In mathematics, a parallelization of a manifold
M
of dimension n is a set of n global linearly independent vector fields.
Given a manifold
M
of dimension n, a parallelization of
M
is a set
{
X
1
,
…
,
X
n
}
of n vector fields defined on all of
M
such that for every
p
∈
M
the set
{
X
1
(
p
)
,
…
,
X
n
(
p
)
}
is a basis of
T
p
M
, where
T
p
M
denotes the fiber over
p
of the tangent vector bundle
T
M
.
A manifold is called parallelizable whenever admits a parallelization.
Every Lie group is a parallelizable manifold.
The product of parallelizable manifolds is parallelizable.
Every affine space, considered as manifold, is parallelizable.
Proposition. A manifold
M
is parallelizable iff there is a diffeomorphism
ϕ
:
T
M
⟶
M
×
R
n
such that the first projection of
ϕ
is
τ
M
:
T
M
⟶
M
and for each
p
∈
M
the second factor—restricted to
T
p
M
—is a linear map
ϕ
p
:
T
p
M
→
R
n
.
In other words,
M
is parallelizable if and only if
τ
M
:
T
M
⟶
M
is a trivial bundle. For example, suppose that
M
is an open subset of
R
n
, i.e., an open submanifold of
R
n
. Then
T
M
is equal to
M
×
R
n
, and
M
is clearly parallelizable.