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...

For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do...

A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however, is not yet completely determined. The following gives as a corollary a first essential answer. It is shown, more generally,that a locally compact topological transformation group, operating effectively on a differentiable space X (which satisfies...

We write a cohomological resolution of the sheaf $𝒮$of solutions of the differential operator $\partial /\partial x_n$ on a manifold $M$ and study the cohomology groups $H^0(M,𝒮)$ and $H^1(M,𝒮)$.