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

TL;DR

This paper demonstrates that isometric embeddings of hyperbolic spaces into Euclidean spaces inherently exhibit complex asymptotic behavior, making simple embeddings impossible regardless of the codimension.

Contribution

It proves that isometric embeddings of hyperbolic spaces into Euclidean spaces must have complex asymptotic properties, highlighting the inherent geometric complexity.

Findings

Any Lipschitz map from a negatively curved Hadamard manifold compresses distant regions.

Isometric embeddings of hyperbolic space into Euclidean space must have complex asymptotic behavior.

No simple isometric embedding of hyperbolic space into Euclidean space exists regardless of 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 $k\geq 2$ 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.