Monodromy

  • An Ehresmann connection on a fiber bundle E is the choice of a complementary horizontal subbundle which can be defined by some  Riemannian metric on the total space.
  • Picard-Lindelöf theorem

How parallel transport relates to connection.

Flat connection implies independence of the parallel transport related to a homotopy euquivalence of loop.

  • Integrability of connection guarantees the action of Der on endmorphism of local system is a Lie-algebra homomorphism.
  • Principal bundle connection:
  • Framed character variety (will typically not be smooth) and the (ordinary) character variety is defined as the GIT quotient1 of the framed character variety by the G-action by conjugation.
  • On the unit ball, all integrable connections are G-gauge equivalent. The general principle is that the curvature is the only local obstruction to a connection being trivial. So locally de Rham moduli space is just a point.