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.

## Contents

* 1 Definition
* 2 Examples
* 3 See also
* 4 References

## Definition

In a Riemannian manifold (M,g), a closed geodesic is a curve γ:ℝ->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 ΛM the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function E:ΛM->ℝ, defined by

E(γ)=∫01gγ(t)(γ˙(t),γ˙(t))dt.

If γ is a closed geodesic of period p, the reparametrized curve t↦γ(pt) is a closed geodesic of period 1, and therefore it is a critical point of E. If γ is a critical point of E, so are the reparametrized curves γm, for each m∈ℕ, defined by γm(t):=γ(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 Sn⊂ℝn+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
* Theorem of the three geodesics
* 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.

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