Parallelization (mathematics)

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

Formal definition

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

• 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 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.

See also

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

Notes

References