Flat vector bundle

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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.

## Contents

* 1 de Rham cohomology of a flat vector bundle
* 2 Flat trivializations
* 3 Examples
* 4 See also

## de Rham cohomology of a flat vector bundle

Let π:E->X denote a flat vector bundle, and ∇:Γ(X,E)->Γ(X,ΩX1⊗E) be the covariant derivative associated to the flat connection on E.

Let ΩX*(E)=ΩX*⊗E denote the vector space (in fact a sheaf of modules over 𝒪X) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of ΩX*(E), and the flatness condition is equivalent to the property d2=0.

In other words, the graded vector space ΩX*(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 ℂ∖{0}, with the connection forms 0 and −12dzz. 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 ΛtopM 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.

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Original source: https://en.wikipedia.org/wiki/Flat vector bundle. Read more

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