On extendability of isometric immersions of spheres (Q674642)

Let \(S^n\) be the standard \(n\)-sphere. This paper is devoted to a classical problem from differential geometry, which consists of classifying isometric immersions from \(S^n\) into Euclidean spaces. It is known that the sphere \(S^n\) of constant curvature one is rigid in \(E^{n+1}\), \(n\geq 2\), and it is not rigid in \(E^{n+2}\). \textit{W. Henke} [Math. Ann. 219, 261-276 (1976; Zbl 0311.53032)] gives an example of a \(C^\infty\) isometric immersion \(F:S^n\to E^{n+2}\) which cannot be extended to a \(C^\infty\) isometric immersion on an open neighborhood of \(D^{n+1}\) in \(E^{n+1}\). The author proves the following Theorem: Suppose that \(S^n\) is the \(n\)-dimensional sphere of constant curvature one, regarded in the usual fashion as the boundary of the flat unit disk \(D^{n+1}\). If \(n\geq 3\) and \(F:S^n\to E^{n+2}\) is a \(C^\infty\) isometric immersion, there exists a unique \(C^\infty\) isometric immersion \(\widetilde{F}:D^{n+1}\to E^{n+2}\) such that \(\widetilde{F}|S^n=F\). This result shows that, in particular, any isometric immersion \(S^n\to E^{n+2}\) is homotopic via isometric immersions to a standard embedding into a hyperplane. The theorem is not true in the class \(C^k\), where \(1\leq k<+\infty\).