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For a connected complete Riemannian manifold of dimension $n\ge 2$ 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.

Let $\pi :\tilde{M}\to M$ be the universal cover and equip it with the pullback Riemannian metric $\tilde{g}={\pi }^{\ast }g$. The covering map is a local isometry, so $\tilde{M}$ has constant sectional curvature one.

We must check completeness of the lifted Riemannian metric. A geodesic on $\tilde{M}$ projects to a geodesic on $M$. By geodesic completeness of $M$, the projected geodesic extends for all real time. Its unique lift with a specified initial point extends the original lifted geodesic, because $\pi$ is a local isometry. Hence $\tilde{M}$ is geodesically complete, and the Hopf-Rinow theorem also gives metric completeness.

The allowed constant-curvature form of Cartan-Ambrose-Hicks theorem says that a linear tangent-space isometry between complete simply connected Riemannian manifolds of the same constant sectional curvature extends to a global isometry. The unit round sphere $S^n$ is complete, simply connected for $n\ge 2$, and has sectional curvature one. Comparing it with $\tilde{M}$ therefore yieldsThis is the covering step in the classification of complete positive constant-curvature manifolds. As usual for this assertion, $n\ge 2$ is understood: in dimension one there are no sectional two-planes, and the universal cover of the circle is the line, not the circle.