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

Global hypoellipticity of sums of squares on compact manifolds