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Parallelization (mathematics)

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In mathematics, a parallelization of a manifold M of dimension n is a set of n global linearly independent vector fields.

Contents

Formal definition

Given a manifold M of dimension n, a parallelization of M is a set { X 1 , , X n } of n vector fields defined on all of M such that for every p M the set { X 1 ( p ) , , X n ( p ) } is a basis of T p M , where T p M denotes the fiber over p of the tangent vector bundle T M .

A manifold is called parallelizable whenever 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 ϕ : T M M × R n such that the first projection of ϕ is τ M : T M M and for each p M the second factor—restricted to T p M —is a linear map ϕ p : T p M R n .

    In other words, M is parallelizable if and only if τ M : T M M is a trivial bundle. For example, suppose that M is an open subset of R n , i.e., an open submanifold of R n . Then T M is equal to M × R n , and M is clearly parallelizable.

    References

    Parallelization (mathematics) Wikipedia