Introduction

We consider compact Riemannian manifolds M. Every tangent vector X to Ì

determines a geodesic c(t) = cx(t\ t Å R, by the condition c(0) = X. If × Ö 0

and if there exists an c o 0 such that c(co) = c(0), then c | [0, ù] is called a closed

geodesic.

Let ô =Î ôÌ : Ã Ì ~ Ì be the tangent bündle of M. On TM we have the

geodesic flow ö,: ÃÌ -» TM. Here, ö,X is given by cx(t).

The geodesics flow is a Hamiltonian flow. More precisely, consider the cotan-

gent bündle ç£: Ã * Ì -* Ì . On Ô* Ì we have a canonical symplectic structure

á* = -dB where 0 is the canonical 1-form given by -Ó v'du' in the local coordi-

nates (w', t') of T^M. Moreover, from the Riemannian metric g* on T*M we

have the function

£*:r*M-»R, **-4g*(**, **).

The cogeodesic flow is the flow of the Hamiltonian System (Ã*Ì , á*, £*). In the

local coordinates of T*M, the Hamiltonian equations read

dt

*v'

* * W ' dt

L»'

2?, 3W

VV'

kj

Using the canonical isomorphism

TM

T

M

Ì

id

-T*M

'Ai'

-*M

(u\

uk)

("0 H

-+!t,*kgik{u)tf)

(«0

we can transport the cogeodesic system (Ã*Ì , á*, £*) into the geodesic system

(TM, á, £). The Hamiltonian equations of this system read, in local coordinates,

dt ~ *"'

That is to say, the flow lines ö, X project under rM into the geodesics on M.

We thus see that the closed geodesics are in 1:1 correspondence with periodic

flow lines on TM — M. Or, in the language of mechanics: Geodesics correspond

to motions of a point in the geodesic system and, in particular, closed geodesics

correspond to periodic motions.

1

http://dx.doi.org/10.1090/cbms/053/01