Sur les algèbres formelles associées par déformation a une variété symplectique

Résumé Sur une varieté symplectique (W, F), on considère une algébre associative formelle (E(N;v),x_v) obtenue par déformation du produit usuel des fonctions et satisfaisant à des hypothèses générales concernant le crochet de Poisson. Cette algèbre est déterminée de manièr unique par l'algèbre de Lie qu'elle engendre. On détermine les dérivations et les automorphismes de l'algèbre associative.     

Ondes et radiations électromagnétiques et gravitationelles en relativité générale

Résumé Théorie des ondes et radiations gravitationelles basée sur l’analogie existant entre les comportements du tenseur de courbure et du tenseur champ électromagnétique en relativité générale. Contribution à la quantification des champs. à Antonio Signorini pour son 70^me anniversaire.     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

Existence and equivalence of twisted products on a symplectic manifold

The twisted products play an important role in Quantum Mechanics [1, 2]. We introduce here a distinction between Vey *_ν-products and strong Vey *_ν-products and prove that each *_ν-product is equivalent to a Vey *_ν-product. If b ₃(W)=0, the symplectic manifold (W, F) admits strong Vey *_ν-products. If b ₂(W)=0, all *_ν-products are equivalent as well as the Vey Lie algebras. In the general case, we characterize the formal Lie algebras which are generated by a *_ν-product and we prove that the existence of a *_ν-product is equivalent to the existence of a formal Lie algebra infinitesimally equivalent to a Vey Lie algebra at the first order.     

First Eigenvalue of the Dirac Operator for a Kähler Manifold of Even Complex Dimension: The Limiting Case

K. D. Kirchberg has given a minoration of the absolute value of the eigenvalues of the Dirac operator for a compact Kähler spin manifold (W,g) with positive scalar curvature and has introduced, in this context, the notion of Kähler twistor-spinor. We prove here that if dim_C W = p 4 is even, in the limiting case, (W, g) is the Kähler product of an odd-dimensional limiting case compact Kähler spin manifold of complex dimension (p-1), by a flat Kähler manifold which is a compact quotient of C.     

Characterization of Lie groups on the cotangent bundle of a Lie group

Characterization, in differential geometric terms, of the groups which can be interpreted as semidirect products of a Lie group G by the group of translations of the dual space of its Lie algebra. Study of the canonical cotangent group of G corresponding to the coadjoint representation. Applications.     

Conformal symplectic geometry,deformations, rigidity and geometrical (KMS) conditions

Deformations admitting a unit element of a local associative algebra defined on the space of functions on a manifold. Definition and properties of the * ^f -products and conformal symplectic geometry. Deformations of a * ^f -products. A theorem of rigidity. Application to statistical mechanics (KMS conditions).     

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.An analysis is made of the twistor-spinors given by , where f is a complex-valued function. There is only one case where is not a Killing spinor. An example is given of a compact spin manifold for which the situation is realized.     

Spin manifolds,killing spinors and universality of the Hijazi inequality

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (–) on the spinors (=(n/4(n–1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension 3, the Hijazi inequality [8] holds for every spinor field such that (P, P) = ²(, ) (=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of . Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the twistor operator.