Isomorphisms of Lie algebras of vector fields
Le problème d'équivalence pour les pseudogroupes de Lie : méthodes intrinsèques
Lie groupoids as generalized atlases
Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings...
Lie groupoids of mappings taking values in a Lie groupoid
Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we...
Local and nice structures of the groupoid of an equivalence relation
Losik cohomology of the Lie algebra of infinitesimal automorphisms of a -structure
Minimal, rigid foliations by curves on
We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space for every dimension and every degree . Precisely, we construct a foliation which is induced by a homogeneous vector field of degree , has a finite singular set and all the regular leaves are dense in the whole of . Moreover, satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if is conjugate to another holomorphic foliation...
Morales-Ramis Theorems via Malgrange pseudogroup
In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.
Morphisms between spaces of leaves viewed as fractions
Nondegenerate cohomology pairing for transitive Lie algebroids, characterization
The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q Aor, Dor) by tensoring by orientation flat line bundle, Q Aor=QA⊗or (M) and D or=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of...
On a general difference Galois theory I
We know well difference Picard-Vessiot theory, Galois theory of linear difference equations. We propose a general Galois theory of difference equations that generalizes Picard-Vessiot theory. For every difference field extension of characteristic , we attach its Galois group, which is a group of coordinate transformation.
On a general difference Galois theory II
We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.
On a partial complex structure
On characterization of Poisson and Jacobi structures
We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.
On formal theory of differential equations. III.
Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy...
On infinite Lie groups
Under some regularity conditions one proves that quotients and kernels of infinitesimal analytic Lie pseudo-groups by invariant fiberings are again infinitesimal Lie pseudo-groups. The regularity conditions are shown to be necessary and sufficient if one wishes both quotient and kernel to be infinitesimal Lie pseudo-groups. One defines and proves the existence of the quotient of an infinitesimal Lie pseudo-group by a normal sub-pseudo group. An equivalence relation for germs of infinitesimal Lie...
On infinitesimal isometries of a hypersurface
On Lie algebroid actions and morphisms
On mappings of a manifold into a Lie group
On spectra of geometric operators on open manifolds and differentiable groupoids.