\begin{definition}[Definition:Metrically Complete Connected Riemannian Manifold]
Let $\struct {M, g}$ be a connected Riemannian manifold.
Let $d_g$ be the Riemannian distance.
Then $M$ is called the '''(metrically) complete manifold''' if every Cauchy sequence in $M$ converges with respect to the distance function $d_g$.
\end{definition}