Regularity of isometric immersions of positively curved Riemannian manifolds and its analogy with CR geometry (Q1093172)

Let M be an n-dimensional Riemannian manifold and F be an isometric immersion of M into \({\mathbb{R}}^{n+1}\). It is shown that under certain conditions on the sign of principal curvatures of F(M), F satisfies an over-determined system of elliptic partial differential equations after one adds the scalar curvature equation. As a corollary, if M is an analytic manifold of positive sectional curvature, F is analytic and uniquely determined by F(P) and dF(P) at a reference point P of M. An analogous problem in CR geometry is proposed.