Parallelization (mathematics)

In mathematics, a parallelization of a manifold $M\,$ 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 $\{X_{1},\dots ,X_{n}\}$ of n smooth vector fields defined on all of $M\,$ such that for every $p\in M\,$ the set $\{X_{1}(p),\dots ,X_{n}(p)\}$ is a basis of $T_{p}M\,$, where $T_{p}M\,$ 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 $\phi \colon TM\longrightarrow M\times {\mathbb {R} ^{n}}\,$ 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 $T_{p}M\,$—is a linear map $\phi _{p}\colon T_{p}M\rightarrow {\mathbb {R} ^{n}}\,$.

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 ${\mathbb {R} ^{n}}\,$, i.e., an open submanifold of ${\mathbb {R} ^{n}}\,$. Then $TM\,$ is equal to $M\times {\mathbb {R} ^{n}}\,$, and $M\,$ is clearly parallelizable.

See also

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

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.