Flat vector bundle

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.