On the hh-principle for Horizontal Immersions in Certain Corank 22 Fat Distributions

In this article we consider a class of fat corank $2$ distribution on a manifold, which includes the holomorphic contact structures. We prove the h-principle for regular horizontal immersion $\Sigma\to (M,\mathcal{D})$ for such a distribution $\mathcal{D}$ on $M$ if $\dim M \ge 4\dim\Sigma + 6$. In particular, we show that $\mathcal{D}$-horizontal maps always exist provided $\dim M \ge \max \{4\dim\Sigma + 6, 5\dim\Sigma-1\}$.Comment: 16 page

Partial Isometries of a Sub-Riemannian Manifold

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 $n\geq m+k$.Comment: 13 pages. This is a revised version of an earlier submission (minor revision