  In this paper, we shall study the parabolic-elliptic Keller-Segel system on
the Poincar{\'e} disk model of the 2D-hyperbolic space. We shall investigate
how the negative curvature of this Riemannian manifold influences the solutions
of this system. As in the 2D-Euclidean case, under the sub-critical condition
$\chi$M < 8$\pi$, we shall prove global well-posedness results with any initial
L 1-data. More precisely, by using dispersive and smoothing estimates we shall
prove Fujita-Kato type theorems for local well-posedness. We shall then use the
logarithmic Hardy-Littlewood-Sobolev estimates on the hyper-bolic space to
prove that the solution cannot blow-up in finite time. For larger mass $\chi$M
> 8$\pi$, we shall obtain a blow-up result under an additional condition with
respect to the flat case, probably due to the spectral gap of the
Laplace-Beltrami operator. According to the exponential growth of the
hyperbolic space, we find a suitable weighted moment of exponential type on the
initial data for blow-up.
