Flat vector bundle

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.

de Rham cohomology of a flat vector bundle

Let $\pi :E\to X$ denote a flat vector bundle, and $\mathrm{\nabla }:\mathrm{\Gamma }(X,E)\to \mathrm{\Gamma }(X,{\mathrm{\Omega }}_X^1\otimes E)$ be the covariant derivative associated to the flat connection on E.

Let ${\mathrm{\Omega }}_X^{*}(E)={\mathrm{\Omega }}_X^{*}\otimes E$ 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 ${\mathrm{\Omega }}_X^{*}(E)$, and the flatness condition is equivalent to the property $d^2=0$.

In other words, the graded vector space ${\mathrm{\Omega }}_X^{*}(E)$ 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 $\mathbb{ℂ}\mathrm{\setminus }\{0\},$ 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 ${\mathrm{\Lambda }}^{\mathrm{t}\mathrm{o}\mathrm{p}}M$ 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.

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.