Characteristic distribution: An application to material bodies

Abstract

Associated to each material body $\mathcal{B}$ there exists a groupoid $Ω\left( \mathcal{B} \right)$ consisting of all the material isomorphisms connecting the points of $\mathcal{B}$. The uniformity character of $\mathcal{B}$ is reflected in the properties of $Ω\left( \mathcal{B} \right)$: $\mathcal{B}$ is uniform if, and only if, $Ω\left( \mathcal{B} \right)$ is transitive. Smooth uniformity corresponds to a Lie groupoid and, specifically, to a Lie subgroupoid of the groupoid $Π^1\left({\mathcal B}, {\mathcal B} \right)$ of 1-jets of $\mathcal B$. We consider a general situation when $Ω\left( \mathcal{B} \right)$ is only an algebraic subgroupoid. Even in this case, we can cover $\mathcal{B}$ by a material foliation whose leaves are transitive. The same happens with $Ω\left( \mathcal{B} \right)$ and the corresponding leaves generate transitive Lie groupoids (roughly speaking, the leaves covering $\mathcal B$). This result opens the possibility to study the homogeneity of general material bodies using geometric instruments.