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
Let ô 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£: à * Ì -* Ì . On Ô* Ì we have a canonical symplectic structure
á* = -dB where 0 is the canonical 1-form given by 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
* * W ' dt
2?, 3W
Using the canonical isomorphism
("0 H
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.
Previous Page Next Page