Parallelization (mathematics)

In mathematics, a parallelization^[1] of a manifold $M\,$ of dimension n is a set of n global 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 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 $T_pM\,$ , where $T_pM\,$ denotes the fiber over $p\,$ of the tangent vector bundle $TM\,$ .

A manifold is called parallelizable whenever 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 $T_pM\,$ —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.^[2]

Notes

↑ Bishop & Goldberg (1968), p. 160
↑ 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

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