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, //
!
) 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.
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