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.

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. 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

• Lyusternik–Fet theorem
• Curve-shortening flow
• Selberg trace formula
• Selberg zeta function
• Zoll surface

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