flat vector bundle

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

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

In other words, the graded vector space $\backslash Omega\_X^*\; (E)$ 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 $\backslash mathbb\; C\backslash backslash\; \backslash \{0\backslash \},$ with the connection forms 0 and $-\backslash frac\{1\}\{2\}\backslash frac\{dz\}\{z\}$. 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 $\backslash Lambda^\{\backslash mathrm\{top\}\}M$ 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.

Category:Vector bundles