  In this paper, we investigate to the existence and uniqueness of periodic
solutions for the parabolic-elliptic Keller-Segel system on whole spaces
detailized by Euclidean space $\mathbb{R}^n\,\,(\hbox{ where }n \geqslant 4)$
and real hyperbolic space $\mathbb{H}^n\,\, (\hbox{where }n \geqslant 2)$. We
work in framework of scritical spaces such as on weak-Lorentz space
$L^{\frac{n}{2},\infty}(\mathbb{R}^n)$ to obtain the results for Keller-Segel
system on $\mathbb{R}^n$ and on $L^{\frac{p}{2}}(\mathbb{H}^n)$ for $n<p<2n$ to
obtain the ones on $\mathbb{H}^n$. Our method is based on the dispersive and
smoothing estimates of the heat semigroup and fixed point arguments. This work
provides also a fully comparison between the asymptotic behaviours of periodic
mild solutions of Keller-Segel system obtained in $\mathbb{R}^n$ and the one in
$\mathbb{H}^n$.
