Genuine deformations of submanifolds

Abstract

We introduce the concept of genuine isometric deformation of an Euclidean submanifold and describe the geometric structure of the submanifolds that admit deformations of this kind. That an isometric deformation is genuine means that the submanifold is not included into a submanifold of larger dimension such that the deformation of the former is given by a deformation of the latter. Our main result says that an Euclidean submanifold together with a genuine deformation in low (but not necessarily equal) codimensions must be mutually ruled, and gives a sharp estimate for the dimension of the rulings. This has several strong local and global consequences. Moreover, the unifying character and geometric nature, as opposed to a purely algebraic one, of our result suggest that it should be the starting point for a deformation theory extending the classical Sbrana - Cartan theory for hypersurfaces to higher codimensions.