Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that forms a simple closed curve. 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 \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. 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.[1] 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

  1. ↑ Grayson, Matthew A. (1989), "Shortening embedded curves" (PDF), Annals of Mathematics, Second Series 129 (1): 71–111, doi:10.2307/1971486, MR 979601.
This article is issued from Wikipedia - version of the Wednesday, November 18, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.