Nous introduisons de nouvelles régularités de Kuo-Verdier $\left(r^e\right)$ et montrons que pour une stratification $C^2$$(a+r^e)...$

It is proved that the normal bundle $\mathscr{H}$ of a distribution $𝒱$ on a riemannian manifold admits a conformal curvature $C$ if and only if $𝒱$ is a conformal foliation. Then $\mathscr{H}$ is conformally flat if and only if $C$ vanishes. Also, the Pontrjagin classes of $\mathscr{H}$ can be expressed in terms of $C$.

For odd-dimensional Poincaré–Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<n/2$) which are $C^m$ (with $m\in \mathbb{N}$) on the conformal compactification $\overline{X}$ of $X$. This set is infinite-dimensional for small $m$ but it becomes finite-dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\overline{X},\partial \overline{X})$ and the kernel of the Branson–Gover [3] differential operators $(L_k,G_k)$ on the conformal infinity $(\partial \overline{X},\left[h_0\right])$. We also relate the set of $C^{n-2k+1}\left({\Lambda }^k\left(\overline{X}\right)\right)$ forms in the kernel of $d+{\delta }_g$ to the conformal...

In this paper we consider a product preserving functor $ℱ$ of order $r$ and a connection $\Gamma$ of order $r$ on a manifold $M$. We introduce horizontal lifts of tensor fields and linear connections from $M$ to $ℱ\left(M\right)$ with respect to $\Gamma$. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.

We classify all $ℱ{ℳ}_{m,n}$-natural operators $:J²⟿J²V^A$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $\left(\Gamma \right):V^{A}Y\to J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y\to M$ corresponding to a Weil algebra A.

For every product preserving bundle functor $T^{\mu }$ on fibered manifolds, we describe the underlying functor of any order $(r,s,q),s\ge r\le q$. We define the bundle $K_{k,l}^{r,s,q}Y$ of $(k,l)$-dimensional contact elements of the order $(r,s,q)$ on a fibered manifold $Y$ and we characterize its elements geometrically. Then we study the bundle of general contact elements of type $\mu$. We also determine all natural transformations of $K_{k,l}^{r,s,q}Y$ into itself and of $T\left(K_{k,l}^{r,s,q}Y\right)$ into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms...

We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $ℰ$ in the Lagrangian Grassmann bundle $M^{\left(1\right)}$ over a $(2n+1)$-dimensional contact manifold $(M,𝒞)$. We develop the theory of characteristics of $ℰ$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $ℰ$. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of...