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.
![]() | Original source: https://en.wikipedia.org/wiki/Closed geodesic.
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