Seminar in Analysis and Geometry

An immersion from a smooth n-dimensional manifold $M$ into $R^q$ is called totally nonparallel if, for every pair of distinct points $x$ and $y$ in $M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. The simplest example is the map $R \to R^2$ sending $x$ to $(x,x^2)$. Given a manifold $M$, what is the minimum dimension $q = q(M)$ such that $M$ admits a totally nonparallel immersion into $R^q$? I will discuss how to apply Eliashberg and Gromov's "removal of singularities" h-principle technique to obtain existence results, and I will talk about some important considerations when studying differential conditions which manifest at pairs (or $k$-tuples) of points.