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, u i 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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