CR Structures On Three Dimensional Contact Manifoldsth2109/ecstatic/slides/Hatice Coban.pdfCR...
Transcript of CR Structures On Three Dimensional Contact Manifoldsth2109/ecstatic/slides/Hatice Coban.pdfCR...
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
CR Structures On Three Dimensional ContactManifolds
Hatice Coban
Middle East Technical University
June 11, 2015
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Presentation Outline
1 Motivation For CR Structures
2 What is a CR manifold?
3 The Main Theorem
4 Proof of the theorem
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The subject arose from a paper of Poincare in which heconsidered whether the Riemann Mapping Theorem could begeneralized from C1 to C2. He showed that the analogousresult does not hold in C2 by using two hypersurfaces of C2 arenot equivalent.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Presentation Outline
1 Motivation For CR Structures
2 What is a CR manifold?
3 The Main Theorem
4 Proof of the theorem
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Different authors use different CR definitions. We use thefollowing definition.
Definition
A CR structure on an odd dimensional manifold M2n+1 is agerm of a complex structure J on O(M × 0) ⊂ M × R
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The weaker and commonly known CR definitions are following:
Embedded CR manifolds: Hypersurface type CR structures.They are hypersurfaces of Cn.Abstract CR manifolds:
1 (M,V ) is a CR manifold if dimM = 2n + 1, V is asubbundle of C⊗ TM with complex dimension dimV = n,V ∩ V = 0 and [V ,V ] ⊂ V .
2 (M,H, J) is a CR manifold if dimM = 2n + 1, H is a realsubbundle of TM with dimH = 2n, J : H → H andJ2 = −Id .If X and Y are in H , then so is [JX ,Y ] + [X , JY ] and
J [JX ,Y ] + [X , JY ] = [JX , JY ]− [X ,Y ]
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The weaker and commonly known CR definitions are following:Embedded CR manifolds: Hypersurface type CR structures.They are hypersurfaces of Cn.
Abstract CR manifolds:
1 (M,V ) is a CR manifold if dimM = 2n + 1, V is asubbundle of C⊗ TM with complex dimension dimV = n,V ∩ V = 0 and [V ,V ] ⊂ V .
2 (M,H, J) is a CR manifold if dimM = 2n + 1, H is a realsubbundle of TM with dimH = 2n, J : H → H andJ2 = −Id .If X and Y are in H , then so is [JX ,Y ] + [X , JY ] and
J [JX ,Y ] + [X , JY ] = [JX , JY ]− [X ,Y ]
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The weaker and commonly known CR definitions are following:Embedded CR manifolds: Hypersurface type CR structures.They are hypersurfaces of Cn.Abstract CR manifolds:
1 (M,V ) is a CR manifold if dimM = 2n + 1, V is asubbundle of C⊗ TM with complex dimension dimV = n,V ∩ V = 0 and [V ,V ] ⊂ V .
2 (M,H, J) is a CR manifold if dimM = 2n + 1, H is a realsubbundle of TM with dimH = 2n, J : H → H andJ2 = −Id .If X and Y are in H , then so is [JX ,Y ] + [X , JY ] and
J [JX ,Y ] + [X , JY ] = [JX , JY ]− [X ,Y ]
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The weaker and commonly known CR definitions are following:Embedded CR manifolds: Hypersurface type CR structures.They are hypersurfaces of Cn.Abstract CR manifolds:
1 (M,V ) is a CR manifold if dimM = 2n + 1, V is asubbundle of C⊗ TM with complex dimension dimV = n,V ∩ V = 0 and [V ,V ] ⊂ V .
2 (M,H, J) is a CR manifold if dimM = 2n + 1, H is a realsubbundle of TM with dimH = 2n, J : H → H andJ2 = −Id .If X and Y are in H , then so is [JX ,Y ] + [X , JY ] and
J [JX ,Y ] + [X , JY ] = [JX , JY ]− [X ,Y ]
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
If (M, ξ, J) are real analytic and satisfy the equation
X ,Y ∈ ξ ⇒ [JX , JY ]− [X ,Y ] = J([JX ,Y ] + [X , JY ]
)∈ ξ
where J = J|ξ on ξ, then J extends to an integrable complexstructure on O(M × 0) ⊂ M × R.
Then the weaker deinition of CR implies our definition.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Presentation Outline
1 Motivation For CR Structures
2 What is a CR manifold?
3 The Main Theorem
4 Proof of the theorem
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Theorem (Y. Ozan, H. Coban)
Let M be a smooth orientable closed 3-manifold and ξ acontact structure on M. Then M admits a CR structure thatinduces a contact structure, which is isotopic to ξ.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
(M2n+1, ξ) is a contact manifold if ker(α) = ξ andα ∧ (dα)n 6= 0.
