Flat vector bundle

In mathematics, flat bundle, among others, in differential geometry and mathematical physics occur.

Definition

A flat bundle is a principal beam which has a flat connection.

A connection is called flat if its curvature vanishes, ie if

Geometric interpretation

By the theorem of Ambrose -Singer the curvature measures the infinitesimal holonomy. For a principal bundle with a flat connection, the holonomy must be so infinitesimal (but not necessarily global) to be trivial, ie homotopic paths have the same holonomy. In particular, the holonomy induces a well-defined representation of the fundamental group of the base into the structure group of the principal bundle.

Holonomy representation

Flat G- bundles over a connected manifold are in bijection with representations

The associate to a representation of flat bundles are obtained - with the help of the effect of on the universal covering - as

With the equivalence relation for.

Sections correspond clearly in the pictures - equivariant, the corresponding section is to be projected on a (any ).