HYPOELLIPTICITY: GEOMETRIZATION AND SPECULATION

MICHAEL CHRIST

Abstract. To any finite collection of smooth real vector fields X_{j} in ℝ^{n} we associate a metric in the phase space T^{∗}ℝ^{n}. The relation between the asymptotic behavior of this metric and hypoellipticity of ∑ X_{j}^{2}, in the smooth, real analytic, and Gevrey categories, is explored.