Flat vector bundle connection

Let $E\to M$ be a vector bundle with connection $\nabla$.

Definition. The connection $\nabla$ is flat if its curvature vanishes identically:

Equivalently, in any local frame the curvature 2-form matrix is zero.

Flatness has two standard geometric consequences:

• On sufficiently small contractible open sets, there exist local frames of $\nabla$-parallel sections (frames $(e_i)$ with $\nabla e_i=0$), so locally the connection looks like the trivial connection in a suitable gauge.
• The associated parallel transport along curves depends only on the homotopy class of the curve with fixed endpoints; loops therefore determine a representation of the fundamental group into the structure group, and the image is captured by the holonomy group .

Viewed on the frame bundle, flatness corresponds to integrability of the induced horizontal distribution (compare integrable horizontal distributions ).

Examples

1. Trivial bundle with the trivial connection. On $M\times\mathbb R^r$, the connection $\nabla=d$ has $R^\nabla=0$, so it is flat.
2. Local systems from representations. Given a representation $\rho:\pi_1(M)\to \mathrm{GL}(r,\mathbb R)$, one can form the associated flat vector bundle (a “local system”) whose parallel transport along loops realizes $\rho$.
3. Flat but with nontrivial holonomy on the circle. On $S^1$, let $E=S^1\times\mathbb R^r$ and define $\nabla = d + A\,d\theta$ with a constant matrix $A$. The curvature is zero (since $d\theta$ is closed and $A$ is constant), but parallel transport around the circle gives holonomy $\exp(2\pi A)$, which can be nontrivial.