Parallelization (mathematics)

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

Formal definition

Given a manifold $M\,$ of dimension n, a parallelization of $M\,$ is a set $\{X_{1},\dots ,X_{n}\}$ of n vector fields defined on all of $M\,$ such that for every $p\in M\,$ the set $\{X_{1}(p),\dots ,X_{n}(p)\}$ is a basis (mathematics) of $T_{p}M\,$, where $T_{p}M\,$ denotes the fiber over $p\,$ of the tangent vector bundle $TM\,$.

See also

• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Connection (mathematics)
• G-structure

References

• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6