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 Ã 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.