Nous caractérisons, en terme de dimension (topologique et de Hausdorff) des fibres des espaces de limites de tangents et du cône de Whitney, les conditions de régularité $b_{\mathrm{cod}q}$ et $b^{*}$ sur une stratification $C^1$. Nous précisons ces résultats lorsque les espaces qui interviennent ne sont pas fractals, en particulier lorsque la stratification est sous-analytique.

For the geometry of oriented $(2,3,5)$ distributions $(M,)$, which correspond to regular, normal parabolic geometries of type $(G_2,P)$ for a particular parabolic subgroup $P<G_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator ${\Theta }_0$ associated to the $7$-dimensional irreducible representation of $G_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator...

We study the cohomology properties of the singular foliation ℱ determined by an action Φ: G × M → M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $ℍ{*}_{p̅}(M/ℱ)$ is finite-dimensional and satisfies the Poincaré duality. This duality includes two well known situations: ∙ Poincaré duality for basic cohomology (the action Φ is almost free). ∙ Poincaré duality for intersection cohomology (the group G is compact...

We classify classical linear connections $A(\Gamma ,\Lambda ,\Theta )$ on the total space $Y$ of a fibred manifold $Y\to M$ induced in a natural way by the following three objects: a general connection $\Gamma$ in $Y\to M$, a classical linear connection $\Lambda$ on $M$ and a linear connection $\Theta$ in the vertical bundle $VY\to Y$. The main result says that if $\dim \left(M\right)\ge 3$ and $\dim \left(Y\right)-\dim \left(M\right)\ge 3$ then the natural operators $A$ under consideration form the $17$ dimensional affine space.

We prove that every natural affinor on ${\left(J^r\left({\odot }^2{T}^{*}\right)\right)}^{*}\left(M\right)$ is proportional to the identity affinor if dim$M\ge 3$.

We define suitable Sobolev topologies on the space ${𝒞}_P(B_k,f)$ of connections of bounded geometry and finite Yang-Mills action and the gauge group and show that the corresponding configuration space is a stratified space. The underlying open manifold is assumed to have bounded geometry.

We determine all natural operators D transforming general connections Γ on fibred manifolds Y → M and torsion free classical linear connections ∇ on M into general connections D(Γ,∇) on the second order jet prolongation J2Y → M of Y → M

Jets of a manifold $M$ can be described as ideals of ${𝒞}^{\infty }\left(M\right)$. This way, all the usual processes on jets can be directly referred to that ring. By using this fact, we give a very simple construction of the contact system on jet spaces. The same way, we also define the contact system for the recently considered $A$-jet spaces, where $A$ is a Weil algebra. We will need to introduce the concept of derived algebra.

This paper is a continuation of [MMR:98], where we give a construction of the canonical Pfaff system $\Omega \left(M_m^{\ell }\right)$ on the space of $(m,\ell )$-velocities of a smooth manifold $M$. Here we show that the characteristic system of $\Omega \left(M_m^{\ell }\right)$ agrees with the Lie algebra of $Aut\left({ℝ}_m^{\ell }\right)$, the structure group of the principal fibre bundle ${\check{M}}_m^{\ell }⟶J_m^{\ell }\left(M\right)$, hence it is projectable to an irreducible contact system on the space of $(m,\ell )$-jets ($=\ell$-th order contact elements of dimension $m$) of $M$. Furthermore, we translate to the language of Weil bundles the structure form...

We study naturality of the Euler and Helmholtz operators arising in the variational calculus in fibered manifolds with oriented bases.

We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety $M\subset {ℝ}^n$ is dense in the Hilbert space $L^2(M,e^{-{\left|x\right|}^2}d\mu )$, where dμ denotes the volume form of M and $d\nu =e^{-{\left|x\right|}^2}d\mu$ the Gaussian measure on M.

Let $M$ be an $m$-dimensional manifold and $A={𝔻}_k^r/I=ℝ\oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T_x^{A}f\in T_x^{A*}M$, $x\in M$ is determined by its values over arbitrary $max\{\mathrm{width}A,m\}$ regular and under the first jet projection linearly independent elements of $T_x^{A}M$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M\simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial result obtained...