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

• Curve-shortening flow
• Selberg trace formula
• Selberg zeta function
• Zoll surface

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.

• 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