1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

332 K. IMPLICIT FUNCTION THEOREM

Let B be the Banach space of Ck,a vector fields on M which when restricted to

8M are tangent to 8M. Define the open subset C =i= {VE B: IV (x) I < inj(g) for

each x EM}. Given f E Ck,a (M; 8M), consider the map 1.fJJ : C---+ Ck,a (M; 8M)

defined by

1.fJJ(V) = {fv: x r--7 exp}(x)(V(f(x)))}.

Similarly to (K.6), we define

(K.8) Oa =i= {VET M : !VI < inj 9 (7r (V)), exp9 (V) EM } c TM.

Given f E c= (M; 8M), consider the subset U1 of a-E Ck,a (f*(Oa)) satisfying

a-(x) ET (8M) for x E 8M. Similarly to the previous subsection, define a Banach

space chart (UJ,<fJJ) for Ck,a (M;aM) by

l.fJJ: U1---+ Ck,a (M; 8M),

'PJ (a-) (x) =i= exp}(x) (a-(x)).

Note that if x E 8M, then 'PJ (a-) (x) E 8M since a-(x) E Tf(x) (8M) and since

8M is totally geodesic in (M, g).

2. Holder spaces and Sobolev spaces on manifolds

In this section we discuss Holder spaces and, briefly, Sobolev spaces on mani-

folds. We also include a discussion of jet bundles.

2.1. Ck,a Holder spaces for tensors.

Let (Mn,g) be a Riemannian manifold, which may be noncompact and may

have boundary. Let 7r : ET ---+ M be a c= real vector bundle with metrics on

its fibers and a compatible connection (such as a tensor bundle). For the tensor

bundle Ee =i= E @@e T* M we have the covariant derivative \7 : Ee ---+ Ee+i · Given an

extended integer k E [O, oo], let C 1 ~c (M, E) be the set of sections¢ of Elint(M) such

that for each 0 :::; C < min { k + 1, oo }, the C-th covariant derivative \le¢ extends

continuously to M as a section of Ee (we shall use the same notation \le¢ for this

extension). We define the seminorms [¢]e =i= supxEM l\le¢(x)I for C ~ 0.

DEFINITION K.6. Given an integer k E [0, oo), the space Ck (M,[) is the

Banach space of all sections¢ E C 1 ~c (M,[) for which the norm

k

ll<Pllk =i= 2::[¢]1

j=O

is finite.

Let ( {U,e};=l, 7/;13 = {x~}~ 1 : U13 ---+ !Rn) be a covering of M by coordinate
charts. Further assume that there are trivializations Erlu 13 ---+ U 13 x ffi.r correspond-

ing to bases of local sections {sb, ... , s~}. Given ¢ E C 1 ~c (M, E), its covariant

derivative may be expressed as