Closed geodesic explained

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: 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

Λ 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: Λ M → R , defined byE(\gamma) = \int ^{1} _{0} g _{\gamma(t)} ( • • \gamma(t), \gamma(t)) 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 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

S ^{n\subsetR} ^{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
• 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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