Editing Geodesic (section)

{{Short description|Straight path on a curved surface or a Riemannian manifold}}
{{About|geodesics in general|geodesics in general relativity|Geodesic (general relativity)|the study of Earth's shape|Geodesy|the application on Earth|Earth geodesic|other uses}}
[[File:Klein quartic with closed geodesics.svg|thumb|[[Klein quartic]] with 28 geodesics <small>(marked by 7 colors and 4 patterns)</small>]]
In [[geometry]], a '''geodesic''' ({{IPAc-en|ˌ|dʒ|iː|.|ə|ˈ|d|ɛ|s|ɪ|k|,_|-|oʊ|-|,_|-|ˈ|d|iː|s|ɪ|k|,_|-|z|ɪ|k}}){{refn|{{Cite dictionary |url=http://www.lexico.com/definition/geodesic |archive-url=https://web.archive.org/web/20200316193343/https://www.lexico.com/definition/geodesic |url-status=dead |archive-date=2020-03-16 |title=geodesic |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}} }}{{refn|{{cite Merriam-Webster|geodesic}}}} is a [[curve]] representing in some sense the locally{{efn|For two points on a sphere that are not antipodes, there are two great circle arcs of different lengths connecting them, both of which are geodesics.}} shortest{{efn|For a [[pseudo-Riemannian manifold]], e.g., a [[Lorentzian manifold]], the definition is more complicated.|name=pseudo}} path ([[arc (geometry)|arc]]) between two points in a [[differential geometry of surfaces|surface]], or more generally in a [[Riemannian manifold]]. The term also has meaning in any  [[differentiable manifold]] with a [[connection (mathematics)|connection]]. It is a generalization of the notion of a "[[Line (geometry)|straight line]]".

The noun ''[[wikt:geodesic|geodesic]]'' and the adjective ''[[wikt:geodetic|geodetic]]'' come from ''[[geodesy]]'', the science of measuring the size and shape of [[Earth]], though many of the underlying principles can be applied to any [[Ellipsoidal geodesic|ellipsoidal]] geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's [[Planetary surface|surface]]. For a [[spherical Earth]], it is a [[line segment|segment]] of a [[great circle]] (see also [[great-circle distance]]).  The term has since been generalized to more abstract mathematical spaces; for example, in [[graph theory]], one might consider a [[Distance (graph theory)|geodesic]] between two [[vertex (graph theory)|vertices]]/nodes of a [[Graph (discrete mathematics)|graph]].

In a [[Riemannian manifold]] or submanifold, geodesics are characterised by the property of having vanishing [[geodesic curvature]]. More generally, in the presence of an [[affine connection]], a geodesic is defined to be a curve whose [[Tangent space|tangent vector]]s remain parallel if they are [[parallel transport|transported]] along it. Applying this to the [[Levi-Civita connection]] of a [[Riemannian metric]] recovers the previous notion.

Geodesics are of particular importance in [[general relativity]].  Timelike [[geodesics in general relativity]] describe the motion of [[free fall]]ing [[test particles]].