In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a

closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition tha ...

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 the prime number theorem.

Technical background

We briefly present some facts from

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...

which are helpful in understanding prime geodesics.

Hyperbolic isometries

Consider the

Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré ...

''H'' of 2-dimensional

. Given a

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations ...

, that is, a

discrete subgroup In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...

Γ of PSL(2, R), Γ

acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...

on ''H'' via linear fractional transformation. Each element of PSL(2, R) in fact defines an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

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 In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

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 ''M''=Γ\''H''. The following description refers to the upper half-plane model of the hyperbolic plane. This is a hyperbolic surface, in fact, a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...

. Each hyperbolic element ''h'' of Γ determines a closed geodesic of Γ\''H'': first, by connecting the geodesic semicircle joining the fixed points of ''h'', we get a geodesic on ''H'' called the axis of ''h'', and by projecting this geodesic to ''M'', 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 classes 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 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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

, and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, as well as

s themselves. These applications often overlap among several different research fields.

Dynamical systems and ergodic theory

In dynamical systems, the closed geodesics represent the periodic

orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

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 to Atle Selberg. In his 1970 Ph.D. thesis,

Grigory Margulis Grigory Aleksandrovich Margulis (russian: Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on ...

proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis,

Peter Sarnak Peter Clive Sarnak (born 18 December 1953) is a South African-born mathematician with dual South-African and American nationalities. Sarnak has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced St ...

proved an analogue of Chebotarev'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 The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...

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 ideals in the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...

of a number field can be split (factored) in a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...

. 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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first r ...

's original definitions of the

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...

of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s of

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is ...

operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...

arithmetic Fuchsian group Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group ...

s, and

Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...