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

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

**E-mail:** clint@gugu.usal.es, fjmuriel@unex.es, jrl@gugu.usal.es
**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