---- Let $M$ be a [[smooth manifold]] of dimension $n$. > [!definition] Definition. ([[local frame, global frame]]) > A **local frame for $M$** is an ordered $n$-tuple of [[vector field|vector fields]] $(X^{(1)},\dots,X^{(n)})$ defined on an open subset $U \subset M$ that is [[linearly independent]] and [[linearly independent vector fields|spans the tangent bundle]] $TU$; i.e., the vectors $X^{(1)}|_{p},\dots,X^{(n)} |_{p}$ form a [[basis]] of $T_{p}M$ for each $p \in U$. We call it **smooth** if each [[vector field]] is smooth. > We call it a **global frame** instead if $U=M$. A manifold $M$ admitting a smooth global frame is called **parallelizable**. ^definition > [!basicexample] > - [[Lie groups are parallelizable]]. This includes $\mathbb{S}^{0}$, $\mathbb{S}^{1}$, $\mathbb{S}^{3} \cong \operatorname{SU}(2)$ ([[special unitary group|see]]), $\mathbb{T}^{n}$, etc. > ^basic-example > [!basicnonexample] > > > $\mathbb{S}^{2n}$ is not parallalizable. To see this, first note that the [[hairy ball theorem]] prohibits parallelizability of $\mathbb{S}^{2n}$, as any [[vector field]] $X^{(i)}$ within a 'smooth [[local frame, global frame|global frame]]' would vanish somewhere so that said 'smooth global frame' fails to be. > > In fact, the only parallelizable sphere other addition to the [[Lie group]] examples above ($\mathbb{S}^{0}, \mathbb{S}^{1}, \mathbb{S}^{3}$) is $\mathbb{S}^{7}$. > > Incidentally, this also shows $\mathbb{S}^{2}, \mathbb{S}^{4},\mathbb{S}^{6},\dots$ are not [[Lie group|Lie groups]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```