# Introducing Totally Nonparallel Immersions

## TL;DR

This paper introduces totally nonparallel immersions of manifolds into Euclidean space, establishing their existence, properties, and optimal dimensions, and connects these to classical problems in topology and geometry.

## Contribution

It defines totally nonparallel immersions, proves their existence in near-minimal dimension, and links them to the vector field problem and symmetric bilinear maps.

## Key findings

- Every n-manifold admits a totally nonparallel immersion into R^{4n-1}.
- For real projective spaces of dimension a power of 2, the minimal dimension is exactly 4n-1.
- This is the first optimal-dimension result for closed manifolds other than S^1 under these conditions.

## Abstract

An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M \to \mathbb{R}^{TN(M)}$. 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 problem 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 space $F_k(M)$ of k-tuples of distinct points of $M$; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, etc. Typically, a map satisfying one of these configuration space conditions induces some $S_k$-equivariant map on the configuration space $F_k(M)$ (or on a bundle thereof) and obstructions can be computed using Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied. Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $\mathbb{R}^{4n-1}$, 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 $TN(\mathbb{R}P^n) = 4n-1$ when $n$ is a power of 2. This is the first optimal-dimension result for any closed manifold $M \neq S^1$, for any of the recently-studied configuration space conditions.