  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.
