Parabolic submanifolds of rank two

TL;DR

This paper classifies parabolic submanifolds of rank two in any codimension, distinguishing ruled from nonruled types, and provides parametrizations and singular set descriptions.

Contribution

It offers a comprehensive classification of parabolic submanifolds, including explicit parametrizations and analysis of their singularities, advancing understanding of their geometric structure.

Findings

Ruled parabolic submanifolds are the only ones that can be isometrically immersed as hypersurfaces.

Nonruled parabolic submanifolds are classified via polar and bipolar parametrizations.

The structure of the singular set of nonruled parabolic submanifolds is characterized.

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.