  We investigate the global existence and exponential decay of mild solutions
for the Boussinesq systems in $L^p$-phase spaces on the framework of real
hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$, where $d \geqslant 2$ and
$1<p\leq d$. We consider a couple of Ebin-Marsden's Laplace and
Laplace-Beltrami operators associated with the corresponding linear system
which provides a vectorial matrix semigoup. Primarily, we show the existence
and the uniqueness of the bounded mild solution for the linear system by using
dispersive and smoothing estimates of the vectorial matrix semigroup. Next,
using the fixed point arguments, we can pass from the linear system to the
semilinear system to establish the existence of the bounded mild solutions. Due
to Gronwall's inequality, we will clarify the exponential stability of such
solutions. Finally, we give an application of stability to the existence of
periodic mild solutions for the Boussinesq systems.
