8 1. FUNDAMENTAL THEOREM S
PROOF. We look for a free isometric embedding u : B
n

Rq
of the following
form:
(1.1.20) u(x) =
(u(x),±x2n).
By (1.1.11)-(1.1.13), u satisfies for 1 fc, I n - 1
( dku 9nnii = 0,
(1.1.21) dnu dnnu =-xn,
[ dkiu 9nn£t = dknu 9zn^ - \dnngki.
By the Cauchy-Kowalevski theorem, there exists an analytic solution u in a neigh-
borhood of the origin 0 E B
n
with the Cauchy data
&U„=o = ^o, dnu\Xn=0 = ui-
Then by the derivation of (1.1.21), the map u solves the Janet system (1.1.11)-
(1.1.13) and satisfies (1.1.14)-(1.1.18). Hence u is an analytic isometric embedding
for g.
Now we need to prove only that u is free in a neighborhood of xn 0. By
the assumption, dku, dnuy dkiu, c^u , 1 /c, I n 1, are linearly independent at
xn = 0. By (1.1.20), the last components of all these vectors are 0 at xn 0 and
dnnU (* 1), where * denotes some vector in
Rq~1.
Hence dkU1 dnu, d^u, dknu,
dnnU, 1 fc, Z n 1, are linearly independent for xn 0.
The following is the main result in this section.
THEOREM
1.1.6. Any n-dimensional analytic Riemannian manifold admits a
local analytic isometric embedding in
RSn
and a local free analytic isometric embed-
ding m R s - + n .
Obviously, Theorem 1.0.1 is a part of Theorem 1.1.6.
PROOF.
We prove Theorem 1.1.6 by an induction on n.
We first consider n = 2. Suppose g is given in a local coordinate system by
9 = 9u(xi,%2)dxi+dx% in
B2,
where gn satisfies gu(xi,0) = 1 and 92#n(^i 0) 0- This can always be arranged,
by Remark 1.1.3. Now let
(1.1.22) uo = (cosxi,sinxi,0), u\ = (0,0,1).
Then
UQ,UI
satisfy (1.1.15)-(1.1.18) and diWo,dn^o, ui a r e linearly independent in
B1.
We may apply Lemma 1.1.4 to get a local analytic isometric embedding of g
inR
3
.
Now we prove the existence of a local free analytic isometric embedding for
n = 2. In this case, sn + n = 5. Set
(1.1.23) i^o = (cosxi,sinxi,0,0), u\ = (0,0,cosxi,sinxi).
Obviously,
UQ,U*
still satisfy (1.1.15)-(1.1.18) and diUQ,duUQ,ul and d\u\ are lin-
early independent. We may apply Lemma 1.1.5 to get a free local analytic isometric
embedding of g in
R5.
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