The Hardy space of slice regular functions on the quaternionic unit ball H^2(B) is a reproducing kernel Hilbert space. In this talk, after an appropriate introduction to the subject, we will see how this property can be exploited to construct a Riemannian metric on B and we will study the geometry arising from this construction. We will also see that, in contrast with the example of the Poincaré metric on the complex unit disc, no Riemannian metric on B is invariant with respect to all slice regular bijective self maps of B. The results presented are obtained in collaboration with Nicola Arcozzi.