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We classify curvature homogeneous hypersurfaces in S⁴ and H⁴. In higher dimesnsion one only has the FKM examples and an isolate one by Tsukada of a hypersurface in H⁵. Besides some simple examples, we show that there exists an isolated hypersurface with a circle of symmetries and and a one parameter family admitting no continuous symmetries. Outside the set of minimal points, which only exists in the case of S⁴, every example is locally and up to covers of this form.
Bryant et al. (Tue,) studied this question.