- THE IMPLICIT FUNCTION THEOREM 331
We call each (Ui, <pi) a Banach space coordinate chart. A Banach manifold
atlas defines a topology '.1' on B , with a subset S of B defined to be open if for every
i E I , <fJi (Sn Ui ) c Xi is open.Let Mn and Nm be smooth closed manifolds and let Ck (M,N) denote the
set of Ck maps from M to N. In the following, we construct a Banach manifold
structure on Ck (M,N). Let 9N be~ C^00 Riemannian metric on N and define the
open neighborhood of the zero-section
(K.6) 0 ~{VE TN: !VI< inj 9 N (7r (V))} c TN,where inj 9 N (q) denotes the injectivity radius of 9N at q E N. We note that for
each q E N the exponential mapexp~N : 0 n TqN ---+ N
is a C^00 diffeomorphism onto its image.Given f E Ck (M,N), consider the set fO c fTN. The set u 1 ~Ck (J*O)
of Ck sections of f*O ---+ M is an open subset of the Banach space Ck (!*TN).We define a B anach space chart (UJ,<fJJ) for Ck(M,N) containing (centered at)
the point f Eck (M,N) by
(K.7a)
(K.7b)<fJJ: U1---+ Ck (M,N),
<fJJ (O") (x) ~ exp}(x) (O" (x)).With the collection { (UJ, <pf)} fECk (M,N) of Banach space local coordinate charts,
we obtain a Banach manifold atlas for Ck (M,N).
Now let Mn and Nm be arbitrary smooth manifolds.DEFINITION K.5. We say that a map f : M ---+ N is in c,~·: (M,N) if 'lj; 0
f o <p-l E Ck CY. •
10 'c for any smooth chart (U, <p = {xi}) of M and any smooth chart
(V, 'lj; = {ya}) of N, whenever this function is defined.
Whenever the manifolds M and N are closed, we denote C 1 ~·: (M,N) by
Ck,a (M,N). If f is smooth, then the map <fJJ : Ck,a (J*O) ---+ Ck,a (M,N)
defined by (K.7b) is a Banach space chart about fin Ck,a (M,N).
Since the IFT (Theorem K.4) is a local result, by using Banach space ch arts,
one easily obtains a version of the IFT for Banach manifolds.
Now we will consider examples of Banach manifolds of maps between manifolds.Let Mn be a compact manifold with boundary 8M. We define Ck,a (M; 8M) to
be the subset of maps Fin C 1 ~·: (M, M) satisfying F(oM) c 8M. We construct
a Banach space coordinate chart for Ck,a (M; 8M) about any smooth map f.
For reasons we shall see below, we change the metric g to a uniformly equivalentmetric g where, in a collar of 8M, the metric g is a product metric with the
property that 8M is totally geodesic in (M, g). Let expx denote the exponential
map with respect to g. Define the injectivity radius inj 9(x) of the manifold
with boundary (M, g) at a point x EM to be the supremum of r such that
expx: exp;^1 (B (x, r)) n Bg(x)(O, r)---+ B (x, r)
is a diffeomorphism, where B ( x, r) ~ {y E M : d9 (y, x) < r}. Define the injectivity
radius of M to be inj(g) = infxEM inj 9 (x). For example, for the upper half-space
{ x E JRn : xn 2': 0}, the injectivity radius of the Euclidean metric is equal to infinity.