CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS 13 PROOF. Define the Riemannian scalar product on TeH\l, M) by (£, ç)»-» (£' ô ?)ï + (^æ* 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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