Geometric structures and Pfaffian groupoids Geometry and Algebra of PDEs - Universitetet i Tromsø

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1 Geometric structures and Pfaffian groupoids Geometry and Algebra of PDEs - Universitetet i Tromsø (joint work with Marius Crainic) 6 June 2017

2 Index 1 G-structures 2 Γ-structures 3 Pfaffian objects 4 Almost Γ-structures

3 Index 1 G-structures 2 Γ-structures 3 Pfaffian objects 4 Almost Γ-structures

4 Introduction G-structures and integrability Atlases and pseudogroups generalise to are used to define Γ-structures

5 Introduction G-structures and integrability Atlases and pseudogroups generalise to are used to define Γ-structures is the integrable version of Almost Γ-structures

6 Introduction G-structures and integrability Atlases and pseudogroups generalise to are used to define Γ-structures is the integrable version of Pfaffian objects are used to define Almost Γ-structures

7 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM.

8 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM. S( is integrable if every ) point x M admits a chart (U, χ) such that (x),..., χ 1 χ (x) Fr(T n x M) is actually in S x Fr(T x M)

9 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM. S( is integrable if every ) point x M admits a chart (U, χ) such that (x),..., χ 1 χ (x) Fr(T n x M) is actually in S x Fr(T x M) G = O(n, R): Riemannian metric g (integrable flat)

10 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM. S( is integrable if every ) point x M admits a chart (U, χ) such that (x),..., χ 1 χ (x) Fr(T n x M) is actually in S x Fr(T x M) G = O(n, R): Riemannian metric g (integrable flat) G = Sp(n): almost symplectic structure ω (integrable dω = 0)

11 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM. S( is integrable if every ) point x M admits a chart (U, χ) such that (x),..., χ 1 χ (x) Fr(T n x M) is actually in S x Fr(T x M) G = O(n, R): Riemannian metric g (integrable flat) G = Sp(n): almost symplectic structure ω (integrable dω = 0) G = GL(n, C): almost complex structure J (integrable N J = 0)

12 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM. S( is integrable if every ) point x M admits a chart (U, χ) such that (x),..., χ 1 χ (x) Fr(T n x M) is actually in S x Fr(T x M) G = O(n, R): Riemannian metric g (integrable flat) G = Sp(n): almost symplectic structure ω (integrable dω = 0) G = GL(n, C): almost complex structure J (integrable N J = 0) G = GL(p, n p): rank p distribution F (integrable involutive)

13 G-structures and integrability Definition A G-structure on M is a G-invariant submanifold S Fr(TM) such that π S : S M is a principal G-subbundle of the frame bundle of TM. S( is integrable if every ) point x M admits a chart (U, χ) such that (x),..., χ 1 χ (x) Fr(T n x M) is actually in S x Fr(T x M) G = O(n, R): Riemannian metric g (integrable flat) G = Sp(n): almost symplectic structure ω (integrable dω = 0) G = GL(n, C): almost complex structure J (integrable N J = 0) G = GL(p, n p): rank p distribution F (integrable involutive) Intrinsic torsion tensors T intr k = progressive obstructions to integrability

14 Index 1 G-structures 2 Γ-structures 3 Pfaffian objects 4 Almost Γ-structures

15 Atlases and pseudogroups A smooth atlas {(U i, χ i )} i has changes of coordinates in Diff loc (R n )

16 Atlases and pseudogroups A smooth atlas {(U i, χ i )} i has changes of coordinates in Diff loc (R n ) A Γ G -atlas has changes of coordinates in Γ G = {φ Diff loc (R n ) Jac x (φ) G x dom(φ)}

17 Atlases and pseudogroups A smooth atlas {(U i, χ i )} i has changes of coordinates in Diff loc (R n ) A Γ G -atlas has changes of coordinates in Γ G = {φ Diff loc (R n ) Jac x (φ) G x dom(φ)} Example: symplectic structure atlas of symplectomorphisms, i.e. elements of Γ Sp(n) = {φ Diffloc (R 2n ) φ ω can = ω can }

18 Atlases and pseudogroups A smooth atlas {(U i, χ i )} i has changes of coordinates in Diff loc (R n ) A Γ G -atlas has changes of coordinates in Γ G = {φ Diff loc (R n ) Jac x (φ) G x dom(φ)} Example: symplectic structure atlas of symplectomorphisms, i.e. elements of Γ Sp(n) = {φ Diffloc (R 2n ) φ ω can = ω can } We can generalise this for every Γ pseudogroup (group-like and sheaf-like set of local diffeomorphisms), even if Γ Γ G or Γ non transitive

19 Atlases and pseudogroups A smooth atlas {(U i, χ i )} i has changes of coordinates in Diff loc (R n ) A Γ G -atlas has changes of coordinates in Γ G = {φ Diff loc (R n ) Jac x (φ) G x dom(φ)} Example: symplectic structure atlas of symplectomorphisms, i.e. elements of Γ Sp(n) = {φ Diffloc (R 2n ) φ ω can = ω can } We can generalise this for every Γ pseudogroup (group-like and sheaf-like set of local diffeomorphisms), even if Γ Γ G or Γ non transitive Example: contact structure atlas of contactomorphisms, i.e. elements of Γ cont = {φ Diff loc (R 2n+1 ) φ ξ can = ξ can }

20 Γ-structure Definition A Γ-atlas on M is an atlas {(U i, χ i )} i with changes of coordinates in the pseudogroup Γ Diff loc (X).

