---
tag: 4MCCP8GG4J
kind: concept
name: Closed geodesic
msc: [53C22, "58E10"]
specializes: [NSYMQMHPDN]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_geodesic
same_as:
  wikidata: Q10288828
  wikipedia: Closed geodesic
---

## Summary

A closed geodesic on a [[Riemannian manifold|7CBT4SM83Y]] is a [[geodesic|NSYMQMHPDN]] that closes up smoothly, returning to its starting point with its starting velocity.
Equivalently, it is a nonconstant critical point of the energy on the space of closed loops.
Closed geodesics are the periodic orbits of the geodesic flow.
Their existence and their number are central questions of the variational theory of geodesics and of Hamiltonian dynamics.

## Explanations

### L1 human

A closed geodesic is a loop on a curved surface that goes straight ahead at every point and comes back to where it started, pointing the same way.
A rubber band stretched around a surface and allowed to settle often sits on one.
On a round sphere the great circles are closed geodesics.
Glue opposite sides of a square to get a flat torus.
On it, a straight line of rational slope closes up into a closed geodesic, while a line of irrational slope never closes.