures give rise to a CY symplectic cone but notnecessarily with respect to the cone metric?
An SU(n)-structure (η, φ,Ω) on N2n+1 is contact if dη = −2φ.
In this case N2n+1 is contact metric with contact form η and wemay consider the symplectic cone over (N2n+1, η) as thesymplectic manifold (N2n+1 × R+,− 1
2 d(r2η)).
If N2n+1 is Sasaki-Einstein, we know that the symplectic coneis CY with the cone metric r2g + dr2 and the Kähler form equalto the conical symplectic form.
Problem
If one thinks the form φ as the pullback to N2n+1 ∼= N2n+1 × 1of the conical symplectic form, which types of contactSU(n)-structures give rise to a CY symplectic cone but notnecessarily with respect to the cone metric?
• A two-parameter family of examples in the sphere bundle inTCP2 [Conti].• A 7-dimensional compact example, quotient of the Lie groupSU(2) n R4, which has a weakly integrable generalizedG2-structure [–, Tomassini].
• A two-parameter family of examples in the sphere bundle inTCP2 [Conti].• A 7-dimensional compact example, quotient of the Lie groupSU(2) n R4, which has a weakly integrable generalizedG2-structure [–, Tomassini].
• A two-parameter family of examples in the sphere bundle inTCP2 [Conti].• A 7-dimensional compact example, quotient of the Lie groupSU(2) n R4, which has a weakly integrable generalizedG2-structure [–, Tomassini].
• A two-parameter family of examples in the sphere bundle inTCP2 [Conti].• A 7-dimensional compact example, quotient of the Lie groupSU(2) n R4, which has a weakly integrable generalizedG2-structure [–, Tomassini].
H: compact Lie groupρ a representation of H on V .Then H nρ V has a left-invariant contact structure if and only ifH nρ V is either SU(2) n R4 or U(1) n C.
Then, if H is compact, the example SU(2) n R4 is unique indimensions > 3.If H is solvable we have
Proposition (Conti, –)
H : 3-dimensional solvable Lie group.There exists H n R4 admitting a contact SU(3)-structure whoseassociated spinor is generalized Killing if and only if the Liealgebra of H is isomorphic to one of the following
H: compact Lie groupρ a representation of H on V .Then H nρ V has a left-invariant contact structure if and only ifH nρ V is either SU(2) n R4 or U(1) n C.
Then, if H is compact, the example SU(2) n R4 is unique indimensions > 3.If H is solvable we have
Proposition (Conti, –)
H : 3-dimensional solvable Lie group.There exists H n R4 admitting a contact SU(3)-structure whoseassociated spinor is generalized Killing if and only if the Liealgebra of H is isomorphic to one of the following
H: compact Lie groupρ a representation of H on V .Then H nρ V has a left-invariant contact structure if and only ifH nρ V is either SU(2) n R4 or U(1) n C.
Then, if H is compact, the example SU(2) n R4 is unique indimensions > 3.If H is solvable we have
Proposition (Conti, –)
H : 3-dimensional solvable Lie group.There exists H n R4 admitting a contact SU(3)-structure whoseassociated spinor is generalized Killing if and only if the Liealgebra of H is isomorphic to one of the following
Let N2n+1 be a (2n + 1)-dimensional manifold endowed with acontact metric structure (η, φ,g) and a spin structurecompatible with the metric g and the orientation.We say that a spinor ψ on N2n+1 is compatible if
η · ψ = i2n+1ψ, φ · ψ = −niψ.
Suppose that S1 acts on N2n+1 preserving both metric andcontact form, so that the fundamental vector field X satisfies
LXη = 0 = LXφ.
and denote by t its norm.The moment map is given by µ = η(X ).
Let N2n+1 be a (2n + 1)-dimensional manifold endowed with acontact metric structure (η, φ,g) and a spin structurecompatible with the metric g and the orientation.We say that a spinor ψ on N2n+1 is compatible if
η · ψ = i2n+1ψ, φ · ψ = −niψ.
Suppose that S1 acts on N2n+1 preserving both metric andcontact form, so that the fundamental vector field X satisfies
LXη = 0 = LXφ.
and denote by t its norm.The moment map is given by µ = η(X ).
