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