Isometric Embeddings of Riemannian Manifolds

The dot in (1) denotes the usual scalar product of R. The notion embedding means, that w is locally an immersion and globally a homeomorphism of M onto the subspace u(M) of R*. If an embedding w : M -• R satisfies (1) on the whole M, we speak of an isometric embedding. If w is an immersion and a solution of (1) in a (possibly small) neighbourhood of any point of M, we speak of a local isometric embedding. A further question is the regularity of the embedding in dependence on the regularity of the metric. And finally, what can be said about the minimal value of <?? We will give some historical remarks. There exists a great number of beautiful papers which handle the isometric embedding problem (local or global) under further assumptions on the manifold M or the metric g (e.g. special values of dimension n, positivity assumptions on the curvature), but we are interested only in the general problem. Janet (1926), Cartan (1927) and Burstin (1931) proved the existence of a local isometric embedding with q = n(n + l)/2 in the analytical case; the essentials of their proofs are suitable applications of the Cauchy-Kowalewski theorem. Up to date a corresponding result (with the same value of q and without further assumptions) is unknown in the nonanalytical case, even for the dimension n = 2. Nash (1954) and Kuiper (1955) proved the existence of a (global) isometric embedding of class C with n < q < 2n -f 1 (in fact their results are more subtle),