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 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
3
http://dx.doi.org/10.1090/cbms/053/02
Previous Page Next Page