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


DimensionalContact

Manifolds

Hatice Coban



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


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



The MainTheorem

Proof of thetheorem




3 The Main Theorem



DimensionalContact

Manifolds

Hatice Coban



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


DimensionalContact

Manifolds

Hatice Coban



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 ]


DimensionalContact

Manifolds

Hatice Coban



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:



J [JX ,Y ] + [X , JY ] = [JX , JY ]− [X ,Y ]


DimensionalContact

Manifolds

Hatice Coban



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:



J [JX ,Y ] + [X , JY ] = [JX , JY ]− [X ,Y ]


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



The MainTheorem

Proof of thetheorem




3 The Main Theorem



DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



The MainTheorem

Proof of thetheorem




3 The Main Theorem



DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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),



DimensionalContact

Manifolds

Hatice Coban



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),



DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



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.


DimensionalContact

Manifolds

Hatice Coban



The MainTheorem

Proof of thetheorem

THANK YOU!

CR Structures On Three Dimensional Contact Manifoldsth2109/ecstatic/slides/Hatice Coban.pdfCR...

Documents

Transcript of CR Structures On Three Dimensional Contact Manifoldsth2109/ecstatic/slides/Hatice Coban.pdfCR...