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.