Parallelization (mathematics)

In mathematics, a parallelization of a manifold ${\displaystyle M\,}$ of dimension n is a set of n global linearly independent vector fields.

Given a manifold ${\displaystyle M\,}$ of dimension n, a parallelization of ${\displaystyle M\,}$ is a set ${\displaystyle \{X_{1},\dots ,X_{n}\}}$ of n vector fields defined on all of ${\displaystyle M\,}$ such that for every ${\displaystyle p\in M\,}$ the set ${\displaystyle \{X_{1}(p),\dots ,X_{n}(p)\}}$ is a basis of ${\displaystyle T_{p}M\,}$, where ${\displaystyle T_{p}M\,}$ denotes the fiber over ${\displaystyle p\,}$ of the tangent vector bundle ${\displaystyle TM\,}$.

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