• Source: Closed geodesic

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



γ
:

R

→
M


{\displaystyle \gamma :\mathbb {R} \rightarrow 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


{\displaystyle \Lambda 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



{\displaystyle E:\Lambda M\rightarrow \mathbb {R} }

, defined by




E
(
γ
)
=

∫

0


1



g

γ
(
t
)


(



γ
˙



(
t
)
,



γ
˙



(
t
)
)


d

t
.


{\displaystyle E(\gamma )=\int _{0}^{1}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,\mathrm {d} t.}


If



γ


{\displaystyle \gamma }

is a closed geodesic of period p, the reparametrized curve



t
↦
γ
(
p
t
)


{\displaystyle t\mapsto \gamma (pt)}

is a closed geodesic of period 1, and therefore it is a critical point of E. If



γ


{\displaystyle \gamma }

is a critical point of E, so are the reparametrized curves




γ

m




{\displaystyle \gamma ^{m}}

, for each



m
∈

N



{\displaystyle m\in \mathbb {N} }

, defined by




γ

m


(
t
)
:=
γ
(
m
t
)


{\displaystyle \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


On the ⁠



n


{\displaystyle n}

⁠-dimensional unit sphere with the standard metric, every geodesic – a great circle – is 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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