= Classification of complete positive constant-curvature manifolds
{title2=$M=S_k^n/\Gamma,\quad k>0$}
For a connected complete <Riemannian manifold> of dimension $n\geq2$ and constant <sectional curvature> $k>0$, its complete simply connected <universal cover> is the round <sphere> of radius $k^{-1/2}$. This is the constant-curvature case of <Cartan-Ambrose-Hicks theorem>. Thus the manifold is a quotient by a finite group of <isometries> acting freely. In even dimension, the determinant embeds this group in $\{1,-1\}$: every determinant-one orthogonal transformation in odd ambient dimension has eigenvalue one and therefore a fixed point. The only possible nonidentity element is the antipodal involution, giving <Real projective space>. In odd dimensions, free cyclic scalar actions on complex unit spheres instead produce <lens spaces>.
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