• Images
• Videos
• Documents
• Books
• Articles

Formal definition

Given a manifold M {\displaystyle M\,} of dimension n, a parallelization of M {\displaystyle M\,} is a set { X 1 , … , X n } {\displaystyle \{X_{1},\dots ,X_{n}\}} of n smooth vector fields defined on all of M {\displaystyle M\,} such that for every p ∈ M {\displaystyle p\in M\,} the set { X 1 ( p ) , … , X n ( p ) } {\displaystyle \{X_{1}(p),\dots ,X_{n}(p)\}} is a basis of T p M {\displaystyle T_{p}M\,} , where T p M {\displaystyle T_{p}M\,} denotes the fiber over p {\displaystyle p\,} of the tangent vector bundle T M {\displaystyle 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 {\displaystyle M\,} is parallelizable iff there is a diffeomorphism ϕ : T M ⟶ M × R n {\displaystyle \phi \colon TM\longrightarrow M\times {\mathbb {R} ^{n}}\,} such that the first projection of ϕ {\displaystyle \phi \,} is τ M : T M ⟶ M {\displaystyle \tau _{M}\colon TM\longrightarrow M\,} and for each p ∈ M {\displaystyle p\in M\,} the second factor—restricted to T p M {\displaystyle T_{p}M\,} —is a linear map ϕ p : T p M → R n {\displaystyle \phi _{p}\colon T_{p}M\rightarrow {\mathbb {R} ^{n}}\,} .

In other words, M {\displaystyle M\,} is parallelizable if and only if τ M : T M ⟶ M {\displaystyle \tau _{M}\colon TM\longrightarrow M\,} is a trivial bundle. For example, suppose that M {\displaystyle M\,} is an open subset of R n {\displaystyle {\mathbb {R} ^{n}}\,} , i.e., an open submanifold of R n {\displaystyle {\mathbb {R} ^{n}}\,} . Then T M {\displaystyle TM\,} is equal to M × R n {\displaystyle M\times {\mathbb {R} ^{n}}\,} , and M {\displaystyle 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

• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
• Milnor, J.W.; Stasheff, J.D. (1974), Characteristic Classes, Princeton University Press

References

1. Bishop & Goldberg (1968), p. 160 - Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6 https://archive.org/details/tensoranalysison00bish ↩

2. Milnor & Stasheff (1974), p. 15. - Milnor, J.W.; Stasheff, J.D. (1974), Characteristic Classes, Princeton University Press ↩