Global analytic hypoellipticity for a class of evolution operators on ${\mathbb{T}}^1\times {\mathbb{S}}^3$

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on ${\mathbb{T}}^1\times {\mathbb{S}}^3$. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition ($\mathcal{P}$) holds, in addition to a Diophantine condition.