nLab flat connection

Definition

A connection on a fibre bundle is flat if its curvature is zero.

The same definition of flatness holds for connections in various algebraic setups and for connections on quasicoherent sheaves.

The condition of flatness is usually expressed via the Maurer-Cartan equation. Flat connections on bundles are also referred to as local systems (by the Riemann-Hilbert correspondence discussed below).

Properties

Riemann-Hilbert correspondence

That flat connections are equivalently representations of the fundamental groupoid of the base space/local systems is known as the Riemann-Hilbert correspondence.

Flatness as integrability

In geometry one says instead of flat connection, integrable connection. The reason is roughly the following: in the theory of systems of differential equations the flatness of the corresponding connection is the condition of the integrability of the system.

(…elaborate on this with equations)

The condition of flatness is usually expressed via the Maurer-Cartan equation, which is in integrable systems theory often called zero curvature equation. For example, the Lax equations can always be written in the form of the zero curvature equation.

Over a complex manifold

Over a complex manifold/complex variety the Koszul-Malgrange theorem identifies holomorphic flat connections on complex vector bundles with holomorphic vector bundles. See there for more

Over a Riemann surface

Existence

In (Milnor) it is shown that a vector bundle over a surface of genus $g$ admits flat connections iff its Euler class is less than $g$ by an absolute value (see also Wood, Bundles with totally disconnected structure group). Sullivan gives a refinement.

Narasimhan–Seshadri theorem

The Narasimhan-Seshadri theorem identifies moduli spaces of flat connections over a Riemann surface with that of certain stable vector bundles.

Isometric embeddings

Maurer-Cartan equation is called also structure equation when used to treat the conditions for isometric embeddings of Riemannian submanifolds in an Euclidean space.

…

Related concepts

• local system, inner local system

• projectively flat connection

• moduli space of flat connections, Narasimhan-Seshadri theorem

• flat ∞-connection

• flat section

References

On the equivalence between flat connections (local systems) with group homomorphisms out of the fundamental group (over each connected component):

• Pierre Deligne, §I.1 of: Equations différentielles à points singuliers réguliers, Lecture Notes Math. 163, Springer (1970) $[$publications.ias:355$]$

and generally with functors out of the fundamental groupoid:

• Alexandru Dimca, Prop. 2.5.1 of: Sheaves in Topology, Universitext, Springer (2004) [doi:10.1007/978-3-642-18868-8]

• Urs Schreiber, Konrad Waldorf, p. 194 of: Parallel Transport and Functors, J. Homotopy Relat. Struct. 4 (2009) 187-244 [arXiv:0705.0452, pdf]

• Florin Dumitrescu, Cor. 3 of: Connections and Parallel Transport, Journal of Homotopy and Related Structures 5 1 (2010) 171–175 [arXiv:0903.0121, pdf]

The latter articles also discuss the generalization of this statement to non-flat connections ,for more see the references at parallel transport here.

See also

• John Milnor, On the existence of a connection with curvature zero, Comm. Math. Helv. 32 (1957/58) 215-223 [dml:139154]

• Dennis Sullivan, A generalization of Milnor’s inequality Comm. Math. Helv. v. 51

• Hélène Esnault, Algebraic Differential Characters of Flat Connections with Nilpotent Residues, in Nils Baas et al., The Abel Symposium 2007 (pdf)