novichyok (novichyok) wrote in geometry_qa,
novichyok
novichyok
geometry_qa

Flat connection vs. representation of the fundamental group

Let M be a closed manifold. A complex representation α: π1(M) → Cn of its fundamental group  gives rise to a flat vector bundle Eα = M×αC whose monodromy is isomorphic to α. Given a continuous family of connections α(t) the bundles Eα(t) are isomorphic (as vector bundles without a connection). Thus it is not difficult to see that there exists one bundle E → M and a continuous family of connections (t) such that the monodromy of  (t) is isomorphic to α(t). Moreover if α(t) is differentiable then (t) can be chosen to be differentiable. I have 2 questions: </p>

    1. Is there any book or paper where these simple facts are proven?

    2. How far this relationship between families of representations and families of connections can be pushed? For example, if α(t) is analytic can one find an analytic family (t)?

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