2 WILHELM KLINGENBERG
Clearly, when considering motions we can restrict ourselves to those of unit
speed. That is, we can restrict the geodesic flow to the unit tangent space TXM of
M.
From the two aspects of a closed geodesic—a closed curve which is a geodesic
and a geodesic (flow line) which is periodic—we elaborate the first one. We will
consider a space formed by all closed curves in which the closed geodesics are
characterized as the critical points of a functional. This idea goes back to Morse.
In our exposition we will give a refined version of Morse's approach which has
several advantages over the old one—in particular, it possesses a canonical 0(2)-
action.
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