Genuine deformations of submanifolds
Luis A. Florit | Dajczer, Marcos
genuine deformation | isometric rigidity | compositions | ruled submanifolds
Since a submanifold of a deformable one is also deformable, to go deeper into the isometric deformation problem for submanifolds one has to discard those deformations that arise this way. Our goal in this paper is twofold. First, to introduce the concept of genuine deformation, and then to give the (quite restricted) geometric structure of the submanifolds that admit deformations of this kind. As a consequence, we have several applications for a new rigidity concept that extends the classical one of isometric rigidity and the one of compositions. In particular, this concept allows us to give generalizations and extensions of several well known results, some of them classical. The unifying character and geometric nature, as opposed to a purely algebraic one, of our main result suggest that it should be the starting point for a deformation theory extending the classical one for hypersurfaces to higher codimension.