21 Γ-structure Definition A Γ-atlas on M is an atlas {(U i, χ i )} i with changes of coordinates in the pseudogroup Γ Diff loc (X). A Γ-structure is an equivalence class of Γ-atlases.

22 Γ-structure Definition A Γ-atlas on M is an atlas {(U i, χ i )} i with changes of coordinates in the pseudogroup Γ Diff loc (X). A Γ-structure is an equivalence class of Γ-atlases. In the previous cases X was R n, in general the atlas can be modelled on X

23 Γ-structure Definition A Γ-atlas on M is an atlas {(U i, χ i )} i with changes of coordinates in the pseudogroup Γ Diff loc (X). A Γ-structure is an equivalence class of Γ-atlases. In the previous cases X was R n, in general the atlas can be modelled on X Theorem Γ G -structures Integrable G-structures on M

24 Γ-structure Definition A Γ-atlas on M is an atlas {(U i, χ i )} i with changes of coordinates in the pseudogroup Γ Diff loc (X). A Γ-structure is an equivalence class of Γ-atlases. In the previous cases X was R n, in general the atlas can be modelled on X Theorem Γ G -structures Integrable G-structures on M Generalisation in two directions: weakening the left-hand side (Γ instead of Γ G ) or the right (skipping the word integrable )

25 Γ-structure Definition A Γ-atlas on M is an atlas {(U i, χ i )} i with changes of coordinates in the pseudogroup Γ Diff loc (X). A Γ-structure is an equivalence class of Γ-atlases. In the previous cases X was R n, in general the atlas can be modelled on X Theorem Γ G -structures Integrable G-structures on M Generalisation in two directions: weakening the left-hand side (Γ instead of Γ G ) or the right (skipping the word integrable ) Theorem Γ-structures Principal Germ(Γ)-bundles over M

26 Index 1 G-structures 2 Γ-structures 3 Pfaffian objects 4 Almost Γ-structures

27 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form)

28 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form) ω Ω 1 (G, t E) form of constant rank, E Rep(G) (G, ω) s t M

29 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form) ω Ω 1 (G, t E) form of constant rank, E Rep(G) ω multiplicative (m ω = pr1 ω + g pr 2 ω) (G, ω) s t M

30 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form) ω Ω 1 (G, t E) form of constant rank, E Rep(G) ω multiplicative (m ω = pr 1 ω + g pr 2 ω) ω s-involutive (ker(ω) ker(ds) T G is involutive) (G, ω) s t M

31 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form) ω Ω 1 (G, t E) form of constant rank, E Rep(G) ω multiplicative (m ω = pr 1 ω + g pr 2 ω) ω s-involutive (ker(ω) ker(ds) T G is involutive) G is Lie-Pfaffian if also ker(ω) ker(dt) = ker(ω) ker(ds) (G, ω) s t M

32 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form) ω Ω 1 (G, t E) form of constant rank, E Rep(G) ω multiplicative (m ω = pr 1 ω + g pr 2 ω) ω s-involutive (ker(ω) ker(ds) T G is involutive) G is Lie-Pfaffian if also ker(ω) ker(dt) = ker(ω) ker(ds) (G, ω) s t M Example: J k Γ X is a Lie-Pfaffian groupoid if Γ Lie pseudogroup on X

33 Pfaffian groupoids (Salazar 13) Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic form) ω Ω 1 (G, t E) form of constant rank, E Rep(G) ω multiplicative (m ω = pr 1 ω + g pr 2 ω) ω s-involutive (ker(ω) ker(ds) T G is involutive) G is Lie-Pfaffian if also ker(ω) ker(dt) = ker(ω) ker(ds) (G, ω) s t M Example: J k Γ X is a Lie-Pfaffian groupoid if Γ Lie pseudogroup on X Theorem (G, ω) Pfaffian groupoid G s M Pfaffian bundle (generalisation of PDEs on jet bundles with Cartan form)

34 Pfaffian actions and principal (Lie)-Pfaffian bundles Analogy: Hamiltonian action of a symplectic groupoid on a symplectic manifold

35 Pfaffian actions and principal (Lie)-Pfaffian bundles Analogy: Hamiltonian action of a symplectic groupoid on a symplectic manifold m P : G s µ P P action of G on P along µ (G, ω) (P, θ) s M t m P µ

