The contact system on the spaces of $(m,\ell)$-velocities

J. Munoz, F. J. Muriel and J. Rodriguez


Abstract. In this paper we define the contact system on the space $M_m^\ell$ of the $(m,\ell)$-velocities of a smooth manifold $M$. For each velocity $p_m^\ell\in M_m^\ell$, the tangent space $T_{p_m^\ell}M_n^\ell$ and the ${\bf R}_m^\ell$-module ${\rm Der}_{\bf R}(C^\infty(M),{\bf R}_m^\ell)$ are canonically isomorphic; as a consequence, $p_m^\ell$ gives rise to a morphism ${p_m^\ell}_*$ between the ${\bf R}_m^{\ell-1}$-modules ${\rm Der}_{\bf R}({\bf R}_m^\ell,{\bf R}_m^{\ell-1})$ and $T_{p_m^{\ell-1}}M_m^{\ell-1}$ which is injective if and only if $p_m^\ell$ is regular. If $X$ is an $m$-dimensional submanifold of $M$ and $p_m^\ell$ is a regular point of $X_m^\ell$, then the image of the above morphism is the tangent space to $X_m^{\ell-1}$ at $p_m^{\ell-1}$; in this sense, $p_m^\ell$ is a frame for $X_m^{\ell-1}$ at $p_m^{\ell-1}$. Each smooth differential form on $M$ can be prolonged to a form on $M_m^\ell$ with values in ${\bf R}_m^\ell$; the inner product of the lift of each $(m+1)$-form $\omega$ on $M$ to $M_m^{\ell-1}$ with the image by each ${p_m^\ell}_*$ of a basis of ${\rm Der}_{\bf R}({\bf R}_m^\ell,{\bf R}_m^{\ell-1})$ gives rise to an ${\bf R}_m^{\ell-1}$-valued $1$-form defined on $M_m^\ell$. The Pfaff system generated by the real components of those $1$-forms, when $\omega$ runs through the set of $(m+1)$-forms on $M$, is the contact system on $M_m^\ell$.

AMSclassification. 58A20

Keywords. Near points, jets, contact system, velocities