Partial Isometries of a Sub-Riemannian Manifold

Details Partial Isometries of a Sub-Riemannian Manifold Partial Isometries of a Sub-Riemannian Manifold

2010-09-27

arxiv.org/abs/1009.5221v3

Mathematics

Differential Geometry

13 pages. This is a revised version of an earlier submission (minor revision)

Scientific paper

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

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