36 Pfaffian actions and principal (Lie)-Pfaffian bundles Analogy: Hamiltonian action of a symplectic groupoid on a symplectic manifold m P : G s µ P P action of G on P along µ (G, ω) (Lie)-Pfaffian groupoid, θ Ω 1 (P, µ m P E) (G, ω) (P, θ) s M t µ

37 Pfaffian actions and principal (Lie)-Pfaffian bundles Analogy: Hamiltonian action of a symplectic groupoid on a symplectic manifold m P : G s µ P P action of G on P along µ (G, ω) (Lie)-Pfaffian groupoid, θ Ω 1 (P, µ E) Pfaffian action: m P θ = pr 1 ω + g pr 2 ω (G, ω) (P, θ) s M t m P µ

38 Pfaffian actions and principal (Lie)-Pfaffian bundles Analogy: Hamiltonian action of a symplectic groupoid on a symplectic manifold m P : G s µ P P action of G on P along µ (G, ω) (Lie)-Pfaffian groupoid, θ Ω 1 (P, µ E) Pfaffian action: m P θ = pr 1 ω + g pr 2 ω Principal (Lie)-Pfaffian bundle: in addition, P π P/G principal G-bundle (G, ω) (P, θ) s M t m P µ π P/G

39 Pfaffian actions and principal (Lie)-Pfaffian bundles Analogy: Hamiltonian action of a symplectic groupoid on a symplectic manifold m P : G s µ P P action of G on P along µ (G, ω) (Lie)-Pfaffian groupoid, θ Ω 1 (P, µ E) Pfaffian action: m P θ = pr 1 ω + g pr 2 ω Principal (Lie)-Pfaffian bundle: in addition, P π P/G principal G-bundle (G, ω) (P, θ) s M t m P µ π P/G Example: j k x f j k x φ = j k f (x) (φ f 1 ) is a Pfaffian J k Γ-action on J k (X, M) J k (X, M) M is a principal Lie-Pfaffian J k Γ-bundle

40 Index 1 G-structures 2 Γ-structures 3 Pfaffian objects 4 Almost Γ-structures

41 Almost Γ-structures Definition (C., Crainic) P J k (X, M) is a k th -order almost Γ-structure on M if P is a principal Lie-Pfaffian J k Γ-subbundle

42 Almost Γ-structures Definition (C., Crainic) P J k (X, M) is a k th -order almost Γ-structure on M if P is a principal Lie-Pfaffian J k Γ-subbundle Theorem Φ Γ-structure J k Φ J k (X, M) k th -order almost Γ-structure

43 Almost Γ-structures Definition (C., Crainic) P J k (X, M) is a k th -order almost Γ-structure on M if P is a principal Lie-Pfaffian J k Γ-subbundle Theorem Φ Γ-structure J k Φ J k (X, M) k th -order almost Γ-structure Theorem 1 st order almost Γ G -structures G-structures

44 Obstructions to integrability Integrable G-structures Γ G -structures

45 Obstructions to integrability Integrable G-structures Γ G -structures Formally integrable G-structures. G-structures with T intr k = 0. G-structures

46 Obstructions to integrability Integrable G-structures Γ G -structures Formally integrable G-structures. th -order almost Γ G -structures. G-structures with T intr k = 0. (k + 1) th -order almost Γ G -structures. G-structures 1 st -order almost Γ G -structures

47 Obstructions to integrability Integrable G-structures Γ G -structures Formally integrable G-structures. th -order almost Γ G -structures. G-structures with T intr k = 0. (k + 1) th -order almost Γ G -structures. G-structures 1 st -order almost Γ G -structures

48 Obstructions to integrability Integrable G-structures Formally integrable G-structures (conjecture) Γ G -structures th -order almost Γ G -structures. G-structures with T intr k = 0 (conjecture). (k + 1) th -order almost Γ G -structures.. G-structures 1 st -order almost Γ G -structures

49 Obstructions to integrability Integrable G-structures Formally integrable G-structures (conjecture) Γ G -structures th -order almost Γ G -structures. G-structures with T intr k = 0 (conjecture). (k + 1) th -order almost Γ G -structures.. G-structures 1 st -order almost Γ G -structures Next: study almost Γ-structures as a subtower of the tower of (principal) Pfaffian bundles J k (X, M) J k 1 (X, M)... J 1 (X, M) ) and find relations with prolongations (generalisation from PDE theory)

50 * References [1] C. Albert, P. Molino, Pseudogroupes de Lie transitifs I. Structures principales, Hermann, Paris (1984) [2] F. Cattafi, M. Crainic, M. Salazar, From PDEs to Pfaffian bundles, in preparation [3] M. Crainic, R. Fernandes, Lectures on integrability of Lie brackets, Geom. Topol. Monogr., 17, Geom. Topol. Publ., Coventry (2011) [4] A. Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comment. Math. Helv. 32 (1958) [5] M. Salazar, Pfaffian Groupoids, PhD Thesis, (2013) [6] O. Yudilevich, Lie Pseudogroups à la Cartan from a Modern Perspective, PhD Thesis, (2016) Thank you for your attention