stability of leaves - cimrui loja fernandes stability of leaves classical results stability of...
TRANSCRIPT
![Page 1: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/1.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Stability of Leaves
Rui Loja Fernandes
Departamento de MatemáticaInstituto Superior Técnico
Seminars of the CIM Scientific Council Meeting 2008
Rui Loja Fernandes Stability of Leaves
![Page 2: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/2.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Goals
Establish stability results for symplectic leaves of Poissonmanifolds;
Understand the relationship between (apparently) distinctstability results in different geometric settings;
Rui Loja Fernandes Stability of Leaves
![Page 3: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/3.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Goals
Establish stability results for symplectic leaves of Poissonmanifolds;
Understand the relationship between (apparently) distinctstability results in different geometric settings;
Rui Loja Fernandes Stability of Leaves
![Page 4: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/4.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Goals
Establish stability results for symplectic leaves of Poissonmanifolds;
Understand the relationship between (apparently) distinctstability results in different geometric settings;
Rui Loja Fernandes Stability of Leaves
![Page 5: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/5.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Outline
1 Classical ResultsFlowsFoliationsGroup actions
2 Stability of symplectic leavesPoisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
3 Universal Stability TheoremBasic problemGeometric Lie theoryUniversal Stability Theorem
Rui Loja Fernandes Stability of Leaves
![Page 6: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/6.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Flows: Stability of periodic orbits
Definition
A periodic orbit of a vector field X ∈ X(M) is called stable ifevery nearby vector field also has a nearby periodic orbit.
Basic Fact: Stability is controled by the Poincaré returnmap h : T → T .Assumptions on dx0h can also guarantee stability.
Rui Loja Fernandes Stability of Leaves
![Page 7: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/7.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Flows: Stability of periodic orbits
Definition
A periodic orbit of a vector field X ∈ X(M) is called stable ifevery nearby vector field also has a nearby periodic orbit.
Basic Fact: Stability is controled by the Poincaré returnmap h : T → T .Assumptions on dx0h can also guarantee stability.
Rui Loja Fernandes Stability of Leaves
![Page 8: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/8.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Flows: Stability of periodic orbits
Definition
A periodic orbit of a vector field X ∈ X(M) is called stable ifevery nearby vector field also has a nearby periodic orbit.
T
Basic Fact: Stability is controled by the Poincaré returnmap h : T → T .Assumptions on dx0h can also guarantee stability.
Rui Loja Fernandes Stability of Leaves
![Page 9: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/9.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Flows: Stability of periodic orbits
Definition
A periodic orbit of a vector field X ∈ X(M) is called stable ifevery nearby vector field also has a nearby periodic orbit.
T
xh(x)
Basic Fact: Stability is controled by the Poincaré returnmap h : T → T .Assumptions on dx0h can also guarantee stability.
Rui Loja Fernandes Stability of Leaves
![Page 10: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/10.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Flows: Stability of periodic orbits
Definition
A periodic orbit of a vector field X ∈ X(M) is called stable ifevery nearby vector field also has a nearby periodic orbit.
T
xh(x)
Basic Fact: Stability is controled by the Poincaré returnmap h : T → T .Assumptions on dx0h can also guarantee stability.
Rui Loja Fernandes Stability of Leaves
![Page 11: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/11.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Flows: Stability of periodic orbits
Definition
A periodic orbit of a vector field X ∈ X(M) is called stable ifevery nearby vector field also has a nearby periodic orbit.
T
xh(x)
Basic Fact: Stability is controled by the Poincaré returnmap h : T → T .Assumptions on dx0h can also guarantee stability.
Rui Loja Fernandes Stability of Leaves
![Page 12: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/12.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Fix a manifold M (dim M = n) and a foliation F (codim(F) = q).F is given by a foliation atlas (Ui , ϕi)i∈I
ϕi : Ui → Rn−q × Rq, ϕij(x , y) = (gij(x , y), hij(y)).
Rui Loja Fernandes Stability of Leaves
![Page 13: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/13.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Fix a manifold M (dim M = n) and a foliation F (codim(F) = q).F is given by a foliation atlas (Ui , ϕi)i∈I
M
i j
j
ϕjo
q q
n-q n-q
-1
U
V
ϕ
R
R
R
R
i
iϕ
ϕ
ϕi : Ui → Rn−q × Rq, ϕij(x , y) = (gij(x , y), hij(y)).
Rui Loja Fernandes Stability of Leaves
![Page 14: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/14.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Fix a manifold M (dim M = n) and a foliation F (codim(F) = q).F is given by a foliation atlas (Ui , ϕi)i∈I
M
i j
j
ϕjo
q q
n-q n-q
-1
U
V
ϕ
R
R
R
R
i
iϕ
ϕ
ϕi : Ui → Rn−q × Rq, ϕij(x , y) = (gij(x , y), hij(y)).
Rui Loja Fernandes Stability of Leaves
![Page 15: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/15.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Theorem (Frobenius)
Folq(M) oo // D : M → Grq(TM)|D is involutive
F // D := TF
=⇒ Folq(M) has a natural Cr compact-open topology
Definition
A leaf L of a foliation F ∈ Folk (M) is called stable if everynearby foliation in Folk (M) has a nearby leaf diffeomorphic to L.
Rui Loja Fernandes Stability of Leaves
![Page 16: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/16.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Theorem (Frobenius)
Folq(M) oo // D : M → Grq(TM)|D is involutive
F // D := TF
=⇒ Folq(M) has a natural Cr compact-open topology
Definition
A leaf L of a foliation F ∈ Folk (M) is called stable if everynearby foliation in Folk (M) has a nearby leaf diffeomorphic to L.
