# Flat vector bundle

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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.

## de Rham cohomology of a flat vector bundle

Let denote a flat vector bundle, and be the covariant derivative associated to the **flat connection** on E.

Let denote the vector space (in fact a sheaf of modules over ) of differential forms on *X* with values in *E*. The covariant derivative defines a degree 1 endomorphism *d*, the **differential** of , and the flatness condition is equivalent to the property .

In other words, the graded vector space 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 with the connection forms 0 and . 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 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.

## See also

- 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.