12 WILHELM KLINGENBERG
is a fibre map. Thus, the composite mapping
Hx{£c)G-^Hx{L(c*TM\
c*TM)) -
L(H°(C*TM); H°(C*TM))
is a positive selfadjoint Operator of dass C*; it therefore defines a Riemannian
metric on the representation Hx(£c) X H°(c*TM) of (a°ylGH(c). Clearly, this
metric does not depend on the particular representation. D
As a preparation for defining a Riemannian metric also on al ôð\{1
My
we
prove an analogue of 1.10.
1.18
PROPOSITION.
The Levi-Chitä covariant derivation V on Ì determines á
Riemannian covariant derivation X
a
» on the bündle a°. in particular, i/ç is á vcctor
field on //'(/ , M), the mapping çð í„«(8). ç is á section in a(). Sincc\ for
á'(ç ) G C''(I, A/), it coincides with the H{)-vector field
VTJ(/)
along c(t), we also
write VTJ instead of í„ï(3).ç .
PROOF.
Denote by Ã(£), î £ 6 , the Christoffel symbol of the Levi-Civitä
connection in the normal coordinates based at ôî G Ì. Let c G C/oc( /, Ì ). Then
we get a bündle mapping
r
c
: ec - L(c*TM, c*TM; c*TM)
by
ÃÃ = ( T * C )
_ 1
Ï Ï T*C)(T*C X T*C) .
The associated mapping (cf. 1.9)
fc:
Hl(ec)
-
Hl(L(c*TM,
c*TM\ c*TM))
-
L(Hx{c*TM),
H°(C*TM);
H()(C*TM))
represents the desired Riemannian connection v .
To compute í
á
ï(è). ç we consider the bündle trivializations of and
á1
over
(exp;1,
%(c)). According to 1.13, ç G
(aly\e)
is represented by (£(/),
TJ,.(/)),
with î = exp^e. 1.16 gives the representation dc£(t) of de. Thus, the principal
part of the representation of í
á
ï(9). ç reads
D2(dcat)).vc(t) + Tc(t)U(t),dMt))-
Substituting 1.16 for 8c£(i) we get
vuc(0 + D2{eci{t)).y\Xt) + Tc(t)(Vc(t),dti(t)).
Thus, if e = c, i.e., if £ = 0, this simplifies to Vijc(/)·
1.19 THEOREM. The Hubert manifold H\I, M) hos á Riemannian metric; for
each element c G C"°°(7, M) C H\l, M), the scalar product on TeH\I, M) s
Hx(c*TM) coincides with the product ( , ,. We denote this Riemannian metric on
H\l,M)by(,)v
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