A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .
Examples
An example with 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 is also parallelizable, as can be seen by expressing it as a
cartesian product of circles. For example, take 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 groupG 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
spheresSn 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
Friedrich Hirzebruch,
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.
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 (that is, non-embedded) manifold with a given stable trivialisation of the
tangent bundle.
A related notion is the concept of a π-manifold.[4] A smooth manifold 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.
^Bishop, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis on Manifolds, New York: Macmillan, p. 160
^Milnor, John W.; Stasheff, James D. (1974), Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, p. 15,
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