# Parallelization (mathematics)

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**.

## Examples[edit]

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

## Properties[edit]

**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]

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

## Notes[edit]

**^**Bishop & Goldberg (1968), p. 160**^**Milnor & Stasheff (1974), p. 15.

## References[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