We present short direct proofs of two known properties of complete flat manifolds. They say that the diffeomorphism classes of m-dimensional complete flat manifolds form a finite set $S_{CF}\left(m\right)$ and that each element of $S_{CF}\left(m\right)$ is represented by a manifold with finite holonomy group.

This paper contains a description of various geometric constructions associated with fibre bundles, given in terms of important algebraic object, the “twisting cochain". Our examples include the Chern-Weil classes, the holonomy representation and the so-called cyclic Chern character of Bismut and others (see [2, 11, 27]), also called the Bismut’s class. The later example is the principal one for us, since we are motivated by the attempt to find an algebraic approach to the Witten’s index formula....