Editing Riemannian manifold (section)

=== Metric space structure ===
Length of curves and the Riemannian distance function <math>d_g : M \times M \to [0,\infty)</math> are defined in a way similar to the finite-dimensional case. The distance function <math>d_g</math>, called the ''geodesic distance'', is always a [[Pseudometric space|pseudometric]] (a metric that does not separate points), but it may not be a metric.{{sfn|Magnani|Tiberio|2020}} In the finite-dimensional case, the proof that the Riemannian distance function separates points uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.
* If <math>g</math> is a strong Riemannian metric on <math>M</math>, then <math>d_g</math> separates points (hence is a metric) and induces the original topology.{{citation needed|date=July 2024}}
* If <math>g</math> is a weak Riemannian metric, <math>d_g</math> may fail to separate points. In fact, it may even be identically 0.{{sfn|Magnani|Tiberio|2020}} For example, if <math>(M, g)</math> is a compact Riemannian manifold, then the <math>L^2</math> weak Riemannian metric on <math>\operatorname{Diff}(M)</math> induces vanishing geodesic distance.{{sfn|Michor|Mumford|2005}}