Partial Isometries of a Sub-Riemannian Manifold

TL;DR

This paper generalizes the Nash-Kuiper $C^1$-immersion theorem to sub-Riemannian manifolds, showing that under certain conditions, maps can be homotoped to partial isometries that preserve the sub-Riemannian structure.

Contribution

It extends the isometric immersion theory to sub-Riemannian manifolds, introducing the concept of partial isometries and proving their existence under dimension constraints.

Findings

Every sub-Riemannian manifold admits a partial isometry into Euclidean space if the dimension condition is met.

Homotopies to partial isometries can be made arbitrarily close to initial maps.

The result generalizes classical isometric embedding theorems to the sub-Riemannian setting.

Abstract

In this paper, we obtain the following generalisation of isometric $C^1$-immersion theorem of Nash and Kuiper. Let $M$ be a smooth manifold of dimension $m$ and $H$ a rank $k$ subbundle of the tangent bundle $TM$ with a Riemannian metric $g_H$. Then the pair $(H,g_H)$ defines a sub-Riemannian structure on $M$. We call a $C^1$-map $f:(M,H,g_H)\to (N,h)$ into a Riemannian manifold $(N,h)$ a {\em partial isometry} if the derivative map $df$ restricted to $H$ is isometric; in other words, $f^*h|_H=g_H$. The main result states that if $\dim N>k$ then a smooth $H$-immersion $f_0:M\to N$ satisfying $f^*h|_H<g_H$ can be homotoped to a partial isometry $f:(M,g_H)\to (N,h)$ which is $C^0$-close to $f_0$. In particular we prove that every sub-Riemannian manifold $(M,H,g_H)$ admits a partial isometry in $\R^n$ provided $n\geq m+k$.