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 $\varphi :TM⟶M\times {\mathbb{ℝ}}^n$ such that the first projection of $\varphi$ is ${\tau }_M:TM⟶M$ and for each $p\in M$ the second factor—restricted to $T_{p}M$—is a linear map ${\varphi }_p:T_{p}M\to {\mathbb{ℝ}}^n$.

In other words, $M$ is parallelizable if and only if ${\tau }_M:TM⟶M$ 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, https://archive.org/details/tensoranalysison00bish 
• Milnor, J.W.; Stasheff, J.D. (1974), Characteristic Classes, Princeton University Press