CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS
13
PROOF.
Define the Riemannian scalar product on TeH\l, M) by (£, ç)»-»
(£' ô?)ï + (^ae* í ç )
0
· That this is a differentiable section in L^(al) follows from
1.16 and 1.18. D
We conclude this section with the
1.20
THEOREM.
The energy integral E: Hl(I, M) -* R is differentiable, with
DE(c).1 = (dcJVv)0.
PROOF.
The differentiability of Å follows from 1.16 and 1.17. To determine its
differential we need only recall from the proof of 1.18 that the local representa-
tion of í ç , for ç Å TCH\L Ì) , yields
VÖO9C.TJ
= í ç . That is, D($(dc, èï)
0
).ç
-3^íç
0
.
1.21
COROLLARY.
The only critical points of Å on H](I, M) are the constant
maps.
PROOF.
Clearly, c const, i.e., 9c = 0 implies DE(c) = 0. Conversely, DE{c).i)
= 9c, í ç )
0
= 0, for all í ç G H°(c*TM\ implies 9c = 0. D
Note. This reflects the fact that H\L M) possesses a canonical retraction onto
Ì the space of constant maps,
//•(/ , M) X / - / / ' ( / , M) ; ({·(/)}, s) h* (c(/(l - s))}.
Only for certain submanifolds of H\L M) do there exist nontrivial critical points
of E. Regarding this, see the next chapter.
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