Let N2n+1 be a (2n + 1)-dimensional manifold endowed with acontact metric structure (η, φ,g) and a spin structurecompatible with the metric g and the orientation.We say that a spinor ψ on N2n+1 is compatible if
η · ψ = i2n+1ψ, φ · ψ = −niψ.
Suppose that S1 acts on N2n+1 preserving both metric andcontact form, so that the fundamental vector field X satisfies
LXη = 0 = LXφ.
and denote by t its norm.The moment map is given by µ = η(X ).
Let N2n+1 be a (2n + 1)-dimensional manifold endowed with acontact metric structure (η, φ,g) and a spin structurecompatible with the metric g and the orientation.We say that a spinor ψ on N2n+1 is compatible if
η · ψ = i2n+1ψ, φ · ψ = −niψ.
Suppose that S1 acts on N2n+1 preserving both metric andcontact form, so that the fundamental vector field X satisfies
LXη = 0 = LXφ.
and denote by t its norm.The moment map is given by µ = η(X ).
N2n+1 with a contact U(n)-structure (g, η, φ) and a compatiblegeneralized Killing spinor ψ.Suppose that S1 acts on N2n+1 preserving both structure andspinor and acts freely on µ−1(0) with 0 regular value. Then theinduced spinor ψπ on N2n+1//S1 is generalized Killing if andonly if at each point of µ−1(0) we have
dt ∈ span < iXφ, η >,
where X is the fundamental vector field associated to theS1-action, and t is the norm of X.
Example
If we apply the previous theorem to SU(2) n R4 we get a newhypo-contact structure on S2 × T3.
N2n+1 with a contact U(n)-structure (g, η, φ) and a compatiblegeneralized Killing spinor ψ.Suppose that S1 acts on N2n+1 preserving both structure andspinor and acts freely on µ−1(0) with 0 regular value. Then theinduced spinor ψπ on N2n+1//S1 is generalized Killing if andonly if at each point of µ−1(0) we have
dt ∈ span < iXφ, η >,
where X is the fundamental vector field associated to theS1-action, and t is the norm of X.
Example
If we apply the previous theorem to SU(2) n R4 we get a newhypo-contact structure on S2 × T3.
N2n+1 with a contact U(n)-structure (g, η, φ) and a compatiblegeneralized Killing spinor ψ.Suppose that S1 acts on N2n+1 preserving both structure andspinor and acts freely on µ−1(0) with 0 regular value. Then theinduced spinor ψπ on N2n+1//S1 is generalized Killing if andonly if at each point of µ−1(0) we have
dt ∈ span < iXφ, η >,
where X is the fundamental vector field associated to theS1-action, and t is the norm of X.
Example
If we apply the previous theorem to SU(2) n R4 we get a newhypo-contact structure on S2 × T3.
From a result by Grantcharov and Ornea the contact reductionof a η-Einstein-Sasaki structure is Sasaki.
Corollary (Conti, –)
N2n+1 with an η-Einstein-Sasaki structure (g, η, φ, ψ), and letS1 act on M preserving the structure in such a way that 0 is aregular value for the moment map µ and S1 acts freely onµ−1(0). Then the Sasaki quotient M//S1 is also η-Einstein ifand only if
From a result by Grantcharov and Ornea the contact reductionof a η-Einstein-Sasaki structure is Sasaki.
Corollary (Conti, –)
N2n+1 with an η-Einstein-Sasaki structure (g, η, φ, ψ), and letS1 act on M preserving the structure in such a way that 0 is aregular value for the moment map µ and S1 acts freely onµ−1(0). Then the Sasaki quotient M//S1 is also η-Einstein ifand only if
From a result by Grantcharov and Ornea the contact reductionof a η-Einstein-Sasaki structure is Sasaki.
Corollary (Conti, –)
N2n+1 with an η-Einstein-Sasaki structure (g, η, φ, ψ), and letS1 act on M preserving the structure in such a way that 0 is aregular value for the moment map µ and S1 acts freely onµ−1(0). Then the Sasaki quotient M//S1 is also η-Einstein ifand only if