---- 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 ---- #### ---- #### 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 > ```