> [!definition] > > Let $X$ be an $E$-[[Manifold|manifold]], $\pi: W \to X$ be a [[Vector Bundle|vector bundle]] modelled on $\real^n$, and $U \subset X$ be [[Open Set|open]]. > > A family of right inverses of $\pi$, $(\xi_i: U \to \widetilde U)_1^k$, is **linearly independent** if the vectors $\bracs{\xi_i(p)}_1^k$ are linearly independent at each $p$. If they also span $\pi^{-1}(p)$ at each $p$, then they form a rough **local frame** for $U$. If they are [[Sections of Vector Bundles|sections]], then they form a local frame for $U$. > > If $U = X$, then $(\xi_i)_1^n$ is a **global frame** for $X$. If $X$ admits a global frame, then $X$ is **parallelisable**. # Extension of Frames (Lee 8.11) Let $(U, \tau, \varphi)$ be a trivialising map and a [[Atlas|chart]] defined on the same domain. > [!theorem] > > Let $(\xi_i)_1^k$ be a linearly independent family of local sections on $U$. For any $p \in U$, there exists a neighbourhood $U_p \in \cn^o(p)$ on which $(\xi_i)_1^k$ can be completed to be a local frame. > > *Proof*. Let $\tilde \xi_{i}: E \to E \times \real^n$ be the coordinate representation of each local section with respect to the bundle chart created from $(U, \tau, \varphi)$. At $p$, the second component of $\{(\tilde \xi_i(p))_2\}_1^k$ forms a linearly independent set, from which we can complete to a basis $\{(\tilde \xi_1(p))_2, \cdots, (\tilde \xi_k(p))_2, v_{k + 1}, \cdots, v_{n}\}$ in $\real^n$. > > For $j > k$, let $(\tilde \xi_j(x))_2$ be identically equal to $v_j$ for all $x \in U$, then $\{(\tilde \xi_i(p))_2\}_1^n$ is a basis for $\real^n$. Combine the vectors $\{(\tilde \xi_i(x))_2\}_1^n$ into a matrix $M_x$. Since the vectors are in $\real^n$ and the [[Determinant|determinant]] is continuous, the map > $ > \begin{CD} > x @>{\{\tilde \xi_j}\}_1^n>> \bracs{(x, v_j)}_1^n > @>{\proj_2}>> \bracs{v_j}_1^n > @>{\text{combine}}>> M_x > @>{\det}>> \det M_x > \end{CD} > $ > is continuous. Hence there exists a neighbourhood $\widehat U_p$ on which $\det M_x$ is non-zero, and $(\xi_i)_1^n$ is a local frame on $U_p = \varphi^{-1}(\widehat U_p)$. > [!theorem] > > Let $\bracs{v_i}_1^k$ be a linearly-independent tuple of vectors in $\pi^{-1}(p)$, then there exists a neighbourhood $U_p \in \cn^o(p)$ and a local frame $\seqf{\xi_i}$ such that $\xi_i|_p = v_i$ for all $i$. > > *Proof*. Firstly we can complete the tuple to a basis. Let > $ > \tilde \xi_i: E \to E \times \real^n \quad \hat p \mapsto (\hat p, \tau(v_i)_2) > $ > be the coordinate representation $\xi_i$ with respect to the bundle chart created from $\tau$ and $\varphi$. Since each $\xi_i$ has constant representation, they are all smooth. As $\seqf{\tau(v_i)_2}$ forms a basis for $\real^n$, and $\tau$ restricted to $\pi(q)$ is a toplinear isomorphism, $\seqf{\xi_i}$ forms a local frame at $U$. > [!theorem] > > Let $X$ be a [[n-Manifold|topological n-manifold]], $A \subset X$ be a [[Closed Set|closed]] subset, and $\seqf{\xi_k}$ be an independent family of smooth vector fields along $A$, then they can be extended to a local frame on a neighbourhood of $A$. > > *Proof.* Let $\{\tilde \xi_i\}_1^n$ be their extensions to a neighbourhood $U$ of $A$. For each $p \in A$, by choosing a trivialising map and a chart at $p$ and using the continuity of the determinant, there exists a neighbourhood $U_p \in \cn^o(p)$ on which $\{\tilde\xi_i\}_1^n$ is a linearly independent set. Therefore $\{\tilde\xi_i\}_1^n$ forms a local frame on $U = \bigcup_{p \in A}U_p \in \cn^o(A)$.