  We show that the family of $m$-dimensional isotropic projections in $\R^{2n}$
is transversal. As an application we show that the Besicovitch-Federer
projection theorem holds for isotropic projections. We also use transversality
to obtain almost sure estimates on the Hausdorff dimension of isotropic
projections of subsets $E \subset \R^{2n}$. These results may also be applied
to gain information on the horizontal projections of the Heisenberg group
$\H^n$.
