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.