Characteristic classes for the deformation of flat connections

Abstract:

In this paper, we study the secondary characteristic classes derived from flat connections. Let M be a differential manifold with flat connection ${\omega _0}$. If f is a diffeomorphism of M, then ${\omega _1} = {f^\ast }{\omega _0}$ is another flat connection. Denote by $\alpha$ the difference of these two connections. Then $\alpha$ and its exterior covariant derivative $D\alpha$ are both tensorial forms on M. To each invariant polynomial $\varphi$ of ${\text {GL}}(n,{\text {R}})$, where $n = \dim M,\varphi (\alpha ;D\alpha )$ is a globally defined form on M. The class $\{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text {R}})$ for $\deg \varphi > 1$ gives rise to an obstruction of the deformability from ${\omega _0}$ to ${\omega _1}$. In particular, we prove that $( + )$ and $( - )$ connections, in the sense of E. Cartan, cannot be deformed to each other.