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.