Parallelization (mathematics)

In mathematics, a parallelization[1] of a manifold ${\displaystyle M\,}$ of dimension n is a set of n global linearly independent vector fields.

Formal definition

Given a manifold ${\displaystyle M\,}$ of dimension n, a parallelization of ${\displaystyle M\,}$ is a set ${\displaystyle \{X_{1},\dots ,X_{n}\}}$ of n vector fields defined on all of ${\displaystyle M\,}$ such that for every ${\displaystyle p\in M\,}$ the set ${\displaystyle \{X_{1}(p),\dots ,X_{n}(p)\}}$ is a basis of ${\displaystyle T_{p}M\,}$, where ${\displaystyle T_{p}M\,}$ denotes the fiber over ${\displaystyle p\,}$ of the tangent vector bundle ${\displaystyle TM\,}$.

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 ${\displaystyle M\,}$ is parallelizable iff there is a diffeomorphism ${\displaystyle \phi \colon TM\longrightarrow M\times {\mathbb {R} ^{n}}\,}$ such that the first projection of ${\displaystyle \phi \,}$ is ${\displaystyle \tau _{M}\colon TM\longrightarrow M\,}$ and for each ${\displaystyle p\in M\,}$ the second factor—restricted to ${\displaystyle T_{p}M\,}$—is a linear map ${\displaystyle \phi _{p}\colon T_{p}M\rightarrow {\mathbb {R} ^{n}}\,}$.

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

1. Bishop & Goldberg (1968), p. 160
2. Milnor & Stasheff (1974), p. 15.

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, J.W.; Stasheff, J.D. (1974), Characteristic Classes, Princeton University Press