39 research outputs found
On the -principle for Horizontal Immersions in Certain Corank Fat Distributions
In this article we consider a class of fat corank distribution on a
manifold, which includes the holomorphic contact structures. We prove the
h-principle for regular horizontal immersion for
such a distribution on if . In
particular, we show that -horizontal maps always exist provided
.Comment: 16 page
Partial Isometries of a Sub-Riemannian Manifold
In this paper, we obtain the following generalisation of isometric
-immersion theorem of Nash and Kuiper. Let be a smooth manifold of
dimension and a rank subbundle of the tangent bundle with a
Riemannian metric . Then the pair defines a sub-Riemannian
structure on . We call a -map into a Riemannian
manifold a {\em partial isometry} if the derivative map restricted
to is isometric; in other words, . The main result states that
if then a smooth -immersion satisfying
can be homotoped to a partial isometry which is
-close to . In particular we prove that every sub-Riemannian manifold
admits a partial isometry in provided .Comment: 13 pages. This is a revised version of an earlier submission (minor
revision