Rui Loja Fernandes Stability of Leaves
![Page 17: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/17.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Theorem (Frobenius)
Folq(M) oo // D : M → Grq(TM)|D is involutive
F // D := TF
=⇒ Folq(M) has a natural Cr compact-open topology
Definition
A leaf L of a foliation F ∈ Folk (M) is called stable if everynearby foliation in Folk (M) has a nearby leaf diffeomorphic to L.
Rui Loja Fernandes Stability of Leaves
![Page 18: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/18.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
Rui Loja Fernandes Stability of Leaves
![Page 19: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/19.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
Rui Loja Fernandes Stability of Leaves
![Page 20: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/20.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
x
γ
y
Rui Loja Fernandes Stability of Leaves
![Page 21: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/21.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
x
γ
y
U
Rui Loja Fernandes Stability of Leaves
![Page 22: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/22.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
x
γ
y
U
S
T
Rui Loja Fernandes Stability of Leaves
![Page 23: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/23.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
iϕ
x
γ
y
U
T
T
S
S
y’x’
yγxL
Construct diffeomorphism f : T → S that satisfies f (x) = y andy ′ = f (x ′) iff x ′ and y ′ are in same plaque. Then:
Rui Loja Fernandes Stability of Leaves
![Page 24: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/24.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
iϕ
x
γ
y
U
T
T
S
S
y’x’
yγxL
Construct diffeomorphism f : T → S that satisfies f (x) = y andy ′ = f (x ′) iff x ′ and y ′ are in same plaque. Then:
HolT ,S(γ) := germx(f ) : (T , x)→ (S, y).
Rui Loja Fernandes Stability of Leaves
![Page 25: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/25.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
Rui Loja Fernandes Stability of Leaves
![Page 26: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/26.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
Rui Loja Fernandes Stability of Leaves
![Page 27: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/27.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
T
Sx
γ
y
Rui Loja Fernandes Stability of Leaves
![Page 28: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/28.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
T
Sx
γ
y
U1
U3
U
U
U
2
4
5
Rui Loja Fernandes Stability of Leaves
![Page 29: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/29.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
The stability of a leaf L is controled by the holonomy of L.
L
T
Sx
γ
y
U1
U3
U
U
U
2
4
5
=T
T
T
T
T
0
1
2
3
4
5=T
HolT ,S(γ) := HolTk ,Tk−1(γ) · · · HolT2,T1(γ) HolT1,T0(γ)
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Facts:Taking germs makes construction independent of choices;If γ, η are curves in L with γ(0) = η(1) then:HolT ,S(γ · η) = HolT ,S(γ) HolS,R(η);If γ and γ′ are homotopic curves in L, then:HolT ,S(γ) = HolT ,S(γ′);
Hence, if we fix x ∈ L we obtain the holonomy homomorphism:
Hol := HolT ,T : π1(L, x)→ Diffx(T ).
Note: The Poincaré return map is a special case of thisconstruction.
Rui Loja Fernandes Stability of Leaves
![Page 31: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/31.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Facts:Taking germs makes construction independent of choices;If γ, η are curves in L with γ(0) = η(1) then:HolT ,S(γ · η) = HolT ,S(γ) HolS,R(η);If γ and γ′ are homotopic curves in L, then:HolT ,S(γ) = HolT ,S(γ′);
Hence, if we fix x ∈ L we obtain the holonomy homomorphism:
Hol := HolT ,T : π1(L, x)→ Diffx(T ).
Note: The Poincaré return map is a special case of thisconstruction.
Rui Loja Fernandes Stability of Leaves
![Page 32: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/32.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Facts:Taking germs makes construction independent of choices;If γ, η are curves in L with γ(0) = η(1) then:HolT ,S(γ · η) = HolT ,S(γ) HolS,R(η);If γ and γ′ are homotopic curves in L, then:HolT ,S(γ) = HolT ,S(γ′);
Hence, if we fix x ∈ L we obtain the holonomy homomorphism:
Hol := HolT ,T : π1(L, x)→ Diffx(T ).
Note: The Poincaré return map is a special case of thisconstruction.
Rui Loja Fernandes Stability of Leaves
![Page 33: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/33.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Facts:Taking germs makes construction independent of choices;If γ, η are curves in L with γ(0) = η(1) then:HolT ,S(γ · η) = HolT ,S(γ) HolS,R(η);If γ and γ′ are homotopic curves in L, then:HolT ,S(γ) = HolT ,S(γ′);
Hence, if we fix x ∈ L we obtain the holonomy homomorphism:
Hol := HolT ,T : π1(L, x)→ Diffx(T ).
Note: The Poincaré return map is a special case of thisconstruction.
Rui Loja Fernandes Stability of Leaves
![Page 34: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/34.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Facts:Taking germs makes construction independent of choices;If γ, η are curves in L with γ(0) = η(1) then:HolT ,S(γ · η) = HolT ,S(γ) HolS,R(η);If γ and γ′ are homotopic curves in L, then:HolT ,S(γ) = HolT ,S(γ′);
Hence, if we fix x ∈ L we obtain the holonomy homomorphism:
Hol := HolT ,T : π1(L, x)→ Diffx(T ).
Note: The Poincaré return map is a special case of thisconstruction.
Rui Loja Fernandes Stability of Leaves
![Page 35: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/35.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Facts:Taking germs makes construction independent of choices;If γ, η are curves in L with γ(0) = η(1) then:HolT ,S(γ · η) = HolT ,S(γ) HolS,R(η);If γ and γ′ are homotopic curves in L, then:HolT ,S(γ) = HolT ,S(γ′);
Hence, if we fix x ∈ L we obtain the holonomy homomorphism:
Hol := HolT ,T : π1(L, x)→ Diffx(T ).
Note: The Poincaré return map is a special case of thisconstruction.
Rui Loja Fernandes Stability of Leaves
![Page 36: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/36.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Differentiating gives the linear holonomy representation:
ρ : π1(L, x)→ GL(ν(L)x), ρ := dx Hol
Denote by H1(π1(L, x), ν(L)x) the first group cohomology.
