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 $\phi_{p} \colon T_pM \rightarrow {\mathbb R^n}\,$.

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]

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