Closed geodesic
In
Definition
In a Riemannian manifold (M,g), a closed geodesic is a curve that is a geodesic for the metric g and is periodic.
Closed geodesics can be characterized by means of a variational principle. Denoting by the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function , defined by
If is a closed geodesic of period p, the reparametrized curve is a closed geodesic of period 1, and therefore it is a critical point of E. If is a critical point of E, so are the reparametrized curves , for each , defined by . Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.
Examples
On the unit sphere with the standard round Riemannian metric, every
See also
- Lyusternik–Fet theorem
- Theorem of the three geodesics
- Curve-shortening flow
- Selberg trace formula
- Selberg zeta function
- Zoll surface
References
- Besse, A.: "Manifolds all of whose geodesics are closed", Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978.