Editing Riemannian manifold (section)

=== Hopf–Rinow theorem ===
In the case of strong Riemannian metrics, one part of the finite-dimensional Hopf–Rinow still holds. 

'''Theorem''': Let <math>(M, g)</math> be a strong Riemannian manifold. Then metric completeness (in the metric <math>d_g</math>) implies geodesic completeness.{{citation needed|date=July 2024}}

However, a geodesically complete strong Riemannian manifold might not be metrically complete and it might have closed and bounded subsets that are not compact.{{citation needed|date=July 2024}} Further, a strong Riemannian manifold for which all closed and bounded subsets are compact might not be geodesically complete.{{citation needed|date=July 2024}}

If <math>g</math> is a weak Riemannian metric, then no notion of completeness implies the other in general.{{citation needed|date=July 2024}}