  This article we investigate the asymptotically almost periodic (AAP-)
solutions of Navier-Stokes equation on the real hyperbolic manifold
$M:=\mathbb{H}^d(\mathbb{R})$ with dimension $d$. Using the dispertive and
smoothing estimates for Stokes equation obtained in the work of Pierfelice [V.
Pierfelice, The incompressible Navier-Stokes equations on non-compact
manifolds, Journal of Geometric Analysis, 27(1) (2017), 577-617], we invoke the
Messera's principle to prove the existence, uniqueness of the AAP-solution for
the Stokes equation in $L^p(\Gamma(TM)))$ space with $p>d$. We then establish
the existence and uniqueness of the small AAP-solutions of the Navier-Stokes
equation by using the fixed point argument. The asymptotic behaviour
(exponential decay and stability) of these small solutions are also related.
