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,æ) = 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 e crf,/ = e cr 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").
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