Parallelizable

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

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

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.