Genuine rigidity of Euclidean submanifolds in codimension two (Q1771168)

An isometric deformation of a Euclidean submanifold is called genuine if the submanifold cannot be included into a submanifold of larger dimension in such a way that the deformation of the former is given by an isometric deformation of the latter. The submanifold is said to be genuinely rigid if it has no genuine deformation. In this paper, the authors study the deformation problem in codimension two for the classes of elliptic and parabolic submanifolds. In spite of having a second fundamental form as degenerate as possible without being flat, i.e., the Gauss map has rank two everywhere, their main result is as follows: If \(f:M^n\to \mathbb R^{n+2}\) is a nowhere surface-like elliptic submanifold that admits no local isometric immersion in \(\mathbb R^{n+1}\), then \(f\) is genuinely rigid. Moreover, an additional unexpected deformation phenomenon for elliptic submanifolds carrying a Kähler structure is described.