  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.
