Flat vector bundle connection - Imported Knowlpedia Content

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

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

R^\nabla = 0.

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).

Equivalent characterizations

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

Examples

Trivial bundle with the trivial connection. On $M\times\mathbb R^r$ , the connection $\nabla=d$ has $R^\nabla=0$ , so it is flat.
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 $\rho$ .
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.