Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.

Definition

In a Riemannian manifold (M,g), a closed geodesic is a curve [math]\displaystyle{ \gamma:\mathbb R\rightarrow M }[/math] that is a geodesic for the metric g and is periodic.

Closed geodesics can be characterized by means of a variational principle. Denoting by [math]\displaystyle{ \Lambda M }[/math] the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function [math]\displaystyle{ E:\Lambda M\rightarrow\mathbb R }[/math], defined by

[math]\displaystyle{ E(\gamma)=\int_0^1 g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,\mathrm{d}t. }[/math]

If [math]\displaystyle{ \gamma }[/math] is a closed geodesic of period p, the reparametrized curve [math]\displaystyle{ t\mapsto\gamma(pt) }[/math] is a closed geodesic of period 1, and therefore it is a critical point of E. If [math]\displaystyle{ \gamma }[/math] is a critical point of E, so are the reparametrized curves [math]\displaystyle{ \gamma^m }[/math], for each [math]\displaystyle{ m\in\mathbb N }[/math], defined by [math]\displaystyle{ \gamma^m(t):=\gamma(mt) }[/math]. Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.

Examples

On the unit sphere [math]\displaystyle{ S^n\subset\mathbb R^{n+1} }[/math] with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Thus, on the sphere, all geodesics are closed. On a smooth surface topologically equivalent to the sphere, this may not be true, but there are always at least three simple closed geodesics; this is the theorem of the three geodesics. Manifolds all of whose geodesics are closed have been thoroughly investigated in the mathematical literature. On a compact hyperbolic surface, whose fundamental group has no torsion, closed geodesics are in one-to-one correspondence with non-trivial conjugacy classes of elements in the Fuchsian group of the surface.

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.