Theorem (Reeb, Thurston, Langevin & Rosenberg)
Let L be a compact leaf and assume that
H1(π1(L, x), ν(L)x) = 0.
Then L is stable.
Rui Loja Fernandes Stability of Leaves
![Page 37: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/37.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Differentiating gives the linear holonomy representation:
ρ : π1(L, x)→ GL(ν(L)x), ρ := dx Hol
Denote by H1(π1(L, x), ν(L)x) the first group cohomology.
Theorem (Reeb, Thurston, Langevin & Rosenberg)
Let L be a compact leaf and assume that
H1(π1(L, x), ν(L)x) = 0.
Then L is stable.
Rui Loja Fernandes Stability of Leaves
![Page 38: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/38.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Foliations: stability of leaves
Differentiating gives the linear holonomy representation:
ρ : π1(L, x)→ GL(ν(L)x), ρ := dx Hol
Denote by H1(π1(L, x), ν(L)x) the first group cohomology.
Theorem (Reeb, Thurston, Langevin & Rosenberg)
Let L be a compact leaf and assume that
H1(π1(L, x), ν(L)x) = 0.
Then L is stable.
Rui Loja Fernandes Stability of Leaves
![Page 39: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/39.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 40: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/40.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 41: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/41.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 42: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/42.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 43: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/43.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 44: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/44.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 45: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/45.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
Fix a manifold M and a Lie group Mα(g, x) := g · x - an action of G on MAction α : G ×M → M ⇔ homomorphism α : G→ Diff(M)
Act(G; M) ⊂ Maps(G; Diff(M))
=⇒ Act(G; M) has a natural Cr compact-open topology
Definition
An orbit O of α ∈ Act(G; M) is called stable if every nearbyaction in Act(G; M) has a nearby orbit diffeomorphic to O.
Rui Loja Fernandes Stability of Leaves
![Page 46: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/46.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 47: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/47.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 48: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/48.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 49: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/49.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
O
x
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 50: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/50.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
O
x
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 51: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/51.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
O
x
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 52: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/52.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
O
x
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 53: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/53.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
The stability of an orbit O is controled by the isotropy of O:Gx := g ∈ G : g · x = x isotropy group at x ∈ O.g ∈ Gx induces a map αg : M → M, y 7→ g · y that fixes x .
O
x
dxαg : TxM → TxM⇒ ρ(g) : ν(O)x → ν(O)x where ν(O)x = TxM/TxO.
Rui Loja Fernandes Stability of Leaves
![Page 54: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/54.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
linear normal isotropy representation:
ρ : Gx → GL(ν(O)x), ρ(g) := dxαg : ν(O)x → ν(O)x
Denote by H1(Gx , ν(O)x) the first group cohomology.
Theorem (Hirsch,Stowe)
Let O be a compact orbit and assume that
H1(Gx , ν(O)x) = 0.
Then O is stable.
Rui Loja Fernandes Stability of Leaves
![Page 55: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/55.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
linear normal isotropy representation:
ρ : Gx → GL(ν(O)x), ρ(g) := dxαg : ν(O)x → ν(O)x
Denote by H1(Gx , ν(O)x) the first group cohomology.
Theorem (Hirsch,Stowe)
Let O be a compact orbit and assume that
H1(Gx , ν(O)x) = 0.
Then O is stable.
Rui Loja Fernandes Stability of Leaves
![Page 56: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/56.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Group actions: stability of orbits
linear normal isotropy representation:
ρ : Gx → GL(ν(O)x), ρ(g) := dxαg : ν(O)x → ν(O)x
Denote by H1(Gx , ν(O)x) the first group cohomology.
Theorem (Hirsch,Stowe)
Let O be a compact orbit and assume that
H1(Gx , ν(O)x) = 0.
Then O is stable.
Rui Loja Fernandes Stability of Leaves
![Page 57: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/57.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Stability of leaves versus orbits
In general, the two theorems are quite different (e.g.,dimension of orbits can vary).If Gx is discrete, dimension of orbits is locally constant.If Gx is discrete and G is 1-connected then π1(O, x) = Gx .=⇒ the theorem for actions follows from the theorem forfoliations.
Rui Loja Fernandes Stability of Leaves
![Page 58: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/58.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Stability of leaves versus orbits
In general, the two theorems are quite different (e.g.,dimension of orbits can vary).If Gx is discrete, dimension of orbits is locally constant.If Gx is discrete and G is 1-connected then π1(O, x) = Gx .=⇒ the theorem for actions follows from the theorem forfoliations.
Rui Loja Fernandes Stability of Leaves
![Page 59: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/59.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Stability of leaves versus orbits
In general, the two theorems are quite different (e.g.,dimension of orbits can vary).If Gx is discrete, dimension of orbits is locally constant.If Gx is discrete and G is 1-connected then π1(O, x) = Gx .=⇒ the theorem for actions follows from the theorem forfoliations.
Rui Loja Fernandes Stability of Leaves
![Page 60: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/60.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
FlowsFoliationsGroup actions
Stability of leaves versus orbits
In general, the two theorems are quite different (e.g.,dimension of orbits can vary).If Gx is discrete, dimension of orbits is locally constant.If Gx is discrete and G is 1-connected then π1(O, x) = Gx .=⇒ the theorem for actions follows from the theorem forfoliations.
