Existence and nonexistence of skew branes

A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a nonzero Euler characteristic χ, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ²/4. We give a version of this result for immersed surfaces in dimension 4. We construct examples of skew spheres of arbitrary odd dimensions, generalizing the construction of skew loops in 3-dimensional space due to Ghomi and Solomon (2002). We conclude with two conjectures that are theorems in 1-dimensional case.