Flat connection

A flat connection has vanishing curvature. On a simply connected patch it is gauge equivalent to the zero connection, because the parallel-transport equation has path-independent solutions.

For $D_i={\partial }_i+A_i$, compatibility for arbitrary initial vectors is equivalent to zero gauge curvature. An invertible parallel bundle frame $S$ obeys $D_{i}S=0$, so $[D_i,D_j]S=F_{ij}S=0$ forces $F_{ij}=0$. Conversely, solve the two ordinary transport equations along one coordinate axis and then along parallel lines. The unused transport residual solves a homogeneous transport equation with zero initial value when the gauge curvature is zero; it therefore vanishes. Thus $A_i=-({\partial }_{i}S)S^{-1}=g^{-1}{\partial }_{i}g$ with $g=S^{-1}$. One particular parallel vector requires only $F_{ij}v=0$, a weaker condition.

The holonomy of a flat connection is invariant under based homotopy of loops, giving a group homomorphism from the fundamental group into the structure group after choosing a frame at the base point. Changing that frame conjugates the representation. Its associated bundle is constructed from the universal cover and this monodromy. It differs from the natural representation of a Riemannian holonomy group on a tangent fiber.

For a connected base and a chosen frame at one point, a flat connection gives a representation of the fundamental group by parallel transport. Conversely, $(\tilde{X}\times G)/{\pi }_1(X)$ with monodromy representation has a flat product connection. Based gauge classes correspond to representations, and unbased classes to conjugacy classes. On a fixed underlying bundle only representations whose associated flat bundle has that bundle type are allowed.

// Destination: geometry-and-topology.bigb