Parabolic submanifolds of rank two Marcos

An immersed submanifold f :M → R , n ≥ 3, into Euclidean space with the induced metric is called of rank two if at any point the kernel of its vector valued second fundamental form has codimension two. Equivalently, we have that the image of the Gauss map in the Grassmannian of non-oriented n-planes Gn is a surface. These submanifolds have been the object of a great deal of work in Riemannian Geometry since long time ago. For instance, see [2] and references therein. This interest is in good part motivated by the fact that their curvature tensor is “as flat as possible” without vanishing altogether. The subspace spanned by the second fundamental form, usually called the first normal space and denoted by N1, of a rank two submanifold satisfies dimN1 ≤ 3 at any point. It turns out that if in substantial codimension, any rank two submanifold is a hypersurface if dimN1 = 1 at any point. Then f is either a Euclidean surface or the cone over a spherical surface, up to a Euclidean factor, if dim N1 = 3 everywhere. Submanifolds in the remaining and much more interesting case, namely, when dimN1 = 2 everywhere, have been divided in three classes: elliptic, hyperbolic and parabolic. A complete parametric description of the elliptic submanifolds was given in [5]. For codimension N − n = 2, it was shown in [6] that elliptic and nonruled parabolic submanifolds are genuinely rigid. This means that given any other isometric immersion f̃ : M → R there is an open dense subset of M such that restricted to any connected component f |U and f̃ |U are either congruent or there are an isometric embedding j: U →֒ N into a Riemannian manifold N and either flat or isometric noncongruent hypersurfaces F, F̃ : N → R such that f |U = F ◦ j and f̃ |U = F̃ ◦ j. Recently, we proved [8] that nonruled parabolic submanifolds in codimension two are not only genuinely rigid but, in fact, isometrically rigid. The goal of this paper is to classify parametrically parabolic submanifolds in any codimension. First, we describe the ones that are ruled and show that they are the only parabolic submanifolds that admit an isometric immersion as a hypersurface. Then, we classify the nonruled ones by two different means. In fact, we provide the polar and bipolar parametrizations, each of which is associated to a parabolic surface and a function on the surface which satisfies a parabolic differential equation. To conclude, we describe the structure of the singular set of the nonruled parabolic submanifolds.