Mean curvature and shape operator of isometric immersions in real-space-forms

According to the well-known Nash's theorem, every Riemannian n -manifold admits an isometric immersion into the Euclidean space E n(n+1)(3n+11)/2 . In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.