(W 2n, ω) is a symplectic manifold if dω = 0 and ωn 6= 0.
Let (N, ξ), ξ = ker(α) be a compact contact manifoldthen (N × R, d(etα)) its symplectization.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
(M2n+1, ξ) is a contact manifold if ker(α) = ξ andα ∧ (dα)n 6= 0.
(W 2n, ω) is a symplectic manifold if dω = 0 and ωn 6= 0.
Let (N, ξ), ξ = ker(α) be a compact contact manifoldthen (N × R, d(etα)) its symplectization.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
(M2n+1, ξ) is a contact manifold if ker(α) = ξ andα ∧ (dα)n 6= 0.
(W 2n, ω) is a symplectic manifold if dω = 0 and ωn 6= 0.
Let (N, ξ), ξ = ker(α) be a compact contact manifoldthen (N × R, d(etα)) its symplectization.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
In the proof, we use the fact that
1 Any closed smooth manifold is diffeomorphic to anonsingular real algebraic set. Indeed more is true: IfM ⊂ Rn is a closed smooth submnaifold then it is isotopicto a nonsingular real algebraic set in Rn+1. Then we mayassume that any closed smooth manifold has a realanalytic structure.
2 Tangent bundle of a nonsingular real algebraic variety isstrongly algebraic. Differential forms on a real algebraicvariety can be approximated by regular, in particular realanalytic forms.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
In the proof, we use the fact that
1 Any closed smooth manifold is diffeomorphic to anonsingular real algebraic set. Indeed more is true: IfM ⊂ Rn is a closed smooth submnaifold then it is isotopicto a nonsingular real algebraic set in Rn+1. Then we mayassume that any closed smooth manifold has a realanalytic structure.
2 Tangent bundle of a nonsingular real algebraic variety isstrongly algebraic. Differential forms on a real algebraicvariety can be approximated by regular, in particular realanalytic forms.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Presentation Outline
1 Motivation For CR Structures
2 What is a CR manifold?
3 The Main Theorem
4 Proof of the theorem
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Let X be the real algebraic set which is diffeomorphic to thesmooth manifold M. Since M embeds into R5 as a smoothmanifold, we may assume that X is an algebraic subset of R6.
Let S1 = (x , y) | x2 + y2 = 1 be the unit circle and considerthe product real algebraic set X × S1 ⊂ R8.From now on we identify M with X . Let α ∈ Ω1(X ) be asmooth 1-form so that ξ = ker(α). Using the coordinate chart
φ : (−1/2, 1/2)→ U ⊂ S1, φ(t) = e2πit , t ∈ (−1/2, 1/2),
define a smooth function f : S1 → R so that f (t) = et if|t| < 1/4 and f (t) = 0 if |t| ≥ 1/3. Then the 2-formω = d(f (t)α) ∈ Ω2(X × S1) is a symplectization of thecontact structure in a neighborhood of X in X × S1.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Let X be the real algebraic set which is diffeomorphic to thesmooth manifold M. Since M embeds into R5 as a smoothmanifold, we may assume that X is an algebraic subset of R6.Let S1 = (x , y) | x2 + y2 = 1 be the unit circle and considerthe product real algebraic set X × S1 ⊂ R8.
From now on we identify M with X . Let α ∈ Ω1(X ) be asmooth 1-form so that ξ = ker(α). Using the coordinate chart
φ : (−1/2, 1/2)→ U ⊂ S1, φ(t) = e2πit , t ∈ (−1/2, 1/2),
define a smooth function f : S1 → R so that f (t) = et if|t| < 1/4 and f (t) = 0 if |t| ≥ 1/3. Then the 2-formω = d(f (t)α) ∈ Ω2(X × S1) is a symplectization of thecontact structure in a neighborhood of X in X × S1.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Let X be the real algebraic set which is diffeomorphic to thesmooth manifold M. Since M embeds into R5 as a smoothmanifold, we may assume that X is an algebraic subset of R6.Let S1 = (x , y) | x2 + y2 = 1 be the unit circle and considerthe product real algebraic set X × S1 ⊂ R8.From now on we identify M with X . Let α ∈ Ω1(X ) be asmooth 1-form so that ξ = ker(α). Using the coordinate chart
φ : (−1/2, 1/2)→ U ⊂ S1, φ(t) = e2πit , t ∈ (−1/2, 1/2),
define a smooth function f : S1 → R so that f (t) = et if|t| < 1/4 and f (t) = 0 if |t| ≥ 1/3. Then the 2-formω = d(f (t)α) ∈ Ω2(X × S1) is a symplectization of thecontact structure in a neighborhood of X in X × S1.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The cotangent bundle of X × S1 is strongly algebraic andtherefore the smooth 1-form f (t)α ∈ Ω1(X × S1) can beapproximated in the C∞-topology by real algebraic (hence realanalytic) 1-forms with arbitrary precision.