Rui Loja Fernandes Stability of Leaves
![Page 61: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/61.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Hamilton’s Equations
R2n with coordinates (q1, . . . , qn, p1, . . . , pn)
Classical Poisson bracket:
f1, f2 =n∑
i=1
(∂f1∂qi
∂f2∂pi− ∂f2
∂qi
∂f1∂pi
)
Hamilton’s equations:qi = ∂h
∂pi
pi = − ∂h∂qi
(i = 1, . . . , n) ⇔ xa = xa, h (a = 1, . . . , 2n)
Rui Loja Fernandes Stability of Leaves
![Page 62: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/62.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Hamilton’s Equations
R2n with coordinates (q1, . . . , qn, p1, . . . , pn)
Classical Poisson bracket:
f1, f2 =n∑
i=1
(∂f1∂qi
∂f2∂pi− ∂f2
∂qi
∂f1∂pi
)
Hamilton’s equations:qi = ∂h
∂pi
pi = − ∂h∂qi
(i = 1, . . . , n) ⇔ xa = xa, h (a = 1, . . . , 2n)
Rui Loja Fernandes Stability of Leaves
![Page 63: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/63.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Hamilton’s Equations
R2n with coordinates (q1, . . . , qn, p1, . . . , pn)
Classical Poisson bracket:
f1, f2 =n∑
i=1
(∂f1∂qi
∂f2∂pi− ∂f2
∂qi
∂f1∂pi
)
Hamilton’s equations:qi = ∂h
∂pi
pi = − ∂h∂qi
(i = 1, . . . , n) ⇔ xa = xa, h (a = 1, . . . , 2n)
Rui Loja Fernandes Stability of Leaves
![Page 64: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/64.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson brackets
, : C∞(R2n)× C∞(R2n)→ C∞(R2n) is R-bilinear andsatisfies:
Skew-symmetry: f , g = −g, f;Jacobi identity: f , g, h+ g, h, f+ h, f, g = 0;Leibniz identity: f · g, h = f , h · g + f · g, h;
Rui Loja Fernandes Stability of Leaves
![Page 65: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/65.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson brackets
, : C∞(R2n)× C∞(R2n)→ C∞(R2n) is R-bilinear andsatisfies:
Skew-symmetry: f , g = −g, f;Jacobi identity: f , g, h+ g, h, f+ h, f, g = 0;Leibniz identity: f · g, h = f , h · g + f · g, h;
Rui Loja Fernandes Stability of Leaves
![Page 66: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/66.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson brackets
, : C∞(R2n)× C∞(R2n)→ C∞(R2n) is R-bilinear andsatisfies:
Skew-symmetry: f , g = −g, f;Jacobi identity: f , g, h+ g, h, f+ h, f, g = 0;Leibniz identity: f · g, h = f , h · g + f · g, h;
Rui Loja Fernandes Stability of Leaves
![Page 67: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/67.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson brackets
, : C∞(R2n)× C∞(R2n)→ C∞(R2n) is R-bilinear andsatisfies:
Skew-symmetry: f , g = −g, f;Jacobi identity: f , g, h+ g, h, f+ h, f, g = 0;Leibniz identity: f · g, h = f , h · g + f · g, h;
Rui Loja Fernandes Stability of Leaves
![Page 68: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/68.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson brackets
, : C∞(M)× C∞(M)→ C∞(M) is R-bilinear andsatisfies:
Skew-symmetry: f , g = −g, f;Jacobi identity: f , g, h+ g, h, f+ h, f, g = 0;Leibniz identity: f · g, h = f , h · g + f · g, h;
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson brackets
, : C∞(M)× C∞(M)→ C∞(M) is R-bilinear andsatisfies:
Skew-symmetry: f , g = −g, f;Jacobi identity: f , g, h+ g, h, f+ h, f, g = 0;Leibniz identity: f · g, h = f , h · g + f · g, h;
A manifold M furnished with a Poisson bracket is called aPoisson manifold.
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Basic examples
Any symplectic manifold (M, ω) is a Poisson manifold:
f , g = −ω(Xf , Xg).
(Xf is the unique vector field such that ιXf ω = df .)The dual of a Lie algebra M = g∗ is a Poisson manifold:
f , g(ξ) = 〈ξ, [dξf , dξg]〉.
Any skew-symmetric matrix (aij) defines a quadraticPoisson bracket on Rn:
xi , xj = aijxixj .
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Basic examples
Any symplectic manifold (M, ω) is a Poisson manifold:
f , g = −ω(Xf , Xg).
(Xf is the unique vector field such that ιXf ω = df .)The dual of a Lie algebra M = g∗ is a Poisson manifold:
f , g(ξ) = 〈ξ, [dξf , dξg]〉.
Any skew-symmetric matrix (aij) defines a quadraticPoisson bracket on Rn:
xi , xj = aijxixj .
Rui Loja Fernandes Stability of Leaves
![Page 72: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/72.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Basic examples
Any symplectic manifold (M, ω) is a Poisson manifold:
f , g = −ω(Xf , Xg).
(Xf is the unique vector field such that ιXf ω = df .)The dual of a Lie algebra M = g∗ is a Poisson manifold:
f , g(ξ) = 〈ξ, [dξf , dξg]〉.
Any skew-symmetric matrix (aij) defines a quadraticPoisson bracket on Rn:
xi , xj = aijxixj .
Rui Loja Fernandes Stability of Leaves
![Page 73: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/73.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Definition
On (M, , ), the hamitonian vector field determined byh ∈ C∞(M) is the vector field Xh ∈ X(M) given by:
Xh(f ) := f , h, ∀f ∈ C∞(M).
Write x ∼ y if there exists a piecewise smooth curve joining xto y made of integral curves of hamiltonian vector fields:
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Definition
On (M, , ), the hamitonian vector field determined byh ∈ C∞(M) is the vector field Xh ∈ X(M) given by:
Xh(f ) := f , h, ∀f ∈ C∞(M).
Write x ∼ y if there exists a piecewise smooth curve joining xto y made of integral curves of hamiltonian vector fields:
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Definition
On (M, , ), the hamitonian vector field determined byh ∈ C∞(M) is the vector field Xh ∈ X(M) given by:
Xh(f ) := f , h, ∀f ∈ C∞(M).
