Hypoellipticity: Geometrization and Speculation

To any finite collection of smooth real vector fields $X_j$ in ${\mathbb{R}}^n$ we associate a metric in the phase space $T^{\ast }{\mathbb{R}}^n$. The relation between the asymptotic behavior of this metric and hypoellipticity of $\sum X_j^2$, in the smooth, real analytic, and Gevrey categories, is explored.