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. The isometric deformation problem for a given isometric immersion f : M → R of a Riemannian manifold into flat Euclidean space with codimension p and a positive integer q is to describe all possible isometric immersions f : M → R. A satisfactory answer to the local version of the problem for every hypersurface (p=1) and q = 1 going back almost a century is due to Sbrana [19] and Cartan [4]. However, basic questions, like the existence of Sbrana-Cartan hypersurfaces of the discrete type or the possibility of smoothly attaching different types of these deformable hypersurfaces, were answered positively only recently; see [10] and also [3] for a special case. The global version of the problem for hypersurfaces was solved in [11] and [18]. Nothing similar to the Sbrana-Cartan theory for codimensions q = p higher than one has yet been obtained. Nevertheless, the classical Beez-Killing rigidity theorem for hypersurfaces, the starting point for the theory, has several generalizations; see [1], [2], [5] and [20]. All these results provide generic algebraic conditions on the second fundamental form of the isometric immersion that imply isometric rigidity, that is, any other isometric immersion must differ by an isometry (rigid motion) of the ambient space. There is a large set of isometric deformations once we allow codimension q > p since one can always compose f : M → R with (local) isometric immersions of R into R. In [7] we found generic algebraic conditions on the second fundamental form