Isometric embedding of a smooth compact manifold with a metric of low regularity
Nash [7] proved that if GEC ~ there is an isometric embedding UECI(X, R N) provided that N>-n+2 and that there is a differentiable embedding of X in R N, in particular i f N~2n. Nash also indicated that the condition N>=n+2 could be weakened to N>=n+I, which was proved by Kuiper [6]. It should be observed that (I. I) in local coordinates means n(n + I)/2 equations for N variables. For G E C k, k=>3, Nash [8] also showed that there is an embedding UECk(X,R N) if N>=n(3n+ll)/2. The condition on N has been improved for smooth metrics to N>--n(n+l)/2+3n+5 by Gromov and Rokhlin [3], who also gave lower estimates for the embedding dimension of the same order of magnitude for k_>-2. This result of Nash was extended by Jacobowitz [5] to HOlder classes H ~ with a > 2 , and he also showed that there are metrics GEH p, f l>2, such that (1.1) has no solution UEH~(35, RN), ~>~, for any N. The result of Nash--Kuiper shows in particular that there is always a local embedding of 35 in R n+l. Borisov [1] has announced that if G is analytic there is a local isometric embedding UEH'(X, R n+l) with any ~< 1 + 1/(n~+n + 1). Thus is close to 1 if n is large. The low regularity seems to be caused by the demand for a low codimension, for by permitting large values of N we shall prove