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
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.