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.

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

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

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

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 ${\displaystyle \mathbb{ℂ}\mathrm{\setminus }\{0\},}$ 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 ${\displaystyle {\mathrm{\Lambda }}^{\mathrm{t}\mathrm{o}\mathrm{p}}M}$ 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.

• 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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