Parallelizable manifold

In mathematics, a parallelizable manifold "M" is a smooth manifold of dimension "n" having vector fields

:"V"₁, ..., "V"_"n",

such that at any point "P" of "M" the tangent vectors

:"V"_{"i", "P"}

provide a basis of the tangent space at "P". Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on "M".

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

Examples

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, any 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" (any translation is a diffeomorphism and therefore these tranlations 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 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.

Notes

*The term "framed manifold" (occasionally "rigged manifold") is most usually applied to an embedded manifold with a given trivialisation of the normal bundle.