Prime geodesic

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.