6
1. FUNDAMENTAL THEOREMS
Since (xi, ,x
n
) are normal coordinates at p, we have Vkdi(0) = 0 for all k and
/. Hence we have dtg{dk, d{) = 0 at the origin.
REMARK
1.1.3. If n = 2, we may simply take M 1 as a geodesic parameterized
by the arc-length parameter x\. Then g is of the form
9 = 9ii(xi,x2)dxl + dxl,
where
#nOi,o) = I, #2011 (#i,o) = o.
These are the geodesic coordinates determined by the geodesic M1. Note gu = 1
and Vidi = 0 on x2 = 0, since x2 0 is an arc-length parameterized geodesic.
Hence d2g\\ = 0 on x2 = 0 by (1.1.5). The geodesic coordinates will be used
frequently in subsequent chapters.
Suppose g is a smooth metric given by (1.1.3) in B
n
. In order to construct a
smooth isometric imbedding of g in
Rq,
we need to find a smooth map u : B
n

Mq
satisfying
(1.1.6) dku-dnu = 01
(1.1.7) dnu-dnu= 1,
(1.1.8) dku-diu = gkU
in B n for any fc, / = 1,..., n 1. Before proceeding, we derive some identities. By a
straightforward calculation, we have
(1.1.9)
and
dn(dku diu) = dknu 9ju + dku 9inw
= 9^(9/^ dnu) + 9z(9/eW #nu) - 29/e/w 9nu,
dnn(dkU C^ll) = 25
fcn
U dinU + 9fcnn^ ' dtU + 9fc7i 9/
n
n ^
(1.1.10) = 2dknu 5/nu - 2dkiu 9nnu
+ dk(diu 9nnw) + d/(dfcii dnnu).
By differentiating (1.1.6)-(1.1.8) with respect to xn, we get for any k,l
l , . . . , n - 1
(1.1.11) dku- dnnu = 0,
(1.1.12) dnu-dnnu = 0,
(1.1.13) 9^ 9nnu = dknu 9Znix - -dnngki-
We used (l.l.lO)-(l.l.ll) to get (1.1.13). We call (1.1.11)-(1.1.13) the Janet system.
There are sn equations in this system, and, as we know, we should require that
q sn.
Now we prescribe Cauchy data for (1.1.11)-(1.1.13) as follows:
(1.1.14) u\Xn=o = u0, dnu\Xn=0 = ui.
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