Editing Prime geodesic

{{Unreferenced|date=December 2009}}
In [[mathematics]], a '''prime geodesic''' on a [[hyperbolic geometry|hyperbolic]] [[Surface (topology)|surface]] is a '''primitive''' [[closed geodesic]], i.e. a geodesic which is a [[curve|closed curve]] that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an [[asymptotic analysis|asymptotic distribution law]] similar to the [[prime number theorem]].

==Technical background==
This section presents some facts from [[hyperbolic geometry]] that are helpful in understanding prime geodesics.

===Hyperbolic isometries===
In the [[Poincaré half-plane model]] ''H'' of 2-dimensional [[hyperbolic geometry]], a [[Fuchsian group]] – that is, a [[discrete subgroup]] Γ of [[projective linear group|PSL(2, '''R''')]] – [[Group action (mathematics)|acts]] on ''H'' via [[linear fractional transformation]]. Each element of PSL(2, '''R''') defines an [[isometry]] of ''H'', so Γ is a group of isometries of ''H''.

There are then 3 types of transformation: hyperbolic, elliptic, and parabolic. (The loxodromic transformations are not present because only [[real number]]s are involved.) Then an element γ of Γ has 2 distinct real fixed points if and only if γ is hyperbolic. See [[Möbius transformation#Classification|Classification of isometries]] and [[Möbius transformation#Fixed points|Fixed points of isometries]] for more details.

===Closed geodesics===
The [[Quotient space (topology)|quotient surface]] ''M''=Γ\''H,'' using the upper half-plane [[Models of the hyperbolic plane|model of the hyperbolic plane]], is a hyperbolic surface – in fact, a [[Riemann surface]]. Each hyperbolic element ''h'' of Γ determines a [[closed geodesic]] of ''M'': first, the geodesic semicircle joining the fixed points of ''h'' forms the axis of ''h,'' which projects to a geodesic on ''M''.

This geodesic is closed because 2 points which are in the same orbit under the action of Γ project to the same point on the quotient, by definition.

It can be shown that this gives a [[bijection|1-1 correspondence]] between closed geodesics on Γ\''H'' and hyperbolic [[conjugacy class]]es in Γ. The prime geodesics are then those geodesics that trace out their image exactly once &mdash; algebraically, they correspond to primitive hyperbolic conjugacy classes, that is, conjugacy classes {γ} such that γ cannot be written as a nontrivial power of another element of Γ.

==Applications of prime geodesics==
The importance of prime geodesics comes from their relationship to other branches of mathematics, especially [[dynamical systems]], [[ergodic theory]], and [[number theory]], as well as [[Riemann surface]]s themselves. These applications often overlap among several different research fields.

===Dynamical systems and ergodic theory===
In dynamical systems, the [[closed geodesic]]s represent the [[Periodic function|periodic]] [[Group action (mathematics)|orbits]] of the [[Geodesic#Geodesic flow|geodesic flow]].

===Number theory===
In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the [[prime number theorem]]. To be specific, we let π(''x'') denote the number of closed geodesics whose norm (a function related to length) is less than or equal to ''x''; then π(''x'') ~ ''x''/ln(''x''). This result is usually credited to [[Atle Selberg]]. In his 1970 Ph.D. thesis, [[Grigory Margulis]] proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, [[Peter Sarnak]] proved an analogue of [[Chebotarev's density theorem]].

There are other similarities to number theory &mdash; error estimates are improved upon, in much the same way that error estimates of the prime number theorem are improved upon. Also, there is a [[Selberg zeta function]] which is formally similar to the usual [[Riemann zeta function]] and shares many of its properties.

Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that [[prime ideal]]s in the [[ring of integers]] of a [[number field]] can be split (factored) in a [[Galois extension]]. See [[Covering map]] and [[Splitting of prime ideals in Galois extensions]] for more details.

===Riemann surface theory===
Closed geodesics have been used to study Riemann surfaces; indeed, one of [[Riemann]]'s original definitions of the [[genus (mathematics)|genus]] of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying the [[eigenvalue]]s of [[Laplacian]] [[operator (mathematics)|operator]]s, [[arithmetic group|arithmetic Fuchsian group]]s, and [[Teichmüller space]]s.

==See also==
*[[Fuchsian group]]
*[[Modular group Gamma]]
*[[Riemann surface]]
*[[Fuchsian model]]
*[[Analytic number theory]]
*[[Zoll surface]]

==References==
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{{DEFAULTSORT:Prime Geodesic}}
[[Category:Riemann surfaces]]
[[Category:Differential geometry]]
[[Category:Dynamical systems]]
[[Category:Number theory]]
[[Category:Geodesic (mathematics)]]
[[Category:Hyperbolic geometry]]