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(I9Rn).
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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