On skew loops, skew branes and quadratic hypersurfaces

A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3-space. The paper extends recent work by M. Ghomi and B. Solomon.