Hypoellipticity: Geometrization and Speculation

Abstract

To any finite collection of smooth real vector fields $X_j$ in $\reals^n$ we associate a metric in the phase space $T^*\reals^n$. 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