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.