Editing Geodesic (section)

==Metric geometry==
In [[metric geometry]], a geodesic is a curve which is everywhere [[locally]] a [[distance]] minimizer. More precisely, a [[curve]] {{nowrap|''γ'' : ''I'' → ''M''}} from an interval ''I'' of the reals to the [[metric space]] ''M'' is a '''geodesic''' if there is a [[mathematical constant|constant]] {{nowrap|''v'' ≥ 0}} such that for any {{nowrap|''t'' ∈ ''I''}} there is a neighborhood ''J'' of ''t'' in ''I'' such that for any {{nowrap|''t''<sub>1</sub>, ''t''<sub>2</sub> ∈ ''J''}} we have

:<math>d(\gamma(t_1),\gamma(t_2)) = v \left| t_1 - t_2 \right| .</math>

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with [[Curve#Lengths of curves|natural parameterization]], i.e. in the above identity ''v''&nbsp;=&nbsp;1 and

:<math>d(\gamma(t_1),\gamma(t_2)) = \left| t_1 - t_2 \right| .</math>

If the last equality is satisfied for all {{nowrap|''t''<sub>1</sub>, ''t''<sub>2</sub> ∈ ''I''}}, the geodesic is called a '''minimizing geodesic''' or '''shortest path'''.

In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a [[length metric space]] are joined by a minimizing sequence of [[rectifiable path]]s, although this minimizing sequence need not converge to a geodesic. The [[Hopf–Rinow theorem#Variations and generalizations|metric Hopf-Rinow theorem]] provides situations where a length space is automatically a geodesic space.

Common examples of geodesic metric spaces that are often not manifolds include [[Metric graph|metric graphs]], (locally compact) metric [[Polyhedral complex|polyhedral complexes]], infinite-dimensional [[Pre-Hilbert space|pre-Hilbert spaces]], and [[Real tree|real trees]].