Curvature homogeneous hypersurfaces in space forms

We provide a classification of curvature homogeneous hypersurfaces in space forms by classifying the ones in $S^4$ and $H^4$. In higher dimensions, besides the isoparametric and the constant curvature ones, there is a single one in $H^5$. Besides the obvious examples, we show that there exists an isolated hypersurface with a circle of symmetries and a one parameter family admitting no continuous symmetries. Outside the set of minimal points, which only exists in the case of $S^4$, every example is, locally and up to covers, of this form.