Prime geodesic
Encyclopedia
In mathematics
, a prime geodesic on a hyperbolic
surface
is a primitive closed geodesic
, 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 the prime number theorem
.
which are helpful in understanding prime geodesics.
H of 2-dimensional hyperbolic geometry
. Given a Fuchsian group
, that is, a discrete subgroup Γ of PSL(2, R)
, Γ acts
on H via linear fractional transformation. Each element of PSL(2, R) in fact 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 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.
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
. 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. this statement is misleading and possibly false, as it fails to distinguish between geodesic loops and closed geodesics.
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.
, and number theory
, as well as Riemann surface
s themselves. These applications often overlap among several different research fields.
s represent the periodic
orbits
of the geodesic flow.
. 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 — 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.
of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying the eigenvalues of Laplacian operators, arithmetic Fuchsian group
s, and Teichmüller space
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a prime geodesic on a hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
is a primitive closed geodesic
Closed geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M.-Examples:On the unit sphere, every great circle is an example of a closed geodesic...
, i.e. a geodesic which is a closed curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
similar to the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
.
Technical background
We briefly present some facts from hyperbolic geometryHyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
which are helpful in understanding prime geodesics.
Hyperbolic isometries
Consider the Poincaré half-plane modelPoincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....
H of 2-dimensional hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
. Given a Fuchsian group
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
, that is, a discrete subgroup Γ of PSL(2, R)
Projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...
, Γ acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on H via linear fractional transformation. Each element of PSL(2, R) in fact defines an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
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
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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 surfaceQuotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
. Each hyperbolic element h of Γ determines a closed geodesic
Closed geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M.-Examples:On the unit sphere, every great circle is an example of 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. this statement is misleading and possibly false, as it fails to distinguish between geodesic loops and closed geodesics.
It can be shown that this gives a 1-1 correspondence
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
between closed geodesics on Γ\H and hyperbolic conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
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 theoryErgodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, as well as Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s themselves. These applications often overlap among several different research fields.
Dynamical systems and ergodic theory
In dynamical systems, the closed geodesicClosed geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M.-Examples:On the unit sphere, every great circle is an example of a closed geodesic...
s represent the periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
orbits
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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 theoremPrime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
. 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
Atle Selberg
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...
. In his 1970 Ph.D. thesis, Grigory Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak
Peter Sarnak
Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...
proved an analogue of Chebotarev's density theorem
Chebotarev's density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only...
.
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 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
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s in the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
of a number field can be split (factored) in a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
. See Covering map
Covering map
In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
and Splitting of prime ideals in Galois extensions
Splitting of prime ideals in Galois extensions
In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory...
for more details.
Riemann surface theory
Closed geodesics have been used to study Riemann surfaces; indeed, one of Riemann's original definitions of the genusGenus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying the eigenvalues of Laplacian operators, arithmetic Fuchsian group
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
s, and Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
s.
See also
- Fuchsian groupFuchsian groupIn mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
- Modular group Gamma
- Riemann surfaceRiemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
- Fuchsian modelFuchsian modelIn mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group \pi_1...
- Analytic number theoryAnalytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
- Zoll surfaceZoll surfaceIn mathematics, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere , equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on S2 obviously has this property, it also has an infinite-dimensional...