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1972•Proceedings of the American Mathematical SocietyOpen access

Integrably Parallelizable Manifolds

Vagn Lundsgaard Hansen

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Abstract

A smooth manifold ${M^n}$ is called integrably parallelizable if there exists an atlas for the smooth structure on ${M^n}$ such that all differentials in overlap between charts are equal to the identity map of the model for ${M^n}$. We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.

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A smooth manifold ${M^n}$ is called integrably parallelizable if there exists an atlas for the smooth structure on ${M^n}$ such that all differentials in overlap between charts are equal to the identity map of the model for ${M^n}$. We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.

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Available abstract

A smooth manifold ${M^n}$ is called integrably parallelizable if there exists an atlas for the smooth structure on ${M^n}$ such that all differentials in overlap between charts are equal to the identity map of the model for ${M^n}$. We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.

Key concepts: Parallelizable manifold, Diffeomorphism, Manifold (fluid mechanics), Torus, Mathematics, Class (philosophy), Pure mathematics, Characteristic class

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