Regularity of certain rigid isometric immersions of \(n\)-dimensional Riemannian manifolds into \({\mathbb{R}}^{n+1}\) (Q1263089)

Let M and N be \(C^{\omega}\) (i.e. real analytic) Riemannian manifolds. The author deals with the general question whether a locally rigid isometric \(C^ k\)-immersion from M into N is \(C^{\omega}\). In this paper the following theorem is proved: Let M be a \(C^{\omega}\) Riemannian manifold of dimension \(n\geq 3\), F an isometric \(C^ 2\)- immersion from M into \({\mathbb{R}}^{n+1}\) and \(P\in M\). If the immersed submanifold F(M) has at least three nonzero principal curvatures at F(P) then F is \(C^{\omega}\) at P. (It is known that the existence of three nonzero principal curvatures is sufficient for F to be locally rigid.) The main idea of the proof is to show that F is a solution of a system of (nonlinear) elliptic partial differential equations of second order, which are \(C^{\omega}\) in its arguments, and then to apply a well-known regularity theorem.