High codimensional $0$-tight maps on spheres

Abstract:

For smooth immersions of 2-manifolds into ${E^M}$, the condition of 0-tightness is equivalent to that of minimal total absolute curvature, but for higher dimensional manifolds these notions are quite different. By a result of Chern and Lashof, a smooth n-sphere embedded in ${E^M}$ with minimal total absolute curvature must bound a convex $(n + 1)$-cell in an affine $(n + 1)$-dimensional subspace, but we show that for any $n > 2$ and any $M > n$ there is a 0-tight polyhedral embedding of the n-sphere into ${E^M}$ with image lying in no hyperplane.