Write x ∼ y if there exists a piecewise smooth curve joining xto y made of integral curves of hamiltonian vector fields:
My
x
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Theorem (Weinstein)
The decomposition of (M, , ) into equivalence classes of ∼:
M =⊔α∈A
Sα.
satisfies:(i) Each Sα is a (immersed) submanifold of M;(ii) Each Sα carries a symplectic structure ωα;(iii) The inclusion iα : Sα → M is a Poisson map.
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Theorem (Weinstein)
The decomposition of (M, , ) into equivalence classes of ∼:
M =⊔α∈A
Sα.
satisfies:(i) Each Sα is a (immersed) submanifold of M;(ii) Each Sα carries a symplectic structure ωα;(iii) The inclusion iα : Sα → M is a Poisson map.
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Theorem (Weinstein)
The decomposition of (M, , ) into equivalence classes of ∼:
M =⊔α∈A
Sα.
satisfies:(i) Each Sα is a (immersed) submanifold of M;(ii) Each Sα carries a symplectic structure ωα;(iii) The inclusion iα : Sα → M is a Poisson map.
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliation
Theorem (Weinstein)
The decomposition of (M, , ) into equivalence classes of ∼:
M =⊔α∈A
Sα.
satisfies:(i) Each Sα is a (immersed) submanifold of M;(ii) Each Sα carries a symplectic structure ωα;(iii) The inclusion iα : Sα → M is a Poisson map.
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliationExample
M = sl∗(2, R) ' R3: x , z = y ; x , y = z; z, y = x .Symplectic foliation: (x , y , z)|x2 + y2 − z2 = c.Foliation is singular (dimension of leaves varies; e.g., conex2 + y2 = z2)
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliationExample
M = sl∗(2, R) ' R3: x , z = y ; x , y = z; z, y = x .Symplectic foliation: (x , y , z)|x2 + y2 − z2 = c.Foliation is singular (dimension of leaves varies; e.g., conex2 + y2 = z2)
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliationExample
M = sl∗(2, R) ' R3: x , z = y ; x , y = z; z, y = x .Symplectic foliation: (x , y , z)|x2 + y2 − z2 = c.Foliation is singular (dimension of leaves varies; e.g., conex2 + y2 = z2)
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliationExample
M = sl∗(2, R) ' R3: x , z = y ; x , y = z; z, y = x .Symplectic foliation: (x , y , z)|x2 + y2 − z2 = c.
Foliation is singular (dimension of leaves varies; e.g., conex2 + y2 = z2)
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Symplectic foliationExample
M = sl∗(2, R) ' R3: x , z = y ; x , y = z; z, y = x .Symplectic foliation: (x , y , z)|x2 + y2 − z2 = c.Foliation is singular (dimension of leaves varies; e.g., conex2 + y2 = z2)
Rui Loja Fernandes Stability of Leaves
![Page 85: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/85.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson bivector
Given Poisson bracket , define the Poisson bivector:
π(df , dg) := f , g.
π ∈ X2(M) = Γ(∧2TM) is a skew-symmetric contravarianttensor;In local coordinates (x1, . . . , xn):
π =∑i<j
πij(x)∂
∂x i ∧∂
∂x j .
Jacobi identity⇔ [π, π] = 0.
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson bivector
Given Poisson bracket , define the Poisson bivector:
π(df , dg) := f , g.
π ∈ X2(M) = Γ(∧2TM) is a skew-symmetric contravarianttensor;In local coordinates (x1, . . . , xn):
π =∑i<j
πij(x)∂
∂x i ∧∂
∂x j .
Jacobi identity⇔ [π, π] = 0.
Rui Loja Fernandes Stability of Leaves
![Page 87: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/87.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson bivector
Given Poisson bracket , define the Poisson bivector:
π(df , dg) := f , g.
π ∈ X2(M) = Γ(∧2TM) is a skew-symmetric contravarianttensor;In local coordinates (x1, . . . , xn):
π =∑i<j
πij(x)∂
∂x i ∧∂
∂x j .
Jacobi identity⇔ [π, π] = 0.
Rui Loja Fernandes Stability of Leaves
![Page 88: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/88.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson bivector
Given Poisson bracket , define the Poisson bivector:
π(df , dg) := f , g.
π ∈ X2(M) = Γ(∧2TM) is a skew-symmetric contravarianttensor;In local coordinates (x1, . . . , xn):
π =∑i<j
πij(x)∂
∂x i ∧∂
∂x j .
Jacobi identity⇔ [π, π] = 0.
Rui Loja Fernandes Stability of Leaves
![Page 89: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/89.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leaves
Poiss(M) ←→ π : M → ∧2(TM)| [π, π] = 0.
=⇒ Poiss(M) has a natural Cr compact-open topology
Definition
A symplectic leaf S of π ∈ Poiss(M) is called stable if everynearby Poisson structure in Poiss(M) has a nearby leafdiffeomorphic to S.
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leaves
Poiss(M) ←→ π : M → ∧2(TM)| [π, π] = 0.
=⇒ Poiss(M) has a natural Cr compact-open topology
Definition
A symplectic leaf S of π ∈ Poiss(M) is called stable if everynearby Poisson structure in Poiss(M) has a nearby leafdiffeomorphic to S.
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leaves
Poiss(M) ←→ π : M → ∧2(TM)| [π, π] = 0.
=⇒ Poiss(M) has a natural Cr compact-open topology
Definition
A symplectic leaf S of π ∈ Poiss(M) is called stable if everynearby Poisson structure in Poiss(M) has a nearby leafdiffeomorphic to S.
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesMain Theorem
Theorem (Crainic & RLF)
Let (M, π) be a Poisson structure and S ⊂ M a compactsymplectic leaf such that:
H2π(M, S) = 0.
Then S is stable.
Again, this result is quite different from the previous ones;The theorem has a more precise version that describes thenearby symplectic leaves diffeomorphic to S;
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesMain Theorem
Theorem (Crainic & RLF)
Let (M, π) be a Poisson structure and S ⊂ M a compactsymplectic leaf such that:
H2π(M, S) = 0.
