Inmathematics, a differentiable manifold${\displaystyle M}$ of dimensionn is calledparallelizable^[1] if there existsmoothvector fields$${\displaystyle \{V_1,\dots ,V_n\}}$$on the manifold, such that at every point${\displaystyle p}$ of${\displaystyle M}$ thetangent vectors$${\displaystyle \{V_1(p),\dots ,V_n(p)\}}$$provide abasis of thetangent space at${\displaystyle p}$. Equivalently, thetangent bundle is atrivial bundle,^[2] so that the associatedprincipal bundle oflinear frames has a global section on${\displaystyle M.}$

A particular choice of such a basis of vector fields on${\displaystyle M}$ is called aparallelization (or anabsolute parallelism) of${\displaystyle M}$.

• An example with${\displaystyle n=1}$ is thecircle: we can takeV₁ to be the unit tangent vector field, say pointing in the anti-clockwise direction. Thetorus of dimension${\displaystyle n}$ is also parallelizable, as can be seen by expressing it as acartesian product of circles. For example, take${\displaystyle n=2,}$ and construct a torus from a square ofgraph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, everyLie groupG is parallelizable, since a basis for the tangent space at theidentity element can be moved around by the action of the translation group ofG onG (every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points inG).
• A classical problem was to determine which of thespheresSⁿ are parallelizable. The zero-dimensional caseS⁰ is trivially parallelizable. The caseS¹ is the circle, which is parallelizable as has already been explained. Thehairy ball theorem shows thatS² is not parallelizable. HoweverS³ is parallelizable, since it is the Lie groupSU(2). The only other parallelizable sphere isS⁷; this was proved in 1958, byFriedrich Hirzebruch,Michel Kervaire, and byRaoul Bott andJohn Milnor, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in thenormed division algebras of the real numbers, complex numbers,quaternions, andoctonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requiresalgebraic topology.
• The product of parallelizablemanifolds is parallelizable.
• Everyorientableclosedthree-dimensional manifold is parallelizable.^[3]

• Any parallelizablemanifold isorientable.
• The termframed manifold (occasionallyrigged manifold) is most usually applied to an embedded manifold with a given trivialisation of thenormal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of thetangent bundle.
• A related notion is the concept of aπ-manifold.^[4] A smooth manifold${\displaystyle M}$ is called aπ-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.

• Chart (topology)
• Differentiable manifold
• Frame bundle
• Kervaire invariant
• Orthonormal frame bundle
• Principal bundle
• Connection (mathematics)
• G-structure

• Bishop, Richard L.; Goldberg, Samuel I. (1968),Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company,ISBN 0-486-64039-6
• Milnor, John W.;Stasheff, James D. (1974),Characteristic Classes, Princeton University Press
• Milnor, John W. (1958),Differentiable manifolds which are homotopy spheres(PDF), mimeographed notes

• Differential topology
• Fiber bundles
• Manifolds
• Vector bundles

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