6
WILHELM KLINGENBERG
1.5
PROPOSITION.
The inclusions H\c*TM) ~ C°(c*TM) » H°(c*TM) are
continuous. More precisely:
(i) ifi G C°(c*TM\ then \\î\\
0
ÉÉîÉÉ^and;
(ii) ifi G H\c*TM), then \\î\\„ i/2\\t\\v
PROOF.
0)
l l €l ,
o=J^€(0»€(0Ä^max€(0
2
Ä = Il€llf.
(ü) Choose /, G / with î\\
K
=  {(/x)  . Then
H€lli=€(0l
2
+
i
(
r ,
 I «^) É
2
Ë €(0I
2
+2/^(^)11^(5) 1 Ë
H€llg+ll€llS+llv€llg2l€llf. •
Note. Besides vector bundles of the type C*T : c*TM * U we will also be led to
consider associated vector bundles like product bundles, bundles of multilinear
mappings, etc.
In each case, such a bündle ð: Å » / carries a Riemannian metric and a
Riemannian connection. This allows us to form, just as in 1.4 for the bündle C*T,
the completions C°(£), H°(E)9 H\E) of the bündle C"°°(£) of piecewise dif
ferentiable sections in ð. 1.5 also holds in this case. The proof is exactly the same.
1.6
DEFINITION.
Let ð: Å » J; ö: F ~ I be vector bundles with Riemannian
metric and Riemannian connection. Let 0 C Å be open such that È
(
=
ð~](ß)
¥= 0,
all r G /.
(i) Denote by
i/l(0)
the set of î G H\E) with £(/) G 0„ all / G /.
(ii) Let/: 0  Fbe a differentiable fibre mapping. That is, for each t G / , /  0,
is differentiable with image in F, = ö
_,(ß)
. Then define / :
i/!(0)

Hl(F);
(£(/)) h* (/o €(0). Our next goal is to show that
Ç!(6)
C #
!
(£ ) is open and/is
differentiable. We begin with the
1.7
PROPOSITION.
With the notions of 1.6, fis á continuous mapping defined on an
open subset of
Hl(E).
PROOF.
Denote by C°(Ö) the set of continuous sections î of ð with {(/) G 0
r
Then there exists a ñ 0 such that £ G C°(0), ç G C°(£), ll€ çÉÉ« , ñ implies
ç(ß) G 0„ all / G J. Hence, C°(6) is open in C°(E). H\&) is the counterimage of
C°(Ö) under the continuous inclusion # \E)  C°(E). Therefore, Ç\â) is
open. To see that / is continuous we note that with ç — î , small, IIç — III^
and
IIVT?
— v£ II0 also become small. Moreover,
H/(ti)/(€)lloll/(n)A{)!L·
Now let Þ(ß) = i\(t)h + Þ(ß)
í
be the decomposition into the horizontal and the
vertical part, respectively. i]{t)h has the local representation (/, ç(ß), 3/,
Tt(dt,
TJ(/))),
and Þ(ß)
¼
is canonically identified with
VTJ(0·
Therefore, if
Mof ^(ï,é/
+ Ti(t)af