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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.\nA *global frame* on $M$ is a list $V_1, \\dots, V_n$ of smooth [[vector fields|0299VCW1TH]] on $M$ with the following property.\nFor every $p \\in M$, the vectors $V_1(p), \\dots, V_n(p)$ form a basis of the tangent space $T_pM$.\nThe manifold $M$ is *parallelizable* if it has a global frame.\n\n## Remarks\n\nEquivalently, the [[tangent bundle|WY6K3P4YCG]] $TM$ is isomorphic, as a smooth vector bundle, to the trivial bundle $M \\times \\R^n$; a global frame gives the isomorphism $(p, a) \\mapsto \\sum_i a_i V_i(p)$.\nExample: the circle $S^1 \\subseteq \\R^2$ is parallelizable.\nThe vector field $V(x, y) = (-y, x)$ is tangent to $S^1$ and nonzero at every point, so it is a global frame for the $1$-dimensional manifold $S^1$.\nEvery Lie group is parallelizable, by translating a basis of the tangent space at the identity.\nNon-example: $S^2$ is not parallelizable.\nA global frame would contain a nowhere-vanishing vector field, and by the hairy ball theorem no such field exists on $S^2$.\nA parallelizable manifold is orientable, so the Möbius band is not parallelizable either.\n",
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