Flat vector bundle

In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection.

de Rham cohomology of a flat vector bundle
Let $$\pi:E \to X$$ denote a flat vector bundle, and $$\nabla : \Gamma(X, E) \to \Gamma\left(X, \Omega_X^1 \otimes E\right)$$ be the covariant derivative associated to the flat connection on E.

Let $$\Omega_X^* (E) = \Omega^*_X \otimes E$$ denote the vector space (in fact a sheaf of modules over $$\mathcal O_X$$) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of $$\Omega_X^*(E)$$, and the flatness condition is equivalent to the property $$d^2 = 0$$.

In other words, the graded vector space $$\Omega_X^* (E)$$ is a cochain complex. Its cohomology is called the de Rham cohomology of E, or de Rham cohomology with coefficients twisted by the local coefficient system E.

Flat trivializations
A trivialization of a flat vector bundle is said to be flat if the connection form vanishes in this trivialization. An equivalent definition of a flat bundle is the choice of a trivializing atlas with locally constant transition maps.

Examples

 * Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over $$\mathbb C\backslash \{0\},$$ with the connection forms 0 and $$-\frac{1}{2}\frac{dz}{z}$$. The parallel vector fields are constant in the first case, and proportional to local determinations of the square root in the second.
 * The real canonical line bundle $$\Lambda^{\mathrm{top}}M$$ of a differential manifold M is a flat line bundle, called the orientation bundle. Its sections are volume forms.
 * A Riemannian manifold is flat if and only if its Levi-Civita connection gives its tangent vector bundle a flat structure.