Parallelizable manifold

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

V₁, ..., 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 V₁ 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 translations induce linear isomorphisms between tangent spaces of points in G).

A classical problem was to determine which of the spheres Sⁿ are parallelizable. The case S¹ is the circle, which is parallelizable as has already been explained. The hairy ball theorem shows that S² is not parallelizable. However S³ is parallelizable, since it is the Lie group SU(2). The only other parallelizable sphere is S⁷; 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, and also for an abstract (i.e. non-embedded) a manifold with a given stable trivialisation of the tangent bundle.