Chapter 1. The Hubert manifold of

H]

-curves

We begin here the construction of a proper Hubert manifold, associated

canonically to a finite-dimensional Riemannian manifold M. The Hubert mani-

fold H\l, M) in question is formed by the maps c: I - Ì of dass H\ I = [0,1];

cf. 1.1. Of greater importance than the space of all

Ç!

-maps c: I -+ Ì are certain

submanifolds of finite codimension which we will introduce in Chapter 2.

Here we describe in füll detail the natural atlas for

Hl(Iy

M). The Charts

(exp"1,

?l(c)), c G H\I, M), are defined with the help of the inverse of the

exponential map, restricted to 'short' vector fields along c; cf. 1.10, 1.12.

Next we describe the two canonical bundles a° and

a]

over Ç*(/, Ì) . The fibre

(a°)~1(c)

over c consists of the i/°-vector fields along c, while

(a})~\c),

consist-

ing of the

H]-vector

fields along c, is identified with the tangent space TcH\l, M)

at c; cf. 1.13. This is then used to show that there exists a canonical Riemannian

metric on

Hl(I,

M) (cf. 1.19), and the energy integral E(c) is differentiable; cf.

1.20. the only critical points of Å are the constant maps.

It is well known that the set C°(7,

Rw)

of continuous curves c: I = [0,1] - R"

in Euclidean «-space

Rn

is a Banach space. C°(7, R") can be viewed as the

completion of the space C'°°(/,

Rw)

of piecewise differentiable curves with respect

to the maximum norm

Hc||eo = sup|c(/)|.

The associated distance is

rfjc,c') = s u p | c ( r ) - c ' ( 0 | .

For an element c Å C°(7, R"), in general, there exists neither the length L(c) nor

the energy integral E(c). Therefore, in differential geometry one considers a

different norm on C"°°(/, R"), i.e., the norm ||eil, derived from the scalar produet

(c, c'), = (c, c')

0

+ (dc,dc')0.

Here, (ey e')0 = fre(t)-e\t)dt and dc(t) = c(t). The completion of C"»{I,W)

with respect to the norm Hell, is denoted by

Hl(I,R").

According to a classical

result of Lebesgue, an //^curve c: I -* R" can be described as an absolutely

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http://dx.doi.org/10.1090/cbms/053/02