- Prime geodesic
In

mathematics , a**prime geodesic**on a hyperbolicsurface is a**primitive closed**, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to thegeodesic prime number theorem .**Technical background**We briefly present some facts from

hyperbolic geometry which are helpful in understanding prime geodesics.**Hyperbolic isometries**Consider the

Poincaré half-plane model "H" of 2-dimensionalhyperbolic geometry . Given aFuchsian group , that is, adiscrete subgroup Γ of PSL(2,**R**), Γ acts on "H" vialinear fractional transformation . Each element of PSL(2,**R**) in fact defines anisometry 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 we are working with

real number s.) Then an element γ of Γ has 2 distinct real fixed points if and only if γ is hyperbolic. See Classification of isometries and Fixed points of isometries for more details.**Closed geodesics**Now consider the quotient surface Γ"H". This is a hyperbolic surface, in fact, a

Riemann surface . Each hyperbolic element of Γ determines a closed geodesic of Γ"H": first, by connecting the geodesic semicircle joining the fixed points, we get a geodesic on "H", and by projecting this, we get a geodesic on Γ"H". 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 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 — 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 , andnumber theory , as well asRiemann surface s themselves. It should be noted that these applications often overlap among several different research fields.**Dynamical systems and ergodic theory**In dynamical systems, the closed geodesics represent the periodic orbits of the 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 toAtle 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 ofChebotarev's density theorem .There are other similarities to number theory — 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 usualRiemann 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 thering of integers of anumber field can be split (factored) in aGalois extension . SeeCovering map andSplitting 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 of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying theeigenvalue s ofLaplacian operator s, arithmetic Fuchsian groups, andTeichmüller space s.**See also***

Fuchsian group

*Modular group Gamma

*Riemann surface

*Fuchsian model

*Analytic number theory

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