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Prime geodesic
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Prime geodesic
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.
This section presents some facts from hyperbolic geometry that are helpful in understanding prime geodesics.
In the Poincaré half-plane model H of 2-dimensional hyperbolic geometry, a Fuchsian group – that is, a discrete subgroup Γ of PSL(2, R) – 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 numbers are involved.) 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.
The quotient surface M=Γ\H, using the upper half-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 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 {γ} such that γ cannot be written as a nontrivial power of another element of Γ.
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 surfaces themselves. These applications often overlap among several different research fields.
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Prime geodesic
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.
This section presents some facts from hyperbolic geometry that are helpful in understanding prime geodesics.
In the Poincaré half-plane model H of 2-dimensional hyperbolic geometry, a Fuchsian group – that is, a discrete subgroup Γ of PSL(2, R) – 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 numbers are involved.) 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.
The quotient surface M=Γ\H, using the upper half-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 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 {γ} such that γ cannot be written as a nontrivial power of another element of Γ.
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 surfaces themselves. These applications often overlap among several different research fields.