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Parallelizable


A hypersphere S^n is parallelizable if there are n vector fields that are linearly independent at each point. There exist only three parallelizable spheres: S^1, S^3, and S^7 (Adams 1958, 1960, Le Lionnais 1983).

More generally, an n-dimensional manifold M is parallelizable if its tangent bundle TM is a trivial bundle (i.e., if TM is globally of the form M×R^n).


See also

Manifold, Sphere, Tangent Bundle, Trivial Bundle, Vector Bundle

Portions of this entry contributed by Steuard Jensen

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References

Adams, J. F. "On the Non-Existence of Elements of Hopf Invariant One." Bull. Amer. Math. Soc. 64, 279-282, 1958.Adams, J. F. "On the Non-Existence of Elements of Hopf Invariant One." Ann. Math. 72, 20-104, 1960.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 49, 1983.Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984.

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Parallelizable

Cite this as:

Jensen, Steuard and Weisstein, Eric W. "Parallelizable." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Parallelizable.html

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