Introducing totally nonparallel immersions

Abstract

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.Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration spaceFk(M) of k-tuples of distinct points of M; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, and several others. Typically, a map satisfying one of these configuration space conditions induces some Sk-equivariant map on the configuration space Fk(M) (or on a bundle thereof) and obstructions can be computed in the form of Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied.Our main result is a Whitney-type theorem: every smooth n-manifold M admits a totally nonparallel immersion into R4n−1, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value TN(RPn)=4n−1 when n is a power of 2. This is the first optimal-dimension result for any closed manifold M besides S1, for any of the recently-studied configuration space conditions.