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. If the surface is given by the quotient of the hyperbolic plane by a Fuchsian group Γ, then there is a 1-1 correspondence between prime geodesics and primitive hyperbolic conjugacy classes {γ} in Γ. Such geodesics are called prime geodesics because they obey an asymptotic distribution law similar to the prime number theorem.

See also: Modular group Gamma