Parabolic submanifolds of rank two

zbMATH Open1205.53059arXiv0904.0094MaRDI QIDQ3560625

M. Dajczer, Pedro Morais

Publication date: 14 May 2010

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/0904.0094

zbMATH Keywords

second fundamental formsingular setruled submanifoldparabolic submanifoldpolar surfacerank of an immersion

Mathematics Subject Classification ID

Higher-dimensional and -codimensional surfaces in Euclidean and related (n)-spaces (53A07) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42) Global submanifolds (53C40)

Related Items (2)

The associated family of an elliptic surface and an application to minimal submanifolds ⋮ Affine submanifolds of rank two

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