Global hypoellipticity of sums of squares on compact manifolds

In this work, we present necessary and sufficient conditions for an operator of the type sum of squares to be globally hypoelliptic on a product of compact Riemannian manifolds $T \times G$, where $G$ is also a Lie group. These new conditions involve the global hypoellipticity of a system of vector fields and are weaker than H\"ormander's condition, at the same time that they generalize the well known Diophantine conditions on the torus. We were also able to provide examples of operators satisfying these conditions in the general setting.