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 [math]\displaystyle{ \pi:E \to X }[/math] denote a flat vector bundle, and [math]\displaystyle{ \nabla : \Gamma(X, E) \to \Gamma\left(X, \Omega_X^1 \otimes E\right) }[/math] be the covariant derivative associated to the flat connection on E.
Let [math]\displaystyle{ \Omega_X^* (E) = \Omega^*_X \otimes E }[/math] denote the vector space (in fact a sheaf of modules over [math]\displaystyle{ \mathcal O_X }[/math]) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of [math]\displaystyle{ \Omega_X^*(E) }[/math], and the flatness condition is equivalent to the property [math]\displaystyle{ d^2 = 0 }[/math].
In other words, the graded vector space [math]\displaystyle{ \Omega_X^* (E) }[/math] 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 [math]\displaystyle{ \mathbb C\backslash \{0\}, }[/math] with the connection forms 0 and [math]\displaystyle{ -\frac{1}{2}\frac{dz}{z} }[/math]. 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 [math]\displaystyle{ \Lambda^{\mathrm{top}}M }[/math] 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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