4 1. FUNDAMENTAL THEOREM S

global isometric embedding of smooth Riemannian manifolds. The main results

are the following.

THEOREM

1.0.1. Any analytic n-dimensional Riemannian manifold admits an

analytic local isometric embedding in

Wn.

THEOREM

1.0.2. Any smooth n-dimensional Riemannian manifold admits a

smooth local isometric embedding in R S n + n .

THEOREM

1.0.3. Any smooth n-dimensional compact Riemannian manifold ad-

mits a smooth isometric embedding in

W1

for q = max{sn + 2n, sn + n + 5}.

We prove Theorems 1.0.1-1.0.3 by solving (1.0.1). For Theorems 1.0.1 and

1.0.2, it suffices to solve the local version (1.0.4).

Theorem 1.0.1 is referred to as the Janet-Cartan Theorem. It was proved by

Janet [123] for n — 2 and Cartan [32] for the general case. In Section 1.1, we

provide a proof of Theorem 1.0.1, based on the Cauchy-Kowalevski Theorem.

In Section 1.2, we prove Theorem 1.0.2, which is due to Gromov and Rokhlin

[71] and also Greene [65]. Here, a crucial step is to solve a perturbation problem

for the isometric embedding. The nonlinear partial differential equation for such

a perturbed problem exhibits a loss of differentiability. By applying the Laplacian

operator, rearranging the resulting equation and then applying the inverse of the

Laplacian operator, we may rewrite this equation in a form which is essentially

elliptic. Then there is no loss of differentiability and we may solve it by a standard

iteration.

Theorem 1.0.3 is often referred to as the Nash Embedding Theorem. It was

first proved by Nash [166], with a bigger dimension q. The present form was due to

Giinther [75]. The proof of Theorem 1.0.3, to be presented in Section 1.3, consists

of two steps. First, for any Riemannian metric g on the manifold M

n

, we find an

embedding u$ :

Mn

— •

Rq

such that g — duo • duo is also a metric. Second, we

modify UQ to get u satisfying du- du = g. During this process, we need to maintain

that u is still an embedding. In this, the technique developed in the second section

plays an important role.

1.1. Local Isometric Embedding of Analytic Metrics

In this section, we discuss the local isometric embedding of analytic Riemann-

ian manifolds and prove Theorem 1.0.1 by solving (1.0.4). The proof is based on

the Cauchy-Kowalevski Theorem, which asserts the local solvability of the Cauchy

problem of certain classes of analytic differential systems with analytic Cauchy data

prescribed on an analytic noncharacteristic hypersurface. However, the Cauchy-

Kowaleski Theorem cannot be applied directly to the differential system (1.0.4). In

the following, we rewrite (1.0.4) for the metric g given in geodesic coordinates as

an equivalent differential system with a naturally prescribed Cauchy data so that

we may apply the Cauchy-Kowaleski Theorem to get a local solution.

We first introduce a concept, which will be useful throughout this chapter. Let

Mn be a C2 n-dimensional manifold and let u : Mn — Rq be a C2 map. For any

given point p G M n , we define the osculating space T2(u) by

(1.1.1) T^(u)= span {diu(p),diju(p); i, j = 1, ...,n} .