  Consider a parallel plane foliation on real finite-dimensional linear vector
space. It induces a foliation on the torus obtained by factorization of the
space by the integer lattice (let us denote the latter foliation by F). Let g
be arbitrary metric on the torus. It induces a complex structure on each leaf
of F such that all the leaves are parabolic: each individual leaf admits a
conformal flat complete metric. We show that for any foliation F as above and
arbitrary smooth metric g the correspondent complete conformal flat metric on
the leaves, which is smooth on individual leaf, can be chosen to be smooth in
the whole torus (in particular, in transversal parameter). We show that under
appropriate Diophantine condition on the foliation for arbitrary smooth metric
g there exists a smooth Euclidean metric on the torus conformal on the leaves
such that the latters are totally geodesic; in other terms, the correspondent
triple consisting of the torus, the foliation and the complex structure on the
leaves is isomorphic to a triple correspondent to a linear foliation on another
torus equipped with the standard complex structure on the leaves. This
Diophantine condition is exact. We state and prove the analytic versions of
these statements.
