Parallelizable manifold

In mathematics, a differentiable manifold of dimension n is called parallelizable[1] if there exist smooth vector fields

on the manifold, such that at any point of the tangent vectors

provide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle,[2] so that the associated principal bundle of linear frames has a section on

A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .



See also


  1. Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 160
  2. Milnor, J.W.; Stasheff, J.D. (1974), Characteristic Classes, p. 15


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