Tangent sphere bundles of natural diagonal lift type.
Tensor fields and connections on cross-sections in the frame bundle of second order of a parallelizable manifold.
Tensor fields of type (0,2) on linear frame bundles and cotangent bundles
Teorema de Gauss-Bonnet para fíbrados de Seifert.
The Bellaterra connection.
A new object is introduced - the "Fischer bundle". It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces"). The definition was inspired by our previous work on the "Fock bundle". An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view....
The Chern connection and curvature of the tangent bundle of an almost complex manifold. (La connexion et la courbure de Chern du fibré tangent d'une variété presque complexe.)
The concept of invariant geometry of second order.
The double exponential map and covariant derivation.
The Frölicher-Nijenhuis bracket on some functional spaces
Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as , for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated...
The geometry of the dual of a vector bundle.
The geometry of the moduli space of CP2 instantons.
The natural operators lifting connections to higher order cotangent bundles
We prove that the problem of finding all -natural operators lifting classical linear connections on -manifolds into classical linear connections on the -th order cotangent bundle of can be reduced to the well known one of describing all -natural operators sending classical linear connections on -manifolds into tensor fields of type on .
The Nonlinear Connections and Harmonicity.
The symplectic geometry of affine connenctions on surfaces.
The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles
Around 1923, Élie Cartan introduced affine connections on manifolds and defined the main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Élie Cartan...
The Yang-Mills equations and the topology of 4-manifolds
Théorie de jauge et symétries des fibrés
Soit un -fibré principal différentiable sur une variété ( un groupe de Lie compact). Étant donné une action d’un groupe de Lie compact sur , on se pose la question de savoir si elle provient d’une action sur le fibré . L’originalité de ce travail est de relier ce problème à l’existence de points fixes pour les actions de que l’on induit naturellement sur divers espaces de modules de -connexions sur .
Théorie de Voronoï géométrique. Propriétés de finitude pour les familles de réseaux et analogues
Nous développons une théorie de Voronoï géométrique. En l’appliquant aux familles classiques de réseaux euclidiens (par exemple symplectiques ou orthogonaux), nous obtenons notamment de nouveaux résultats de finitude concernant les configurations de vecteurs minimaux et les réseaux particuliers (par exemple parfaits) de ces familles. Les méthodes géométriques introduites sont également illustrées par l’étude d’objets voisins (formes de Humbert) ou analogues (surfaces de Riemann).
Theory of conformally Berwald Finsler spaces and its applications to -metrics.
Torsion and the second fundamental form for distributions
The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry.