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.
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Definition
In a Riemannian manifold (M,g), a closed geodesic is a curve that is a geodesic for the metric g and is periodic.
Closed geodesics can be characterized by means of a variational principle. Denoting by the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function , defined by
If is a closed geodesic of period p, the reparametrized curve is a closed geodesic of period 1, and therefore it is a critical point of E. If is a critical point of E, so are the reparametrized curves , for each , defined by . Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.
Examples
On the unit sphere 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
- ^ Grayson, Matthew A. (1989), "Shortening embedded curves" (PDF), Annals of Mathematics, Second Series, 129 (1): 71–111, doi:10.2307/1971486, MR 0979601.
- 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