One of the fundamental problems in the theory of submanifolds is the immersibility of a Riemannian manifold in a Euclidean space. According to the 1956 famous Nash embedding theorem, every Riemannian manifold can be isometrically embedded in some Euclidean spaces with sufficiently high codimension. The Nash theorem was aimed with the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help in the study of Riemannian geometry. However, this hope has not materialized until the early of 1990s. The main reason for this is the lack of controls of the extrinsic properties of the submanifolds by the known intrinsic invariants. This difficulty leads to the following fundamental problem in differential geometry.