Formal deformations of symplectic manifolds with boundary.
Formal geometric quantization
Let be a compact Lie group acting in a Hamiltonian way on a symplectic manifold which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map is proper so that the reduced space is compact for all . Then, we can define the “formal geometric quantization” of asThe aim of this article is to study the functorial properties of the assignment .
Generalized distance functions of Riemannian manifolds and the motions of gyroscopic systems.
Geometric Quantization and Multiplicities of Group Representations.
Geometric quantization and no-go theorems
A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle. We prove that for a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist. This is a...
Geometric quantization of integrable systems with hyperbolic singularities
We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.
Geometric Quantization on Presymplectic Manifolds.
Geometrische Krystallographie [Book]
Geometry of the Kepler system in coherent states approach
Integrability for representations appearing in geometric pre-quantization
Integrable systems and group actions
The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.
Invariants symplectiques et semi-classiques des systèmes intégrables avec singularités
On définit les notions de feuilletages classiques et semi-classiques pour les systèmes complètement intégrables avec singularités. Les résultats de classification standard (telles les coordonnées actions-angles semi-classiques) sont rappelés. Le cas du feuilletage classique de type foyer-foyer est examiné en détail, où des nouveaux invariants semi-globaux apparaissent. Ces invariants sont identifiés dans les conditions de Bohr-Sommerfeld singulières qui donnent le spectre conjoint au voisinage d’une...
Krychlový obsah šikmého rovnoběžnostěnu bez užívání sférické trigonometrie
La trilogie du moment
A toute deux-forme fermée, sur une variété connexe, on associe une famille d’extensions centrales du groupe de ses automorphismes par son tore des périodes. On discute ensuite quelques propriétés de cette construction.
Lagrangian Intersections in Contact Geometry.
Legendrian distributions with applications to relative Poincaré series.
Locally conformal symplectic manifolds.
Logarithmic Poisson cohomology: example of calculation and application to prequantization
In this paper we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure. We prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kähler differentials. Therefore it induces a differential complex from which we derive the notion of logarithmic Poisson cohomology. We prove that Poisson cohomology and logarithmic Poisson cohomology are equal when the Poisson structure is log symplectic. We give an example of...
L'oscillateur relativiste et les fonctions de Mathieu
Maslov indices on the metaplectic group
We use the properties of to construct functions associated with the elements of the lagrangian grassmannian (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between and a subset of , equipped with appropriate algebraic and topological structures.