Parallelizable manifold
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In mathematics, a parallelizable manifold M is a smooth manifold of dimension n having vector fields
- V1, ..., Vn,
such that at any point P of M the tangent vectors
- Vi, 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.
[edit] 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 G's action on itself by translation.
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.
[edit] Notes
- The term framed manifold (occasionally rigged manifold) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle.
