---
tag: X93AEE88PX
kind: statement
role: theorem
name: Bangert–Franks theorem on closed geodesics on the sphere
msc: ["58E10", 53C22, "37E30"]
uses: [Q61VFHCH2J, P9CM4GQ2T0, 4MCCP8GG4J]
about: [4MCCP8GG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_geodesic
---

## Statement

Let $g$ be a smooth [[Riemannian metric|Q61VFHCH2J]] on the two-dimensional [[sphere|P9CM4GQ2T0]] $S^2$.
Then $(S^2, g)$ has infinitely many geometrically distinct [[closed geodesics|4MCCP8GG4J]].

## Remarks

The proof combines two results.
Franks treated, with a theorem on area-preserving homeomorphisms of the open annulus, the metrics for which a simple closed geodesic gives an annulus of section for the geodesic flow.
Bangert treated the remaining metrics by variational methods on the loop space.
The [[theorem of the three geodesics|AB9FNZD18B]] gives a simple closed geodesic to start from.

The [[Gromoll–Meyer theorem|XC8BRYJN93]] does not apply here, because the Betti numbers of the free loop space of $S^2$ are bounded.
The hypothesis that the metric is Riemannian matters: Katok constructed non-reversible Finsler metrics on $S^2$ with only finitely many closed geodesics.

## Explanations

### L1 human

However one bends a sphere, it carries infinitely many different closed loops that go straight ahead, though they may cross themselves.
On a round sphere this is clear, since every great circle is one.
The theorem says that no smooth deformation, however irregular, can destroy all but finitely many of them.

### L2 human

The result answers a long-standing question for the simplest closed surface.
It mixes two methods: dynamics of the geodesic flow, through a surface of section and a return map of an annulus, and Morse theory of the energy on loop spaces.
For higher-dimensional spheres the corresponding question for Riemannian metrics is open.