Closed geodesic

Remarks

Equivalently, a closed geodesic is a smooth map $\gamma \colon \R / T\Z \to M$ that satisfies the geodesic equation and is not constant. A geodesic $\gamma$ is closed if and only if $\gamma(T) = \gamma(0)$ and $\dot\gamma(T) = \dot\gamma(0)$ for some $T > 0$, by uniqueness of geodesics with given initial point and velocity. The closed geodesics are the critical points with positive energy of the energy functional $E(\gamma) = \frac{1}{2} \int_0^1 \abs{\dot\gamma}^2 \, dt$ on the free loop space of $M$. Each closed geodesic of period $T$ gives infinitely many critical points, its iterates $t \mapsto \gamma(kt)$ for $k \ge 1$, which is why counting geometrically distinct ones is hard.

Example: on the unit sphere $S^2 \subset \R^3$, the great circle $\gamma(t) = (\cos t, \sin t, 0)$ is a geodesic with period $2\pi$, so it is a closed geodesic.

Non-example: on the flat torus $\R^2 / \Z^2$, the geodesic $\gamma(t) = (t, \sqrt{2}\, t) \bmod \Z^2$ is not periodic. Indeed $\gamma(t + T) = \gamma(t)$ would need $T \in \Z$ and $\sqrt{2}\, T \in \Z$, which is impossible for $T > 0$.