Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus

Let L=∂/∂t+∑j=1N(aj+ibj)(t)∂/∂xj be a vector field defined on the torus TN+1≃RN+1/2πZN+1, where aj, bj are real-valued functions and belonging to the Gevrey class Gs(T1), s>1, for j=1,…,N. We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets.