Flat vector bundle explained
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
denote a flat vector bundle, and
\nabla:\Gamma(X,E)\to\Gamma\left(X,
⊗ E\right)
be the
covariant derivative associated to the
flat connection on E.
Let
denote the
vector space (in fact a
sheaf of
modules over
) of
differential forms on
X with values in
E. The covariant derivative defines a degree-1
endomorphism d, the
differential of
, and the flatness condition is equivalent to the property
.
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
with the
connection forms 0 and
. The parallel vector fields are constant in the first case, and proportional to local determinations of the square root in the second.
of a
differential manifold M is a flat line bundle, called the
orientation bundle. Its sections are
volume forms.
See also