  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 $n\geq m+k$.
