High Codimensional 0-Tight Maps on Spheres
by Thomas F. Banchoff

The condition of 0-tightness for smooth immersions of 2-manifolds in E^M is equivalent to minimal total absolute curvature, but these notions do not coincide in higher dimensions. By a result of Chern and Lashof, if a smooth n-sphere embedded in E^M has minimal total absolute curvature, then it must bound a convex (n+1)-cell in an affine (n+1)-dimensional subspace. We will show that for any n > 2 and M > n, there exists a 0-tight polyhedral embedding of the n-sphere into E^M whose image lies in no hyperplane.