G\'eom\'etrie de l'espace tangent sur l'hyperbolo\"ide quantique

We introduce the tangent space on a quantum hyperboloid. We define an action of this tangent space on the corresponding "quantum function space" ${\cal A}$, what converts the elements of the tangent space into "braided vector fields". The tangent space is shown to be a projective ${\cal A}$-module and we define a quantum (pseudo)metric and a quantum connection (partially defined) on it.