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In [[differential geometry]] and [[dynamical systems]], a '''closed geodesic''' on a [[Riemannian manifold]] is the projection of a closed orbit of the [[geodesic|geodesic flow]] on the manifold.


==Definition==
In a [[Riemannian manifold]] (''M'',''g''), a closed geodesic is a curve <math>\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>\Lambda M</math> the space of smooth 1-periodic curves on ''M'', closed geodesics of period 1 are precisely the [[critical point (mathematics)|critical points]] of the energy function <math>E:\Lambda M\rightarrow\mathbb R</math>, defined by
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<math>E(\gamma)=\int_0^1 g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,\mathrm{d}t.</math>
 
If <math>\gamma</math> is a closed geodesic of period ''p'', the reparametrized curve <math>t\mapsto\gamma(pt)</math> is a closed geodesic of period 1, and therefore it is a critical point of ''E''. If <math>\gamma</math> is a critical point of ''E'', so are the reparametrized curves <math>\gamma^m</math>, for each <math>m\in\mathbb N</math>, defined by <math>\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>S^n\subset\mathbb R^{n+1}</math> with the standard round Riemannian metric, every great circle is an example of a closed geodesic. 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 class]]es of elements in the [[Fuchsian group]] of the surface.
 
==See also==
*[[Selberg trace formula]]
*[[Zoll surface]]
*[[geodesic]]
 
==References==
 
*[[Arthur Besse|Besse, A.]]: "Manifolds all of whose geodesics are closed", ''Ergebisse Grenzgeb. Math.'', no. 93, Springer, Berlin, 1978.
*[[Wilhelm Klingenberg|Klingenberg, W.]]: "Lectures on closed geodesics", Grundlehren der Mathematischen Wissenschaften, Vol. 230. Springer-Verlag, Berlin-New York, 1978. x+227 pp. ISBN 3-540-08393-6
 
[[Category:Differential geometry]]
[[Category:Dynamical systems]]
[[Category:Geodesic (mathematics)]]

Latest revision as of 13:41, 21 July 2014


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