> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] modelled on $E$, $\pi$ be a [[Vector Bundle|vector bundle]] over $X$ modelled on $F$, and $\pi^*$ be its [[Dual Bundle|dual bundle]]. Let $U \subset X$ be open, then a local **coframe** for $\pi$ over $U$ is a [[Frame|frame]] of $\pi^*$, and a global coframe for $\pi$ over $U$ is a global frame of $\pi^*$. > [!theorem] > > Let $X$ be a $C^p$-manifold modelled on $E$, and $\pi$ be a vector bundle over $X$ modelled on $\real^n$. Let $\seqf{\xi_i}$ be a frame over $U$, then for each $p$, $\seqf{\xi_i(p)}$ is a [[Basis|basis]] for $\pi_p$. Let $\seqf{\xi_i^*(p)}$ be the corresponding dual basis, then $\seqf{\xi_i^*}$ is a coframe over $U$, known as the **frame dual to** $\seqf{\xi_i}$. > > The frame $\seqf{\xi_i}$ is $C^p$ if and only if its dual $\seqf{\xi_i^*}$ is $C^p$. > > > *Proof*. Let $(U, \varphi)$ be a chart at $p$, $\tau: \pi^{-1}(U) \to U \times \real^n$ be a trivialising map. Let $\{\td\xi_i\}_1^n$ be the local representations of $\seqf{\xi_i}$ with respect to $\varphi$ and $\tau$, and $\{\td\xi^*_j\}_1^n$ be the local representations of $\seqf{\xi_j^*}$ with respect to $\varphi$ and $\tau^*$. Then > $ > \delta_i^j = \angles{\xi_i, \xi^*_j} = \langle\tilde \xi_i, \td\xi^*_j\rangle > $ > If $\td\xi_i = a_j^i e^j$ and $\td\xi_j^* = b_k^j\varepsilon^k$, then the matrices $(a^i_j)$ and $(b^j_k)$ are inverses of each other. As > - $\seqf{\xi_i}$ is $C^p$ if and only if each $a_j^i$ is $C^p$, > - $\seqf{\xi_j^*}$ is $C^p$ if and only if $b_k^j$ is $C^p$, > - The inverse is smooth, meaning that each $a_j^i$ is $C^p$ if and only if each $b_k^j$ is $C^p$. > > $\seqf{\xi_i}$ is $C^p$ if and only if $\seqf{\xi_i^*}$ is $C^p$.