In this paper we extend a recent work \cite{HuyXuan2020} to study the forward
asymptotically almost periodic (AAP-) mild solution of Navier-Stokes equation
on the real hyperbolic manifold Hd(R) with dimension d≥2. Using the dispertive and smoothing estimates for Stokes equation
\cite{Pi} we invoke the Massera-type principle to prove the existence and
uniqueness of the AAP- mild solution for the Stokes equation in
Lp(Γ(TM))) space with p>d. We then establish the existence and
uniqueness of the small AAP- mild 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. Our results extend also
\cite{FaTa2013} for the forward asymptotic mild solution of the Navier-Stokes
equation on the curved background.Comment: 21 page