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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