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 $C^{\infty }(E₁ₓ,E₂ₓ)$, 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...

We prove that the problem of finding all $ℳf_m$-natural operators $C:Q⇝Q{T}^{r*}$ lifting classical linear connections $\nabla$ on $m$-manifolds $M$ into classical linear connections $C_M\left(\nabla \right)$ on the $r$-th order cotangent bundle $T^{r*}M=J^r{(M,ℝ)}_0$ of $M$ can be reduced to the well known one of describing all $ℳf_m$-natural operators $D:Q⇝{⨂}^{p}T\otimes {⨂}^q{T}^{*}$ sending classical linear connections $\nabla$ on $m$-manifolds $M$ into tensor fields $D_M\left(\nabla \right)$ of type $(p,q)$ on $M$.

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...

Soit $\xi$ un $G$-fibré principal différentiable sur une variété $M$ ($G$ un groupe de Lie compact). Étant donné une action d’un groupe de Lie compact $\Gamma$ sur $M$, on se pose la question de savoir si elle provient d’une action sur le fibré $\xi$. L’originalité de ce travail est de relier ce problème à l’existence de points fixes pour les actions de $\Gamma$ que l’on induit naturellement sur divers espaces de modules de $G$-connexions sur $\xi$.

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).

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.

By a torsion of a general connection $\Gamma$ on a fibered manifold $Y\to M$ we understand the Frölicher-Nijenhuis bracket of $\Gamma$ and some canonical tangent valued one-form (affinor) on $Y$. Using all natural affinors on higher order cotangent bundles, we determine all torsions of general connections on such bundles. We present the geometrical interpretation and study some properties of the torsions.

We introduce the concept of a dynamical connection on a time-dependent Weil bundle and we characterize the structure of dynamical connections. Then we describe all torsions of dynamical connections.

A total connection of order $r$ in a Lie groupoid $\Phi$ over $M$ is defined as a first order connections in the $(r-1)$-st jet prolongations of $\Phi$. A connection in the groupoid $\Phi$ together with a linear connection on its base, ie. in the groupoid $\Pi \left(M\right)$, give rise to a total connection of order $r$, which is called simple. It is shown that this simple connection is curvature-free iff the generating connections are. Also, an $r$-th order total connection in $\Phi$ defines a total reduction of the $r$-th prolongation of $\Phi$ to $\Phi \times \Pi \left(M\right)$....