Then S is stable.
Again, this result is quite different from the previous ones;The theorem has a more precise version that describes thenearby symplectic leaves diffeomorphic to S;
Rui Loja Fernandes Stability of Leaves
![Page 94: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/94.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesMain Theorem
Theorem (Crainic & RLF)
Let (M, π) be a Poisson structure and S ⊂ M a compactsymplectic leaf such that:
H2π(M, S) = 0.
Then S is stable.
Again, this result is quite different from the previous ones;The theorem has a more precise version that describes thenearby symplectic leaves diffeomorphic to S;
Rui Loja Fernandes Stability of Leaves
![Page 95: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/95.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
M = su∗(3) ' su(3) (via the Killing form) with linear Poissonstructure:
Symplectic leaves are the conjugacy classes of SU(3):
A ∼
iλ1 0 00 iλ2 00 0 iλ3
(λ1+λ2+λ3 = 0, 0 ≤ λ1 ≤ λ2)
Leaves have:(i) Dimension 6 (flag);(ii) Dimension 4 (CP(2));(iii) Dimension 0 (the origin);
All leaves satisfy criterion: H2(M, S) = 0.
Rui Loja Fernandes Stability of Leaves
![Page 96: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/96.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
M = su∗(3) ' su(3) (via the Killing form) with linear Poissonstructure:
Symplectic leaves are the conjugacy classes of SU(3):
A ∼
iλ1 0 00 iλ2 00 0 iλ3
(λ1+λ2+λ3 = 0, 0 ≤ λ1 ≤ λ2)
Leaves have:(i) Dimension 6 (flag);(ii) Dimension 4 (CP(2));(iii) Dimension 0 (the origin);
All leaves satisfy criterion: H2(M, S) = 0.
Rui Loja Fernandes Stability of Leaves
![Page 97: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/97.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
M = su∗(3) ' su(3) (via the Killing form) with linear Poissonstructure:
Symplectic leaves are the conjugacy classes of SU(3):
A ∼
iλ1 0 00 iλ2 00 0 iλ3
(λ1+λ2+λ3 = 0, 0 ≤ λ1 ≤ λ2)
Leaves have:(i) Dimension 6 (flag);(ii) Dimension 4 (CP(2));(iii) Dimension 0 (the origin);
All leaves satisfy criterion: H2(M, S) = 0.
Rui Loja Fernandes Stability of Leaves
![Page 98: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/98.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
M = su∗(3) ' su(3) (via the Killing form) with linear Poissonstructure:
Symplectic leaves are the conjugacy classes of SU(3):
A ∼
iλ1 0 00 iλ2 00 0 iλ3
(λ1+λ2+λ3 = 0, 0 ≤ λ1 ≤ λ2)
Leaves have:(i) Dimension 6 (flag);(ii) Dimension 4 (CP(2));(iii) Dimension 0 (the origin);
All leaves satisfy criterion: H2(M, S) = 0.
Rui Loja Fernandes Stability of Leaves
![Page 99: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/99.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
All leaves of su∗(3) (including singular leaves) are stable;The same result applies for g∗, where g is any semi-simpleLie algebra of compact type;This is related to (and explains!) a famous linearizationtheorem of Conn;If g is semi-simple and non-compact stability, in general,does not hold (e.g., sl(2, R)).
Rui Loja Fernandes Stability of Leaves
![Page 100: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/100.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
All leaves of su∗(3) (including singular leaves) are stable;The same result applies for g∗, where g is any semi-simpleLie algebra of compact type;This is related to (and explains!) a famous linearizationtheorem of Conn;If g is semi-simple and non-compact stability, in general,does not hold (e.g., sl(2, R)).
Rui Loja Fernandes Stability of Leaves
![Page 101: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/101.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
All leaves of su∗(3) (including singular leaves) are stable;The same result applies for g∗, where g is any semi-simpleLie algebra of compact type;This is related to (and explains!) a famous linearizationtheorem of Conn;If g is semi-simple and non-compact stability, in general,does not hold (e.g., sl(2, R)).
Rui Loja Fernandes Stability of Leaves
![Page 102: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/102.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Stability of symplectic leavesExample
All leaves of su∗(3) (including singular leaves) are stable;The same result applies for g∗, where g is any semi-simpleLie algebra of compact type;This is related to (and explains!) a famous linearizationtheorem of Conn;If g is semi-simple and non-compact stability, in general,does not hold (e.g., sl(2, R)).
Rui Loja Fernandes Stability of Leaves
![Page 103: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/103.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 104: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/104.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 105: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/105.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 106: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/106.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 107: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/107.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 108: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/108.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 109: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/109.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Ordinary Geometry
Differential forms:Ωk (M) = Γ(∧kT ∗M);DeRham differential:d : Ω•(M)→ Ω•+1(M),ιX dω = LX ω − dιX ω;DeRham cohomology:H•
DR(M) := Ker d/ Im d;
Poisson Geometry
Multivector fields:Xk (M) = Γ(∧kTM);Lichnerowitz differential:dπ : X•(M)→ X•+1(M),dπθ := [θ, π];Poisson cohomology:H•
π(M) := Ker dπ/ Im dπ;
Rui Loja Fernandes Stability of Leaves
![Page 110: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/110.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Geometric interpretations of H•π(M) in low degrees:
H0π(M) = Z (C∞(M)) - Casimirs;
H1π(M) = Poisson vect. fields/hamiltonian vect. fields -
infinitesimal outer Poisson automorphisms;H2
π(M) = Tπ Poiss(M) - infinitesimal (formal) deformationsof π;
Relative Poisson cohomology H•π(M, S): replace X•(M) by
multivector fields along the symplectic leaf S:
X•(M, S) := Γ(∧•TSM).
Rui Loja Fernandes Stability of Leaves
![Page 111: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/111.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Geometric interpretations of H•π(M) in low degrees:
H0π(M) = Z (C∞(M)) - Casimirs;
H1π(M) = Poisson vect. fields/hamiltonian vect. fields -
infinitesimal outer Poisson automorphisms;H2
π(M) = Tπ Poiss(M) - infinitesimal (formal) deformationsof π;
Relative Poisson cohomology H•π(M, S): replace X•(M) by
multivector fields along the symplectic leaf S:
X•(M, S) := Γ(∧•TSM).
Rui Loja Fernandes Stability of Leaves
![Page 112: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/112.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Geometric interpretations of H•π(M) in low degrees:
H0π(M) = Z (C∞(M)) - Casimirs;
H1π(M) = Poisson vect. fields/hamiltonian vect. fields -
infinitesimal outer Poisson automorphisms;H2
π(M) = Tπ Poiss(M) - infinitesimal (formal) deformationsof π;
Relative Poisson cohomology H•π(M, S): replace X•(M) by
multivector fields along the symplectic leaf S:
X•(M, S) := Γ(∧•TSM).
Rui Loja Fernandes Stability of Leaves
![Page 113: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/113.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Geometric interpretations of H•π(M) in low degrees:
H0π(M) = Z (C∞(M)) - Casimirs;
H1π(M) = Poisson vect. fields/hamiltonian vect. fields -
infinitesimal outer Poisson automorphisms;H2
π(M) = Tπ Poiss(M) - infinitesimal (formal) deformationsof π;
Relative Poisson cohomology H•π(M, S): replace X•(M) by
multivector fields along the symplectic leaf S:
X•(M, S) := Γ(∧•TSM).
Rui Loja Fernandes Stability of Leaves
![Page 114: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/114.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Poisson geometrySymplectic leavesStability of symplectic leavesPoisson cohomology
Poisson cohomology
Geometric interpretations of H•π(M) in low degrees:
H0π(M) = Z (C∞(M)) - Casimirs;
H1π(M) = Poisson vect. fields/hamiltonian vect. fields -
infinitesimal outer Poisson automorphisms;H2
π(M) = Tπ Poiss(M) - infinitesimal (formal) deformationsof π;
Relative Poisson cohomology H•π(M, S): replace X•(M) by
multivector fields along the symplectic leaf S:
X•(M, S) := Γ(∧•TSM).
Rui Loja Fernandes Stability of Leaves
![Page 115: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/115.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Basic problem
Is there a general setup to deal with these kind of stabilityproblems?
A positive answer to this question should lead to:(i) A universal stability theorem which would yield the stability
theorems stated above.(ii) A way to handle with stronger notions of stability.
Rui Loja Fernandes Stability of Leaves
![Page 116: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/116.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Basic problem
Is there a general setup to deal with these kind of stabilityproblems?
A positive answer to this question should lead to:(i) A universal stability theorem which would yield the stability
theorems stated above.(ii) A way to handle with stronger notions of stability.
Rui Loja Fernandes Stability of Leaves
![Page 117: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/117.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Basic problem
Is there a general setup to deal with these kind of stabilityproblems?
A positive answer to this question should lead to:(i) A universal stability theorem which would yield the stability
theorems stated above.(ii) A way to handle with stronger notions of stability.
Rui Loja Fernandes Stability of Leaves
![Page 118: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/118.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Basic problem
Is there a general setup to deal with these kind of stabilityproblems?
A positive answer to this question should lead to:(i) A universal stability theorem which would yield the stability
theorems stated above.(ii) A way to handle with stronger notions of stability.
Rui Loja Fernandes Stability of Leaves
![Page 119: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/119.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroids
Definition
A Lie algebroid is a vector bundle A→ M with:(i) a Lie bracket [ , ]A : Γ(A)× Γ(A)→ Γ(A);(ii) a bundle map ρ : A→ TM (the anchor);
such that:
[α, fβ]A = f [αβ]A + ρ(α)(f )β, (f ∈ C∞(M), α, β ∈ Γ(A)).
Im ρ ⊂ TM is a integrable (singular) distribution⇓
Lie algebroids have a characteristic foliation
Rui Loja Fernandes Stability of Leaves
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Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroids
Definition
A Lie algebroid is a vector bundle A→ M with:(i) a Lie bracket [ , ]A : Γ(A)× Γ(A)→ Γ(A);(ii) a bundle map ρ : A→ TM (the anchor);
such that:
[α, fβ]A = f [αβ]A + ρ(α)(f )β, (f ∈ C∞(M), α, β ∈ Γ(A)).
Im ρ ⊂ TM is a integrable (singular) distribution⇓
Lie algebroids have a characteristic foliation
Rui Loja Fernandes Stability of Leaves
![Page 121: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/121.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroids
Definition
A Lie algebroid is a vector bundle A→ M with:(i) a Lie bracket [ , ]A : Γ(A)× Γ(A)→ Γ(A);(ii) a bundle map ρ : A→ TM (the anchor);
such that:
[α, fβ]A = f [αβ]A + ρ(α)(f )β, (f ∈ C∞(M), α, β ∈ Γ(A)).
Im ρ ⊂ TM is a integrable (singular) distribution⇓
Lie algebroids have a characteristic foliation
Rui Loja Fernandes Stability of Leaves
![Page 122: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/122.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroidsExamples
Flows. For X ∈ X(M), the associated Lie algebroid is:A = M × R, [f , g]A := fX (g)− gX (f ), ρ(f ) = fX .
Leaves of A are the orbits of X .
Foliations. For F ∈ Folk (M), the associated Lie algebroidis:
A = TF , [X , Y ]A = [X , Y ], ρ =id.Leaves of A are the leaves of F .
