A class of austere submanifolds

Abstract

Austerity is a pointwise algebraic condition on the second fundamental form of an Euclidean submanifold and requires that the nonzero principal curvatures in any normal direction occur in pairs with opposite signs. These submanifolds have been introduced by Harvey and Lawson in the context of special Lagrangian submanifolds.

The main purpose of this paper is to classify all austere submanifolds whose Gauss maps have rank two. This condition means that the image of the Gauss map in the corresponding Grassmannian is a surface. The hypersurface case is due to Dajczer and Gromoll and the three dimensional case to Bryant. We show that any such submanifold is, roughly, a subbundle of the normal bundle of a surface whose ellipse of curvature of a certain order is a circle. We also characterize austere submanifolds which are Kaehler manifolds.