= Upward stability of Riemannian completeness
{title2=$\widetilde g\geq g,\quad g\text{ complete}\Longrightarrow\widetilde g\text{ complete}$}
Pointwise domination of <Riemannian metrics> gives domination of their distances. A Cauchy sequence for the larger distance is Cauchy for the complete smaller one. Smooth positive metrics induce the same manifold topology, so its limit is also a limit in the larger distance. The <Hopf-Rinow theorem> turns metric completeness into <geodesic completeness>. Mere positivity of the second metric, without domination, is insufficient.
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