Now, let β ∈ Ω1(X × S1) be a real algebraic 1-form so close tof (t)α that its restriction to X is isotopic to α, as a contactform, and the 2-form dβ is still symplectic on an openneighborhood U = X × (−ε, ε) of X in X × S1. Moreover, thesymplectic form on U is real algebraic and thus real analytic.Note that the restriction of the Euclidean metric of R8 to thesubmanifold X × S1 is a Riemannian metric g = (gij) with realanalytic coefficients.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The cotangent bundle of X × S1 is strongly algebraic andtherefore the smooth 1-form f (t)α ∈ Ω1(X × S1) can beapproximated in the C∞-topology by real algebraic (hence realanalytic) 1-forms with arbitrary precision.Now, let β ∈ Ω1(X × S1) be a real algebraic 1-form so close tof (t)α that its restriction to X is isotopic to α, as a contactform, and the 2-form dβ is still symplectic on an openneighborhood U = X × (−ε, ε) of X in X × S1. Moreover, thesymplectic form on U is real algebraic and thus real analytic.
Note that the restriction of the Euclidean metric of R8 to thesubmanifold X × S1 is a Riemannian metric g = (gij) with realanalytic coefficients.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
The cotangent bundle of X × S1 is strongly algebraic andtherefore the smooth 1-form f (t)α ∈ Ω1(X × S1) can beapproximated in the C∞-topology by real algebraic (hence realanalytic) 1-forms with arbitrary precision.Now, let β ∈ Ω1(X × S1) be a real algebraic 1-form so close tof (t)α that its restriction to X is isotopic to α, as a contactform, and the 2-form dβ is still symplectic on an openneighborhood U = X × (−ε, ε) of X in X × S1. Moreover, thesymplectic form on U is real algebraic and thus real analytic.Note that the restriction of the Euclidean metric of R8 to thesubmanifold X × S1 is a Riemannian metric g = (gij) with realanalytic coefficients.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Since ω and g are real analytic, there is a real analyticcompatible J almost complex structure such thatω(u, v) = g(Ju, v).
The almost complex structure on U gives a complex linedistribution on T∗X , which is nothing but the contact structuregiven by the 1-form ker(β|X ). We know that this contactstructure is isotopic to ker(α) = ξ. This complex linedistribution on T∗X is integrable by dimension reasons.The complex structure J need not to be integrable on U. Wecan modify the almost complex structure on a neighborhood ofX , without changing it on X , to an integrable complexstructure on a possibly smaller open subset of U, but stillcontaining X .This finishes the proof.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Since ω and g are real analytic, there is a real analyticcompatible J almost complex structure such thatω(u, v) = g(Ju, v).The almost complex structure on U gives a complex linedistribution on T∗X , which is nothing but the contact structuregiven by the 1-form ker(β|X ). We know that this contactstructure is isotopic to ker(α) = ξ. This complex linedistribution on T∗X is integrable by dimension reasons.
The complex structure J need not to be integrable on U. Wecan modify the almost complex structure on a neighborhood ofX , without changing it on X , to an integrable complexstructure on a possibly smaller open subset of U, but stillcontaining X .This finishes the proof.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Since ω and g are real analytic, there is a real analyticcompatible J almost complex structure such thatω(u, v) = g(Ju, v).The almost complex structure on U gives a complex linedistribution on T∗X , which is nothing but the contact structuregiven by the 1-form ker(β|X ). We know that this contactstructure is isotopic to ker(α) = ξ. This complex linedistribution on T∗X is integrable by dimension reasons.The complex structure J need not to be integrable on U. Wecan modify the almost complex structure on a neighborhood ofX , without changing it on X , to an integrable complexstructure on a possibly smaller open subset of U, but stillcontaining X .
This finishes the proof.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
Since ω and g are real analytic, there is a real analyticcompatible J almost complex structure such thatω(u, v) = g(Ju, v).The almost complex structure on U gives a complex linedistribution on T∗X , which is nothing but the contact structuregiven by the 1-form ker(β|X ). We know that this contactstructure is isotopic to ker(α) = ξ. This complex linedistribution on T∗X is integrable by dimension reasons.The complex structure J need not to be integrable on U. Wecan modify the almost complex structure on a neighborhood ofX , without changing it on X , to an integrable complexstructure on a possibly smaller open subset of U, but stillcontaining X .This finishes the proof.
CR StructuresOn Three
DimensionalContact
Manifolds
Hatice Coban
MotivationFor CRStructures
What is a CRmanifold?
The MainTheorem
Proof of thetheorem
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