Given a manifold ${\displaystyle M\,}$  of dimension n, a parallelization of ${\displaystyle M\,}$  is a set ${\displaystyle \{X_{1},\dots ,X_{n}\}}$  of n smooth 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 ${\displaystyle TM\,}$ .

A manifold is called parallelizable whenever it admits a parallelization.

• Every Lie group is a parallelizable manifold.
• The product of parallelizable manifolds is parallelizable.
• Every affine space, considered as manifold, is parallelizable.

Proposition. A manifold ${\displaystyle M\,}$  is parallelizable iff there is a diffeomorphism ${\displaystyle \phi \colon TM\longrightarrow M\times {\mathbb {R} ^{n}}\,}$  such that the first projection of ${\displaystyle \phi \,}$  is ${\displaystyle \tau _{M}\colon TM\longrightarrow M\,}$  and for each ${\displaystyle p\in M\,}$  the second factor—restricted to ${\displaystyle T_{p}M\,}$ —is a linear map ${\displaystyle \phi _{p}\colon T_{p}M\rightarrow {\mathbb {R} ^{n}}\,}$ .

In other words, ${\displaystyle M\,}$  is parallelizable if and only if ${\displaystyle \tau _{M}\colon TM\longrightarrow M\,}$  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.^[2]

• Chart (topology)
• Differentiable manifold
• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Connection (mathematics)
• G-structure
• Web (differential geometry)

• 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