Nash-Kuiper Theorem

The Nash-Kuiper theorem is a flexibility theorem for continuously differentiable isometric embeddings of Riemannian manifolds into Euclidean space (Nash 1954, Kuiper 1955). One form states that, for a compact smooth manifold without boundary equipped with a Riemannian metric tensor and , every smooth embedding satisfying

for every nonzero manifold tangent vector can be approximated uniformly arbitrarily closely by isometric embeddings. The displayed inequality is the strictly short condition. Uniform approximation controls the positions of image points, rather than requiring their derivatives to be close.

In particular, a torus obtained by identifying opposite sides of a Euclidean square admits a isometric embedding in . Its intrinsic metric is preserved even though its image can be highly corrugated. The differentiability qualification is essential for this conclusion (Borrelli et al. 2012).