---
tag: 7WX14GWD4M
kind: concept
name: Parallelizable manifold
msc: [57R25, 57R22]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Parallelizable_manifold
same_as:
  wikidata: Q1014612
  wikipedia: Parallelizable manifold
---

## Summary

A smooth manifold is parallelizable if it carries a global frame.
This is a family of [[vector fields|0299VCW1TH]] that form a basis of the tangent space at every point.
Equivalently, its [[tangent bundle|WY6K3P4YCG]] is trivial.
Lie groups are parallelizable, while the $2$-sphere is not, by the hairy ball theorem.
Among the spheres, exactly $S^0$, $S^1$, $S^3$ and $S^7$ are parallelizable, a fact tied to the real division algebras.

## Explanations

### L1 human

On 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.
The plane and the torus allow this: use the two coordinate directions everywhere.
The 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.