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} .
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