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In this note, we extend the theory of Chern-Cheeger-Simons to constructcanonical invariants for a one-parameter family of flat connections on a smoothmanifold. These invariants lie in degrees $(2p-2)$-cohomology with$\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in avariation of paths (parametrising flat connections), in degrees at least three.

Tertiary classes for a one-parameter variation of flat connections on a smooth manifold