Geometric structures and Pfaffian groupoids Pure and Applied Differential Geometry - Katholieke Universiteit Leuven

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1 Geometric structures and Pfaffian groupoids Pure and Applied Differential Geometry - Katholieke Universiteit Leuven (joint work with Marius Crainic) 23 August 2017

2 Introduction (integrable) G-structures Atlases and pseudogroups generalise to are used to define Γ-structures are the integrable versions of Pfaffian objects are used to handle Almost Γ-structures

3 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): 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) Intrinsic torsion tensors T intr k = progressive obstructions to integrability

4 Atlases and pseudogroups Definition A Γ G -atlas is a smooth atlas with changes of coordinates in Γ G = {φ Diff loc (R n ) Jac x (φ) G x dom(φ)} Example: symplectic structure atlas of symplectomorphisms, ie elements of Γ Sp(n) = {φ Diffloc (R 2n ) φ ω can = ω can } Remark We can generalise this for every Γ pseudogroup, even if Γ Γ G or Γ non transitive Example: contact structure atlas of contactomorphisms, ie elements of Γ cont = {φ Diff loc (R 2n+1 ) φ ξ can = ξ can }

5 Γ-structures Definition A Γ-atlas on M is an atlas A = {(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 Key idea: consider J 1 A J 1 (R n, M) = Fr(M) Theorem (generalisation of left-hand side) Γ-structures Principal Germ(Γ)-bundles over M

6 Pfaffian groupoids (Salazar) Inspiration: Isolate the abstract properties of jets of pseudogroups ω Ω 1 (G, t E), E Rep(G) ω multiplicative: (m ω) (g,h) = (pr 1 ω) (g,h) + g (pr 2 ω) (g,h) ω s-involutive: ker(ω) ker(ds) T G is involutive Lie-Pfaffian if also ker(ω) ker(dt) = ker(ω) ker(ds) (G, ω) s t M Example: J k Γ X Intuition Pfaffian groupoid = PDE on jet bundle (G s M Pfaffian bundle ) such that solutions can be multiplied Analogy: Symplectic groupoids (G, ω) (ω multiplicative+symplectic)

7 Pfaffian actions and principal (Lie)-Pfaffian bundles Inspiration: Hamiltonian actions of symplectic groupoids m P : G s µ P P action of G on P (G, ω) Pfaffian groupoid, θ Ω 1 (P, µ E) Pfaffian action: (m P θ) (g,p) = (pr 1 ω) (g,p) + g (pr 2 ω) (g,p) Principal Pfaffian: P π P/G principal bundle Principal Lie-Pfaffian: ker(µ) ker(θ) = ker(π) ker(θ) (G, ω) (P, θ) s M t m P Example: J k (X, M) M is a principal Lie-Pfaffian J k Γ-bundle Remark: Given the Γ-atlas A, J k A inherits the same structure µ π P/G

8 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 Corollary A Γ-structure J k A J k (X, M) k th -order almost Γ-structure Theorem 1 st order almost Γ G -structures G-structures Key idea: consider P J 1 (R n, M) = Fr(M) and J 1 Γ G = R n R n G

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

10 Obstructions to integrability Integrable G-structures Formally integrable G-structures (in progress) Γ G -structures th -order almost Γ G -structures T intr k G-structures with T intr k+1 G-structures with T intr k (in progress) (k + 1) th -order almost Γ G -structures = 0 obstructions? (in progress) k th -order almost Γ G -structures G-structures (with T intr 1 ) 1 st -order almost Γ G -structures

11 Future results Theorems in progress (using Morita equivalences, prolongations, etc) Aim: define inductive obstructions Tk intr for almost Γ-structures through a suitable (Spencer) cohomology H = 0 for every ( th -order almost) Γ-structure If H k = 0, a k-order almost Γ-structure induces a (k + 1)-order one If H k vanishes analytically, a k-order structure induces a Γ-structure Alternative solution to (formal) integrability problem for G-structures

12 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, 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