Introducing totally nonparallel immersions

An immersion of a smooth n-dimensional manifold M→Rq is called totally nonparallel if, for every distinct x,y∈M, the tangent spaces at f(x) and f(y) contain no parallel lines; equivalently, they span a 2n-dimensional space. Given a manifold M, we seek the minimum dimension TN(M) such that there exists a totally nonparallel immersion M→RTN(M). In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion and embedding problems for real projective spaces, and nonsingular symmetric bilinear maps.