= Completeness of homogeneous Riemannian manifolds
In a <homogeneous Riemannian manifold>, choose $r>0$ so that one closed metric ball $\overline B(o,r)$ is compact. Such a ball exists from local compactness and the metric topology. Every radius-$r$ ball is isometric to it. A <Cauchy sequence> eventually lies in one such compact ball, has a convergent subsequence, and therefore converges. This proves metric completeness; the <Hopf-Rinow theorem> gives <geodesic completeness> and minimizing <geodesics> between points.
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