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 ω 0 {\omega _0} . If f is a diffeomorphism of M, then ω 1 = f ∗ ω 0 {\omega _1} = {f^\ast }{\omega _0} is another flat connection. Denote by α \alpha the difference of these two connections. Then α \alpha and its exterior covariant derivative D α D\alpha are both tensorial forms on M. To each invariant polynomial φ \varphi of GL ( n , R ) {\text {GL}}(n,{\text {R}}) , where n = dim ⁡ M , φ ( α ; D α ) n = \dim M,\varphi (\alpha ;D\alpha ) is a globally defined form on M. The class { φ ( α ; D α ) } ∈ H ( M ; R ) \{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text {R}}) for deg ⁡ φ > 1 \deg \varphi > 1 gives rise to an obstruction of the deformability from ω 0 {\omega _0} to ω 1 {\omega _1} . In particular, we prove that ( + ) ( + ) and ( − ) ( - ) connections, in the sense of E. Cartan, cannot be deformed to each other.