  The main goal of this paper is to address global hypoellipticity issues for
the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in
\mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a
closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator
on $M$, smoothly depending on the periodic variable $t$. In the case of
separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we
give necessary and sufficient conditions for the global hypoellipticity of $L$.
In particular, we show that, under suitable conditions, the famous (P)
condition of Niremberg-Treves is neither necessary nor sufficient to guarantee
the global hypoellipticity of $L$.
