An invariant K\"ahler metric on the tangent disk bundle of a space-form

We find a family of K"ahler metrics invariantly defined on the radius $r_0>0$ tangent disk bundle ${\mathcal{T}}_{M,r_0}$ of any given real space-form $M$ or any of its quotients by discrete groups of isometries. Such metrics are complete in the non-negative curvature case and non-complete in the negative curvature case. If $\dim M=2$ and $M$ has constant sectional curvature $K\mathrm{\ne }0$, then the K"ahler manifolds ${\mathcal{T}}_{M,r_0}$ have holonomy $\mathrm{S}\mathrm{U}(2)$; hence they are Ricci-flat. For $M=S^2$, just this dimension, the metric coincides with the Stenzel metric on the tangent manifold ${{\cal T}_{S^2}}$, giving us a new most natural description of this well-know metric.