Closed geodesic

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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 flow on the manifold.

Definition

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

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

E(\gamma )=\int _{0}^{1}g_{{\gamma (t)}}({\dot \gamma }(t),{\dot \gamma }(t))\,{\mathrm {d}}t.

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

Examples

On the unit sphere S^{n}\subset {\mathbb R}^{{n+1}} 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 classes of elements in the Fuchsian group of the surface.

See also

References

  • Besse, A.: "Manifolds all of whose geodesics are closed", Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978.
  • 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
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