1.1. LOCAL ISOMETRIC EMBEDDING O F ANALYTIC METRIC S 7
By (1.1.6)-(1.1.9), u0 and m satisfy in M71'1
(1.1.15) dku0 -diu0 =gki(-,0),
(1.1.16) dku0 -ui = 0,
(1.1.17) dkiu0 -ux = --dngki(-,0),
(1.1.18) u i - u i = l.
Although we have used differentiation in obtaining the Janet system (1.1.11)-
(1.1.13) from (1.1.6)-(1.1.8), we will show the equivalence of the Janet system to the
original system (1.1.6)-(1.1.8). Suppose u is a C 3 solution to the Cauchy problem
of the Janet system (1.1.11)-(1.1.13) with (1.1.14)-(1.1.18). Then (1.1.12), together
with (1.1.18), immediately implies (1.1.7). In a similar way, (1.1.6) follows from
(1.1.11) and (1.1.16). To prove (1.1.8), we first note (1.1.15) implies gik — diu-dku =
0 at xn = 0. Considering the initial condition (1.1.16) and (1.1.17), we have, by
dn(gki - dku • diu) = -2dkiu0 • m + 2dkiu0 • m - dk(ui • diu0) - di(ui • dku0) = 0,
at xn = 0. Next by (1.1.10), (1.1.13) and (1.1.11), we have
dnn(gki ~ dku • diu) = -dk(diu • dnnu) - dt(dku • dnnu) = 0.
Hence, (1.1.8) is valid. Therefore, we have proved the equivalence of the system
(1.1.6)-(1.1.8) and (1.1.11)-(1.1.13) with the initial condition (1.1.14)-(1.1.18).
If dku, dkiu, dnu, 1 /c, / n — 1, are linearly independent, we can solve dnnu
from (1.1.11)-(1.1.13) to get
(1.1.19) dnnu = F(x, dku, dnu, dkiu, dknu) near xn = 0,
where F is smooth in x and analytic in other arguments and /c, / run over 1, • • • , n—1.
Moreover, F is analytic in x if g is an analytic metric. Note that there are sn
equations in (1.1.11)-(1.1.13). If q = s
, then dnnu can be solved uniquely in
(1.1.11)-(1.1.13). For q sni solutions for dnnu may not be unique.
If g is an analytic metric, the Cauchy-Kowalevski theorem implies that (1.1.19)
always admits an analytic solution in a neighborhood of the origin with the given
Cauchy data (1.1.14). Hence we have the following result.
L E M M A 1.1.4. Let g be an analytic metric of the form (1.1.3) inW1 c W1 and
q sn. Suppose that there exist analytic functions UQ,UI : B n _ 1 — M9 satisfying
(1.1.15)-(1.1.18) and that dku$, dkiUo, u\, 1 fc, / n — 1, are linearly independent
in B n _ 1 . Then g, restricted to a neighborhood of the origin 0 G B n
analytic isometric embedding in W1.
Sometimes we are interested in free isometric embeddings.
L E M M A 1.1.5. Let g be an analytic metric of the form (1.1.3) in B n C R n
and q sn + n. Suppose that there exist analytic functions UQ,U\ : B n _ 1 — * W1"1
satisfying (1.1.15)-(1.1.18) and that dkUQ, dkiUo, u\,dku\, 1 /c, / n— 1, are
linearly independent in B n _ 1 . Then g, restricted to a neighborhood of the origin
0 G B n , admits a free analytic isometric embedding in R 9 .