Ruled Austere Submanifolds of Dimension Four

TL;DR

This paper classifies 4-dimensional austere submanifolds in Euclidean space that are ruled by 2-planes, revealing their geometric structures and providing explicit constructions for different algebraic types.

Contribution

It provides a complete classification of 2-ruled austere 4-folds into three types and constructs explicit examples, including holomorphic and helicoid-based submanifolds.

Findings

Type A: ruling map is holomorphic and constructed from a holomorphic curve.

Type B: submanifolds are generalized helicoids or products of helicoids.

Type C: includes generalized helicoids in R^7 and other examples.

Abstract

We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R^n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R^6 or the product of two classical helicoids in R^3. If M is 2-ruled of Type C, then M is either a one of the above, or a generalized helicoid in R^7. We also construct examples of 2-ruled austere hypersurfaces in R^5 with degenerate Gauss map.