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

on the manifold, such that at every 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 global 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, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis on Manifolds, New York: Macmillan, p. 160
  2. ^ Milnor, John W.; Stasheff, James D. (1974), Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, p. 15, ISBN 0-691-08122-0
  3. ^ Benedetti, Riccardo; Lisca, Paolo (2019-07-23). "Framing 3-manifolds with bare hands". L'Enseignement Mathématique. 64 (3): 395–413. arXiv:1806.04991. doi:10.4171/LEM/64-3/4-9. ISSN 0013-8584. S2CID 119711633.
  4. ^ Milnor, John W. (1958), Differentiable manifolds which are homotopy spheres (PDF)