  In this paper we study the forward asymptotically almost periodic (AAP-) mild
solutions of Navier-Stokes equations on the real hyperbolic manifold
$\mathcal{M}=\mathbb{H}^d(\mathbb{R})$ with dimension $d \geq 2$. Using the
dispersive and smoothing estimates for the Stokes equation we invoke the
Massera-type principle to prove the existence and uniqueness of the AAP- mild
solution for the inhomogeneous Stokes equations in $L^p(\Gamma(T\mathcal{M})))$
space with $1<p\leq d$. Next, we establish the existence and uniqueness of the
small AAP- mild solutions of the Navier-Stokes equations by using the fixed
point argument and the results of inhomogeneous Stokes equations. The
asymptotic behaviour (exponential decay and stability) of these small solutions
are also related. This work, together with our recent work [P.T. Xuan, N.T. Van
and B. Quoc, {\it On Asymptotically Almost Periodic Solution of Parabolic
Equations on real hyperbolic Manifolds}, J. Math. Anal. Appl., Vol. 517, Iss. 1
(2023), pages 1-19], provide a full existence and asymptotic behaviour of AAP-
mild solutions of Navier-Stokes equations in $L^p(\Gamma(T\mathcal{M})))$
spaces for all $p>1$.
