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We define a Gauss map $$\gamma :M\rightarrow \mathbb {S}^{6}$$ γ : M → S 6 of an oriented hypersurface M of the unit sphere $$\mathbb {S}^{7}$$ S 7 and prove that $$\gamma $$ γ is harmonic if and only if M has CMC. Results on the geometry and topology of CMC hypersurfaces of $$\mathbb {S}^{7}$$ S 7 , under hypothesis on the image of $$\gamma $$ γ , are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of $$\mathbb {CP}^{3}$$ CP 3 and obtain similar results for CMC hypersurfaces of this space.

Gauss map and the topology of constant mean curvature hypersurfaces of $$\mathbb {S}^{7}$$ and $$\mathbb {CP}^{3}$$