1.1. LOCAL ISOMETRIC EMBEDDING O F ANALYTIC METRICS 5
Such a definition is independent of coordinates.
DEFINITION
1.1.1. The map u is free at the point p G Mn if dim(T^(iz)) =
sn + n, or diu(p),dijii(p), i,j 1, ...,n, are linearly independent as vectors in R9.
Moreover, u is a free map if u is free at each point in
Mn.
It is easy to see that if u :
Mn

R9
is free, then q sn + n and w must be an
immersion. Finally, if 0 : M n Nn is a C2 diffeomorphism and u : 7Vn Rq is
free, then the composition u o / : M
n

R9
is also free.
The map
(1.1.2) (zi, , xn) G Rn •—• (xl5 , x
n
, -x 2 , xix2, , -x 2 ) G R s - + n
gives the simplest example of a free map from Rn to R S n + n . Prom (1.1.2) and the
local charts of manifolds, it is easy to see that every C2 differential manifold Mn
has a local free map into R
S n + n
.
Now we proceed to deal with (1.0.4). If u satisfies (1.0.4), then {d\u, , dnu}
are linearly independent.
The differential system (1.0.4) is highly degenerate. We will transform it to an
equivalent differential system which is easier to analyze. We first express the metric
in a special form, adopting the notation x
(xf,xn)
= (xi, , x
n
_i, xn).
LEMMA 1.1.2. Let (Mn,g) be a smooth n-dimensional Riemannian manifold.
Then for any p G
Mn,
there exists a local coordinate system (xi,--« ,x
n
) in a
neighborhood N(p) of p where g is of the form
n-l
(1.1.3) g= ^2
gki(xf,Xn)dxkdxi
+
dx2n,
k,l=l
with
(1.1.4) 9ki(0) = Skh dngki(0)=0 for any k,l = 1, •• , n - 1.
PROOF .
We start with a normal coordinate system (xi, , xn) centered at p,
and let M
n _ 1
= {xn = 0} and e be the unit normal field along M
n _ 1
in
Mn.
For
any q G M
n _ 1
, consider the geodesic c = c(t) in
Mn
with the initial conditions
c(0) = q and c7(0) = e(q). Then (xi, ,xn_i,£) forms a local coordinate system
in a neighborhood of p.
First, we note g(dt^dt) = 1 since each t-curve is an arc-length parametrized
geodesic. Next, we have for any k = 1, , n 1
dtg(du dk) = g(Vtdt, dk) + g(du Vtdk) = g(du Vfeft) = \dkg(du dt) = 0.
Hence g(dt, dk) = 0 since it is zero at t = 0. Therefore, the metric g is of the form
(1.1.3) in the coordinates (xi, ,x
n
_i,£).
To prove (1.1.4), we have for any /c, / = 1, , n 1
dtg(dk, d,) = g(Vtdk, dt) + g(dk, Vtdt)
= g(Vkdt,dl)+g(dk,Vldt)
( L 1
'
5 )
= dkg(dt, dt) + dl9(dk, %) - g(dt, vfcft) - g(Vtdk, dt)
= -g(duVkdt)-g(Vidk,dt).
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