Flat vector bundle

Let ${\displaystyle \pi :E\to X}$ denote a flat vector bundle, and ${\displaystyle \nabla$ :\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 ${\displaystyle \Omega _{X}^{*}(E)=\Omega _{X}^{*}\otimes E}$ denote the vector space (in fact a sheaf of modules over ${\displaystyle {\mathcal {O}}_{X}}$) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of ${\displaystyle \Omega _{X}^{*}(E)}$, and the flatness condition is equivalent to the property ${\displaystyle d^{2}=0}$.

In other words, the graded vector space ${\displaystyle \Omega _{X}^{*}(E)}$ 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.

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.

• Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over ${\displaystyle \mathbb {C} \backslash \{0\},}$ with the connection forms 0 and ${\displaystyle -{\frac {1}{2}}{\frac {dz}{z}}}$. 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 ${\displaystyle \Lambda ^{\mathrm {top} }M}$ 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.

• Vector-valued differential forms
• Local system, the more general notion of a locally constant sheaf.
• Orientation character, a characteristic form related to the orientation line bundle, useful to formulate Twisted Poincaré duality
• Picard group whose connected component, the Jacobian variety, is the moduli space of algebraic flat line bundles.
• Monodromy, or representations of the fundamental group by parallel transport on flat bundles.
• Holonomy, the obstruction to flatness.