# nLab flat connection

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Definition

A connectiojn 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 refereed to as local systems

## Properties

### 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 Riemann surface: 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.

Revised on May 10, 2013 18:46:52 by Urs Schreiber (82.169.65.155)