In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic: one whose parametrization is not obtained by going repeatedly around a shorter closed geodesic. Informally, one follows the geodesic until its motion begins to repeat; the geodesic is prime if this first full return already completes the entire cycle, rather than repeating a shorter cycle several times. For comparison, a great circle on a sphere traversed once is analogous to a prime geodesic, whereas the same great circle traversed twice is not. Prime geodesics play, for hyperbolic surfaces, a role analogous to that of prime numbers in number theory: every closed geodesic is obtained by iterating a prime geodesic, and their asymptotic distribution is described by the prime geodesic theorem.
Definition
Let <math>X</math> be a hyperbolic surface. A closed geodesic <math>\gamma</math> on <math>X</math> is called prime or primitive if it is not an iterate <math>\gamma_0^n</math> of another closed geodesic <math>\gamma_0</math> with <math>n \ge 2</math>. Under this correspondence, prime geodesics are exactly the conjugacy classes represented by primitive hyperbolic elements, that is, elements that are not nontrivial powers of other elements of <math>\Gamma</math>.
Prime geodesic theorem
Let
:<math>\pi_X(x)=\#\{P : P \text{ is a prime geodesic on } X,\ N(P)\le x\}.</math>
For a finite-area hyperbolic surface, the prime geodesic theorem states that
:<math>\pi_X(x)\sim \frac{x}{\log x}\qquad (x\to\infty).</math>
Equivalently, the number of prime geodesics of length at most <math>L</math> is asymptotic to <math>e^L/L</math> as <math>L\to\infty</math>.
Selberg zeta function
Prime geodesics enter the theory of hyperbolic surfaces through the Selberg zeta function, an Euler product taken over prime geodesics:
:<math>Z_X(s)=\prod_{P}\prod_{k=0}^{\infty}\left(1-N(P)^{-s-k}\right),</math>
where <math>P</math> ranges over the prime geodesics on <math>X</math>.
In a broader Riemannian and dynamical systems context, the term prime geodesic is also used for a closed geodesic that is not an iterate of a shorter one, even when the surface is not hyperbolic.
See also
- Zoll surface