  A general method for the construction of smooth flat connections on
3-manifolds is introduced. The procedure is strictly connected with the
deduction of the fundamental group of a manifold M by means of a Heegaard
splitting presentation of M. For any given matrix representation of the
fundamental group of M, a corresponding flat connection A on M is specified. It
is shown that the associated classical Chern-Simons invariant assumes then a
canonical form which is given by the sum of two contributions: the first term
is determined by the intersections of the curves in the Heegaard diagram, and
the second term is the volume of a region in the representation group which is
determined by the representation of pi_1(M) and by the Heegaard gluing
homeomorphism. Examples of flat connections in topologically nontrivial
manifolds are presented and the computations of the associated classical
Chern-Simons invariants are illustrated.
