Parallelization (mathematics)

Given a manifold $M\backslash ,$ of dimension n, a parallelization of $M\backslash ,$ is a set $\backslash \{X\_1,\; \backslash dots,X\_n\backslash \}$ of n smooth vector fields defined on all of $M\backslash ,$ such that for every $p\backslash in\; M\backslash ,$ the set $\backslash \{X\_1(p),\; \backslash dots,X\_n(p)\backslash \}$ is a basis of $T\_pM\backslash ,$, where $T\_pM\backslash ,$ denotes the fiber over $p\backslash ,$ of the tangent vector bundle $TM\backslash ,$.

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

In other words, $M\backslash ,$ is parallelizable if and only if $\backslash tau\_\{M\}\backslash colon\; TM\; \backslash longrightarrow\; M\backslash ,$ is a trivial bundle. For example, suppose that $M\backslash ,$ is an open subset of $\{\backslash mathbb\; R^n\}\backslash ,$, i.e., an open submanifold of $\{\backslash mathbb\; R^n\}\backslash ,$. Then $TM\backslash ,$ is equal to $M\backslash times\; \{\backslash mathbb\; R^n\}\backslash ,$, and $M\backslash ,$ is clearly parallelizable.{{harvtxt|Milnor|Stasheff|1974}}, p. 15.

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

{{Reflist}}

• {{citation|last1=Bishop|first1=R.L.|author1-link=Richard L. Bishop|last2=Goldberg|first2=S.I.|title=Tensor Analysis on Manifolds|publisher=The Macmillan Company|year=1968|edition=First Dover 1980|isbn=0-486-64039-6|url-access=registration|url=https://archive.org/details/tensoranalysison00bish}}
• {{citation | last1=Milnor|first1=J.W.|last2=Stasheff|first2=J.D.|author2-link=Jim Stasheff | title = Characteristic Classes| publisher=Princeton University Press | year=1974}}

Category:Differential geometry

Category:Fiber bundles

Category:Vector bundles