Rui Loja Fernandes Stability of Leaves
![Page 123: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/123.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroidsExamples
Flows. For X ∈ X(M), the associated Lie algebroid is:A = M × R, [f , g]A := fX (g)− gX (f ), ρ(f ) = fX .
Leaves of A are the orbits of X .
Foliations. For F ∈ Folk (M), the associated Lie algebroidis:
A = TF , [X , Y ]A = [X , Y ], ρ =id.Leaves of A are the leaves of F .
Rui Loja Fernandes Stability of Leaves
![Page 124: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/124.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroidsExamples
Actions. For α ∈ Act(G; M), the associated Lie algebroid is:A = M × g, ρ =infinitesimal action,[f , g]A(x) = [f (x), g(x)]g + Lρ(f (x))g(x)− Lρ(g(x))f (x).
Leaves of A are the orbits of α (for G connected).
Poisson structures. For π ∈ Poiss(M), the associated Liealgebroid is:
A = T ∗M, ρ = π],[df , dg]A = df , g, (f , g ∈ C∞(M)).
Leaves of A are the symplectic leaves of π.
Rui Loja Fernandes Stability of Leaves
![Page 125: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/125.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Lie algebroidsExamples
Actions. For α ∈ Act(G; M), the associated Lie algebroid is:A = M × g, ρ =infinitesimal action,[f , g]A(x) = [f (x), g(x)]g + Lρ(f (x))g(x)− Lρ(g(x))f (x).
Leaves of A are the orbits of α (for G connected).
Poisson structures. For π ∈ Poiss(M), the associated Liealgebroid is:
A = T ∗M, ρ = π],[df , dg]A = df , g, (f , g ∈ C∞(M)).
Leaves of A are the symplectic leaves of π.
Rui Loja Fernandes Stability of Leaves
![Page 126: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/126.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
For a fixed vector bundle A there is a natural compact-opentopology on the set Algbrd(A) of Lie algebroid structureson A.
A leaf L of A is called stable if every nearby Lie algebroidstructure in Algbrd(A) has a nearby leaf diffeomorphic to L.
There are natural A-cohomology theories. For a leaf L, onecan define the relative A-cohomology with coefficients inthe normal bundle ν(L), denoted H•(A|L; ν(L)).
Rui Loja Fernandes Stability of Leaves
![Page 127: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/127.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
For a fixed vector bundle A there is a natural compact-opentopology on the set Algbrd(A) of Lie algebroid structureson A.
A leaf L of A is called stable if every nearby Lie algebroidstructure in Algbrd(A) has a nearby leaf diffeomorphic to L.
There are natural A-cohomology theories. For a leaf L, onecan define the relative A-cohomology with coefficients inthe normal bundle ν(L), denoted H•(A|L; ν(L)).
Rui Loja Fernandes Stability of Leaves
![Page 128: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/128.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
For a fixed vector bundle A there is a natural compact-opentopology on the set Algbrd(A) of Lie algebroid structureson A.
A leaf L of A is called stable if every nearby Lie algebroidstructure in Algbrd(A) has a nearby leaf diffeomorphic to L.
There are natural A-cohomology theories. For a leaf L, onecan define the relative A-cohomology with coefficients inthe normal bundle ν(L), denoted H•(A|L; ν(L)).
Rui Loja Fernandes Stability of Leaves
![Page 129: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/129.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
Theorem (Crainic & RLF)
Let L be a compact leaf of a Lie algebroid A, and assume thatH1(A, L; ν(L)) = 0. Then L is stable.
The theorem says: infinitesimal stability⇒ stability.Likewise, the proof is a “infinite dimensional transversalityargument”.
All other stability theorems can be deduced from this one.This explains the appearence of different cohomologies.
The Lie algebroid approach allows the study of strongernotions of stability...
Rui Loja Fernandes Stability of Leaves
![Page 130: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/130.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
Theorem (Crainic & RLF)
Let L be a compact leaf of a Lie algebroid A, and assume thatH1(A, L; ν(L)) = 0. Then L is stable.
The theorem says: infinitesimal stability⇒ stability.Likewise, the proof is a “infinite dimensional transversalityargument”.
All other stability theorems can be deduced from this one.This explains the appearence of different cohomologies.
The Lie algebroid approach allows the study of strongernotions of stability...
Rui Loja Fernandes Stability of Leaves
![Page 131: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/131.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
Theorem (Crainic & RLF)
Let L be a compact leaf of a Lie algebroid A, and assume thatH1(A, L; ν(L)) = 0. Then L is stable.
The theorem says: infinitesimal stability⇒ stability.Likewise, the proof is a “infinite dimensional transversalityargument”.
All other stability theorems can be deduced from this one.This explains the appearence of different cohomologies.
The Lie algebroid approach allows the study of strongernotions of stability...
Rui Loja Fernandes Stability of Leaves
![Page 132: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/132.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Basic problemGeometric Lie theoryUniversal Stability Theorem
Stability Theorem
Theorem (Crainic & RLF)
Let L be a compact leaf of a Lie algebroid A, and assume thatH1(A, L; ν(L)) = 0. Then L is stable.
The theorem says: infinitesimal stability⇒ stability.Likewise, the proof is a “infinite dimensional transversalityargument”.
All other stability theorems can be deduced from this one.This explains the appearence of different cohomologies.
The Lie algebroid approach allows the study of strongernotions of stability...
Rui Loja Fernandes Stability of Leaves
![Page 133: Stability of Leaves - CIMRui Loja Fernandes Stability of Leaves Classical Results Stability of symplectic leaves Universal Stability Theorem Summary Flows Foliations Group actions](https://reader036.vdocuments.net/reader036/viewer/2022081621/6121e40553a6c514f468fef3/html5/thumbnails/133.jpg)
Classical ResultsStability of symplectic leavesUniversal Stability Theorem
Summary
Moral: There is a general framework to deal with stability of“leaf-type” problems.
Rui Loja Fernandes Stability of Leaves