### $\mathcal{D}$-modules, contact valued calculus and Poincaré-Cartan form

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Let $G$ be a bundle functor of order $(r,s,q)$, $s\ge r\le q$, on the category $\mathcal{F}{\mathcal{M}}_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma $ on an $\mathcal{F}{\mathcal{M}}_{m,n}$-object $Y\to M$ we construct a general connection $\mathcal{G}(\Gamma ,\lambda ,\Lambda )$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda $ on $M$ and an $s$-th order linear connection $\Lambda $ on $Y$. Then we construct a general connection $\mathcal{G}(\Gamma ,{\nabla}_{1},{\nabla}_{2})$ on $GY\to Y$ by means of auxiliary classical linear connections ${\nabla}_{1}$ on $M$ and ${\nabla}_{2}$ on $Y$. In the case $G={J}^{1}$ we determine all general connections $\mathcal{D}(\Gamma ,\nabla )$ on ${J}^{1}Y\to Y$ from...

We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map ${f}_{*}{|}_{Y}:Y\subseteq {J}^{r}(D,M)\to {J}^{r}(D,N)$ is generically (for $f:M\to N$) transverse to a submanifold $Z\subseteq {J}^{r}(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\to P$ to the preimage ${\left({j}^{s}f\right)}^{-1}\left(A\right)$ of a submanifold $A\subseteq {J}^{s}(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction ${g|}_{{\left({j}^{s}f\right)}^{-1}\left(A\right)}$ is also generic. We also present an example of $A$ where the...

Suppose that, for each point x in a given subset E ⊂ Rn, we are given an m-jet f(x) and a convex, symmetric set σ(x) of m-jets at x. We ask whether there exist a function F ∈ Cm,w(Rn) and a finite constant M, such that the m-jet of F at x belongs to f(x) + Mσ(x) for all x ∈ E. We give a necessary and sufficient condition for the existence of such F, M, provided each σ(x) satisfies a condition that we call "Whitnet w-convexity".

The theory of variational bicomplexes is a natural geometrical setting for the calculus of variations on a fibred manifold. It is a well–established theory although not spread out very much among theoretical and mathematical physicists. Here, we present a new approach to infinite order variational bicomplexes based upon the finite order approach due to Krupka. In this approach the information related to the order of jets is lost, but we have a considerable simplification both in the exposition...

The paper studies the relation between asymptotically developable functions in several complex variables and their extensions as functions of real variables. A new Taylor type formula with integral remainder in several variables is an essential tool. We prove that strongly asymptotically developable functions defined on polysectors have ${C}^{\infty}$ extensions from any subpolysector; the Gevrey case is included.

Weil algebra morphisms induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle passes to jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure...

We describe all M fm-natural operators S: Q ↝ Symp P1 transforming classical linear connections ∇ on m-dimensional manifolds M into almost symplectic structures S(∇) on the linear frame bundle P1M over M.

Second order anti-holonomic jets as anti-symmetric parts of second order semi-holonomic jets are introduced. The anti-holonomic nature of the Lie bracket is shown. A general result on universality of the Lie bracket is proved.