In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that forms a simple closed curve. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.
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
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 by
E
(
γ
)
=
∫
0
1
g
γ
(
t
)
(
γ
˙
(
t
)
,
γ
˙
(
t
)
)
d
t
.
If
γ
is a closed geodesic of period p, the reparametrized curve
t
↦
γ
(
p
t
)
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
γ
m
, for each
m
∈
N
, defined by
γ
m
(
t
)
:=
γ
(
m
t
)
. Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.
On the unit sphere
S
n
⊂
R
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.
