On the complexity of isometric immersions of hyperbolic spaces in any codimension

Abstract

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a Lipschitz map $F:M^m\to {\mathbb{R}}^n$, where $M^m$ is a Hadamard manifold whose curvature lies between negative constants. The main result of this paper is that $F$ must perform a substantial compression: For every $r>0$ and integer $k\ge 2$ there exist $k$ geodesic balls of radius $r$ in $M^m$ that are arbitrarily far from each other, but whose images under $F$ are bunched together arbitrarily close in the Hausdorff sense of ${\mathbb{R}}^n$. In particular, every isometric embedding ${\mathbb{H}}^m\to {\mathbb{R}}^n$ of hyperbolic space must have a complex asymptotic behavior, regardless of how high the codimension is. Hence, there is no truly simple way to realize ${\mathbb{H}}^m$ isometrically inside any Euclidean space.