# 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 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 .

## Contents

## Examples[edit]

- An example with
*n*= 1 is the circle: we can take*V*_{1}to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension*n*is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take*n*= 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, every Lie group*G*is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of*G*on*G*(every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in*G*). - A classical problem was to determine which of the spheres
*S*^{n}are parallelizable. The zero-dimensional case*S*^{0}is trivially parallelizable; the case*S*^{1}is the circle, which is parallelizable as has already been explained. The hairy ball theorem shows that*S*^{2}is not parallelizable; however*S*^{3}is parallelizable, since it is the Lie group SU(2). The only other parallelizable sphere is*S*^{7}; this was proved in 1958, by Michel Kervaire, and by Raoul Bott and John Milnor, in independent work; the parallelizable spheres correspond precisely to elements of unit norm in the normed division algebras of the real numbers, complex numbers, quaternions, and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires algebraic topology. - The product of parallelizable manifolds is parallelizable.
- Every orientable three-dimensional manifold is parallelizable.

## Remarks[edit]

- Any parallelizable manifold is orientable.
- The term
*framed manifold*(occasionally*rigged manifold*) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle, and also for an abstract (i.e. non-embedded) manifold with a given stable trivialisation of the tangent bundle. - A related notion is the concept of a
**π-manifold**^{[3]}. A smooth manifold*M*is called a π-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.

## See also[edit]

- Chart (topology)
- Differentiable manifold
- Frame bundle
- Kervaire invariant
- Orthonormal frame bundle
- Principal bundle
- Connection (mathematics)
- G-structure

## Notes[edit]

## References[edit]

- Bishop, R.L.; Goldberg, S.I. (1968),
*Tensor Analysis on Manifolds*(First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6 - Milnor, J.W.; Stasheff, J.D. (1974),
*Characteristic Classes*, Princeton University Press - Milnor, J.W. (1958),
*Differentiable manifolds which are homotopy spheres*, mimeographed notes