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 $$T_pM\,$$, where $$T_pM\,$$ 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 $$T_pM\,$$—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.