4
WILHELM KLINGENBERG
continuous curve for which dc(t) = c(t) exists for almost all t and dc(t) is Square
integrable, i.e., (3c, 3c)0 oo.
We recall that c: 7 - R" is called absolutely continuous if, for every å 0,
there exists a ä 0 such that 0 t0 · · · t2k+1 1 and 2f=01 '2/+é ~ '2/1 *
imply Ó,| c(i2|.+ 1) - c(/2i)| å.
Note that, in particular,
Hl(I, Rn)
is a subset of C°(7,
Rw).
Actually, as one can
easily see, the inclusion is continuous; cf. 1.5. The functional E(c) = (3c, 3c0/2
on
C'°°(I,Rn)
possesses an extension to the Hubert Space
H\l,Rn)
and there it
defines an R-valued C°°-function. Its differential is given by
DE{C).T\
= (c, ç)
0
.
We now extend these constructions to the case of curves on a «-dimensional
Riemannian manifold M. We assume Ì to be complete, 7 = [0,1].
1.1
DEFINITION,
(i) Denote by C"°°(7, M) the set of piecewise differentiable
curves c: 7 - M.
(ii) By C°(I, M) denote the space of continuous curves c: 7 - M, endowed
with the metric
rfjc,c') = sup/(c(0,c'(0).
Here, d is the metric derived from the Riemannian metric on M.
(iii) Á curve c: 7 Ì is said to be of dass 77* if, for a chart (w, M') of Ì and
7' = c-\M'\ the mapping ß G 7'H w ° c(t) G
Rw
is in
#l(7',Rw).
By
HX(I,
M)
we denote the set of /f
!-maps
c: 7 M.
Note. In 1.1(iii) the particular choice of the chart does not play a role. Indeed, if
ö : (7 C
Rn
-* U' C R" is a diffeomorphism and w: / t/ is of dass
77!,
then
ö ï Ì : / -* ß/ ' is also of dass H\
We have the following canonical inclusions: C"°°(7, Ëß) «= * H\I, M) **
C°(7, Ì) .
1.2
PROPOSITION.
C"°°(7, M) is á dense subspace of the complete metric space
{C°(7, M), /„}.
PROOF.
Á curve c Å C°(7, Ì ) can be covered by finitely many Charts. Thus,
the proposition is reduced to the well-known fact that
C/QO(I\R")
is dense in
C°(7\ R"), Ã an interval in 7.
On H\I, Ì ) we can consider the energy integral E.
1.3
PROPOSITION.
For c G
i7!(7,
M),
£(c) = ^(e(0,e(0*
w we// defined.
PROOF.
Let w © c: 7'
Rn
be a local representation of c 17' with respect to
some chart («, AT). Then we can define 2E(c \ Ã) to be the integral over à of the
function g(u ° c(t))((u ï c)'(/), (w°cy(0). Note that « ï
c
G Ç
÷(Ã,¸Ñ)
pos-
sesses a Square integrable derivative. In the same way as for curves of dass
C00,
one shows that this is independent of the particular choice of the chart.
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