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  "body": "\n## Summary\n\nA smooth manifold is parallelizable if it carries a global frame.\nThis is a family of [[vector fields|0299VCW1TH]] that form a basis of the tangent space at every point.\nEquivalently, its [[tangent bundle|WY6K3P4YCG]] is trivial.\nLie groups are parallelizable, while the $2$-sphere is not, by the hairy ball theorem.\nAmong the spheres, exactly $S^0$, $S^1$, $S^3$ and $S^7$ are parallelizable, a fact tied to the real division algebras.\n\n## Explanations\n\n### L1 human\n\nOn a parallelizable manifold you can set up \"north\" and \"east\" (and further directions) at every point at once, smoothly, and with no point where they collapse.\nThe plane and the torus allow this: use the two coordinate directions everywhere.\nThe round $2$-sphere does not: by the hairy ball theorem every tangent vector field on it vanishes somewhere, so not even one direction can be chosen everywhere.\n",
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