Twisted-Austere Submanifolds in Euclidean Space - Ivey, Thomas A.; Karigiannis, Spiro (Symmetry, Integrability and Geometry: Methods and Applications, 2021)
A twisted-austere -fold (, ) in ℝⁿ consists of a -dimensional submanifold of ℝⁿ together with a closed 1-form on , such that the second fundamental form A of and the 1-form satisfy a particular system of coupled nonlinear second-order PDE. Given such an object, the ''twisted conormal bundle'' *+ is a special Lagrangian submanifold of ℂⁿ. We review the twisted austere condition and give an explicit example. Then we focus on twisted austere 3-folds. We give a geometric description of all solutions when the ''base'' is a cylinder, and when is austere. Finally, we prove that, other than the case of a generalized helicoid in ℝ⁵ discovered by Bryant, there are no other possibilities for the base . This gives a complete classification of twisted-austere 3-folds in Rⁿ.