Parallelizable manifold

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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 .

Examples[edit]

  • 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, 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 Sn are parallelizable. The zero-dimensional case S0 is trivially 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; 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]

Notes[edit]

  1. ^ Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 160
  2. ^ Milnor, J.W.; Stasheff, J.D. (1974), Characteristic Classes, p. 15
  3. ^ Milnor, John (1958), Differentiable manifolds which are homotopy spheres

References[edit]