In mathematics, a vector bundle is said to be flat if it is endowed with an linear connection with vanishing curvature, ie. a flat connection.
Let π : E → X denote a flat vector bundle, and ∇ : Γ ( X , E ) → Γ ( X , Ω X 1 ⊗ E ) be the covariant derivative associated to the flat connection on E.
Let Ω X ∗ ( E ) = Ω X ∗ ⊗ E denote the vector space (in fact a sheaf of modules over O X ) of differential forms on X with values in E. The covariant derivative defines a degree 1 endomorphism d, the differential of Ω X ∗ ( E ) , and the flatness condition is equivalent to the property d 2 = 0 .
In other words, the graded vector space Ω 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 C ∖ { 0 } , with the connection forms 0 and − d z z . The parallel vector fields are constant in the first case, and proportional to local determinations of the complex logarithm in the second.The real canonical line bundle Λ t o p 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.