  We show that on a closed smooth manifold $M$ equipped with $k$ fiber bundle
structures whose vertical distributions span the tangent bundle, every smooth
diffeomorphism $f$ of $M$ sufficiently close to the identity can be written as
a product $f=f_1... f_k$, where $f_i$ preserves the $i^{\text{th}}$-fiber. The
factors $f_i$ can be chosen smoothly in $f$. We apply this result to show that
on a certain class of closed smooth manifolds every diffeomorphism sufficiently
close to the identity can be written as product of commutators and the factors
can be chosen smoothly. Furthermore we get concrete estimates on how many
commutators are necessary.
