In mathematics, a parallelization of a manifold M {\displaystyle M\,} of dimension n is a set of n global smooth linearly independent vector fields.
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.
See also Chart (topology) Differentiable manifold Frame bundle Orthonormal frame bundle Principal bundle Connection (mathematics) G-structure Web (differential geometry)
Notes
References Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6 Milnor, John W.; Stasheff, James D. (1974). Characteristic Classes. Annals of Mathematics Studies. Vol. 76. Princeton University Press; University of Tokyo Press. ISBN 978-0-691-08122-9. MR 0440554.
