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 ? = {ç 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 » (ô 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).