In mathematics, a vector bundle is said to be flat if it is endowed with an linear connection with vanishing curvature, ie. a flat connection.
Let
π
:
E
→
X
denote a flat vector bundle, and
∇
:
Γ
(
X
,
E
)
→
Γ
(
X
,
Ω
X
1
⊗
E
)
be the covariant derivative associated to the flat connection on E.
Let
Ω
X
∗
(
E
)
=
Ω
X
∗
⊗
E
denote the vector space (in fact a sheaf of modules over
O
X
) of differential forms on X with values in E. The covariant derivative defines a degree 1 endomorphism d, the differential of
Ω
X
∗
(
E
)
, and the flatness condition is equivalent to the property
d
2
=
0
.
In other words, the graded vector space
Ω
X
∗
(
E
)
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.
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.
Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over
C
∖
{
0
}
,
with the connection forms 0 and
−
d
z
z
. The parallel vector fields are constant in the first case, and proportional to local determinations of the complex logarithm in the second.
The real canonical line bundle
Λ
t
o
p
M
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.
