We investigate the global existence, uniqueness and exponential stability of almost periodic (AP) and pseudo almost periodic (PAP) mild solutions to the Boussinesq systems on the non-compact Riemannian manifold \((\textbf{M},g)\) , which satisfies certain bounded and negative curvature conditions. By using exponential dispersive and smoothing estimates for the scalar heat and Stokes semigroups, we prove the boundedness and well-posedness of mild solutions defined on the whole line time-axis (i.e., \(t \in \mathbb {R}\) ) for linear systems. Moreover, we establish Massera-type principles ensuring the existence of AP and PAP mild solutions for linear systems. Next, we obtain the well-posedness of such solutions for semilinear systems by employing results from the linear case and fixed-point arguments. Finally, we prove the exponential stability of these mild solutions using Gronwall’s inequality.
