Parallelization (mathematics)

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

In other words, ${\displaystyle M}$ is parallelizable if and only if ${\displaystyle {\tau }_M:TM⟶M}$ 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.