CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS

9

The family 3i(c), c Å C,oc(7, M), is an open covering of H](I, M) since

C'X(L M) is dense in H](L M) C C°(7, M); see 1.2. 1.11 shows that the natural

atlas is differentiable.

To see that H\I, M) has a countable base it suffices to show that the natural

atlas has a countable subatlas. Actually, it suffices to show that, for a sequence

{MA} of relatively compact open subsets Mk of Ì with UkMk — M, Mk C

Mk + \ there exists a countable subatlas of the natural atlas covering H\l, Mk).

To see this we show that for fixed k and any integer / 0 the set H\l, Mk)! —

{c Å H\L Mk)\ E(c) /} can be covered by a finite subset of the natural atlas.

To prove this we choose t — i(Mk) 0 to be the injectivity radius on Mk. Let

m — m{L l) be an integer satisfying 18/ mr. Then e Å H](L Mk)1 implies, for

tG[(j- \)]/nuj/kl

"('(^)·"")!â(0""À)''^/'«"·'"^4·Ë

Hence, e j [(./ - \)/m. j/m] lies entirely in an t/3-ball.

There exists a finite set Ñ of points on Mk such that the é/3-balls around these

points will cover Mk. Given e Å H\L Mk)1. we can find a sequence {/?, /?,„}

in Ñ such that e(j/m) Å Bl/}(Pj). For each of these finitely many sequences we

choose a c Å C'°°(L M) such that c\[{j - \)/nu j/m] is the minimizing geo-

desic from/;-_, to/?,·. Then e Å H\L Mk) implies e Å ^l(c). for one of these c's.

RIMARK.

Given a differentiable mapping/: Ì -* Í from a manifold Ì into a

manifold Á Ë we obtain a mapping

H\Lf):H\l.M)^Hl(I.N):

( ö ) ) ð ( / ï ö ) ) .

This mapping is differentiable. And if g: Í -^ L is another differentiable map-

ping we get H\L go f)- H\L g)o H\L f). Thus, //

!

(Ë ) constitutes a

covariant functor from the category {finite-dimensional manifold and differentia-

ble mappings} into the category {Hubert manifolds and differentiable mappings}.

Associated with the manifolds H\L M) there are in a natural way two vector

bundles á': H'\H\L M)*TM) - H\L M\ r = 0. 1. The total space is the

union of the spaces H\c*TM\ c Å H\L Ì), á1 is canonically isomorphic to the

tangent bündle ô„.(/

Ë/ )

of H\l, M). To make this precise we define for î Å C:\

Àan Ë open neighborhood of Ì in TM as above,

7 ^ exp = T^exp ï

(Tr\T,hTM)-]

: TriM - TcxpiM,

T^exp = T^exp ï (Ê\Ôîí ÔÌ)'

1

: T^M - T^M.

Here, 7ô| TVJM : Ã^,Ã Ì - ÃôßÌ and Ê | Ã|ÃÃÌ : TivTM - 7^Ë ß are the canon-

ical isomophisms. Under our assumptions, T$A exp and 7^2exp are linear isomor-

phisms.

Note. In the above definitions, we have written exp instead of exp^ = exp | TpM,

where ñ = ô£. There can be no misunderstanding as to where the base point of

the vector is. Á similar simplification will also be used further down.