In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60: 383-396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58: 545-556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58: 683-689, 1955) says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,\ensuremathα with \ensuremathα \ensuremath> 2/3 in (Borisov in Vestn. Leningr. Univ. 14(13): 20-26, 1959; Borisov in Vestn. Leningr. Univ. 15(19): 127-129, 1960). On the other hand he announced in (Borisov in Doklady 163: 869-871, 1965) that the Nash-Kuiper statement can be extended to local C1,\ensuremathα embeddings with \ensuremathα \ensuremath< (1 + n + n2)-1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared in (Borisov in Sib. Mat. Zh. 45(1): 25-61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.

Nonlinear Partial Differential Equations Abel Symposia Volume 7, 2012, pp 83-116