8 WILHELM KLINGENBERG
is a fibre map of some 6' X fi' C ß X (5, È' open, into the bündle
L(TT; i):L(£;F)-*/.
Consider the associated continous mapping r:
Hl(6'
X 0')
H}(L(E;
F)).
Then, with T2f: //'(è') - H\L{E\ F)\
const||r(£, 11)11,1111 €11,.
Since Ñ(î,ae) = 0 we get ||f(£, ç)||, - 0 with ÉÉ£-ç||,-0 , i.e., r
2
f G
H\L(H\E)\ H\F))) has the properties of a differential, ?/ = Ã2/~.
There exists an open neighborhood 0 of the 0-section of ô: ÃÌ - Ì with the
property that exp \(È ñ= ÔñÌ Ð È) is a, diffeomorphism onto its image = open
neighborhood of ñ GM. For Ì compact, we can choose 0 to be the å-ball bündle
of ô, some small å 0, i.e., 0^ = Be(0p), all ñ G Ì.
1.10
DEFINITION.
Let 0 be as above. For c G C'°°(/, Ì ) denote by 6C the
subset of c*TM, formed by the 0C, = 0C Ð Tc,, which corresponds under r*c to
TC,0M ç è.
Define
(t) exVc:H\Qc)-*H\LM)
by (£(0) é- (expc(0T*c£(/)) and denote the image by %(c).
1.11
PROPOSITION,
(f) is bijective. Let c,d G C'°°(/, Ì ).
exp^1
ï e x p ^ e x p ^ ^ c ) Ð %(/)) - exp^(%(/) Ð %(c))
w á diffeomorphism between open sets in the Hubert Spaces
Hx(c*TM)
and
Hl(d*TM).
PROOF.
%(c) consists precisely of those e G Ç\I, M) with e(t) G expc(/)(0£(o).
This shows that (f) is a bijection.
For each t G / we form
ecrf,/ = ecr n (exP
°
T*c)"!
°
(exP
°
T*d)&d,t
and put U
0 / I
6 ^ , = 0ci/ if 0Ct#f, * 0 for all / G /. Otherwise put Öc
d
= 0 .
0Cd is an open subset of 0C, and
H\^d) = exp-'i^ic) Ð %(/)).
The map
fdc: (exp ï
ô**/)"1
ï (exp ï
T
*
c
) : 0cd - /*ÃÌ
is a fibre map,
exp^1
© expc = /£c. Hence, 1.9 applies. D
1.12
THEOREM.
The set H\I, M) of the
Hx-mapping
c: I - Ì is á Hubert
manifold; its differentiable structure is given by the natural atlas
{exp"1,
%(c); c G
C'°°(/, M)}.
PROOF.
The Charts are modelled on a proper separable Hubert Space with
typical representative H\c*TM) ss Ç\l, R").
Previous Page Next Page