Parallelization (mathematics) explained

M of dimension n is a set of n global smooth linearly independent vector fields.

Formal definition

Given a manifold

M of dimension n, a parallelization of M is a set \{X _{1,} ... ,X _{n\}} of n smooth vector fields defined on all of M such that for every p\in M the set \{X _{1(p),} ... ,X _{n(p)\}} is a basis of T _{pM} , where T _{pM} denotes the fiber over p of the tangent vector bundle TM .

A manifold is called parallelizable whenever it admits a parallelization.

Examples

• Every Lie group is a parallelizable manifold.
• The product of parallelizable manifolds is parallelizable.
• Every affine space, considered as manifold, is parallelizable.

Properties

Proposition. A manifold

M is parallelizable iff there is a diffeomorphism \phi \colon TM \longrightarrow M x { R ^{n}} such that the first projection of \phi is \tau _{M} \colon TM \longrightarrow M and for each p\in M the second factor—restricted to T _{pM} —is a linear map \phi _{p} \colon T _{pM} → { R ^{n}} .

In other words,

M is parallelizable if and only if \tau _{M} \colon TM \longrightarrow 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 TM is equal to M x { R ^{n}} , and M is clearly parallelizable.^[2]

See also

• Chart (topology)
• Differentiable manifold
• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Connection (mathematics)
• G-structure
• Web (differential geometry)

Notes and References

1. , p. 160
2. , p. 15.