10 WILHELM KLINGENBERG Define, for c G C'°°(/, M) and r = 0,1, i- l c :i/ l (0 c )X^(c*rM)-(«T 1 (c) by (€(0,1,(0) - ((À ·^(ï.2«ñ).ô*^(0). Here, the right-hand side is viewed as an Ç '-mapping / -* TM, which under ô goes into the base //'-curve (exp ° T*C|(0) belonging to %(c). 1.13 LEMMA. The family {(öÃ^exp 1 , 9l(c)) c Å C"°°(/, M)} consitutes á bün- dle atlas for á bündle ar over Hl(I, M) associated with the natural atlas of H](I, M). The typical fibre of the bündle is the separable Hubert space Hr(I 9 Rn). The bündle xl is canonically isomorphic to the tangent bündle ôÇ\(ß M) . PROOF. First consider the case r 1. Then we see that, for c, d in C/00(/, M), *\.ä° Öº!Ã= Hl(®cä) X H\c*TM) - Hl(edJ X Hl(d*TM) is of the form (exp 1 ï expc, r(exp ! ï â ÷ñÃ)) = ( £ 7/^), with/c/ as in the proof of 1.11. This shows that the above atlas is precisely the tangent atlas associated with the natural atlas of Hl(I, M). When r = 0, we observe that the composition maps ö0 ä ö ^ are again of the form (fd c , % i C ), and the composition mapping Tf H \®cd) ^ H\L{c*TM\ d*TM)) - L(H°{c*TM) H{\d*TM)) is differentiable see 1.8. D Note. The previous result shows that we obtain an intrinsic description of the tangent space TeH\I, M) of H\I, M) at an arbitrary dement e Å Hl(I % M) by considering the vector space of //l-maps ç : / -* TM satisfying ô ï ç = e . That is, ç is an i/l-vector field along the //]-curve e. Before we prove that we also have a natural scalar product on TeH\L M\ we show that natural Charts exist for every e Å H\L M). This will follow from the next lemma. To formulate our result we put é = Å TH\L Ì): ç(ß) Å (?}, with é C TM as before. 1.14 LEMMA. The mapping ^= T //V.M) X exp : i ? C TH\LM) -* H\LM) X H\L M)\ ç(ß) ì » ï ç(À),ò÷ñç(ß)) is differentiable. It maps á suffieiently small open neighborhood (?' C i o/ //?e zero section of ô^é (ËË/ ) cwio Ï Ë o/?ew neighborhood of the diagonal of H\l, Ì) × #'(/ , M).
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