Parallelization (mathematics)

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In mathematics, a parallelization[1] of a manifold of dimension n is a set of n global smooth linearly independent vector fields.

Formal definition[edit]

Given a manifold of dimension n, a parallelization of is a set of n smooth vector fields defined on all of such that for every the set is a basis of , where denotes the fiber over of the tangent vector bundle .

A manifold is called parallelizable whenever it admits a parallelization.



Proposition. A manifold is parallelizable iff there is a diffeomorphism such that the first projection of is and for each the second factor—restricted to —is a linear map .

In other words, is parallelizable if and only if is a trivial bundle. For example, suppose that is an open subset of , i.e., an open submanifold of . Then is equal to , and is clearly parallelizable.[2]

See also[edit]



  • 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