> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] modelled on $E$, and $\pi$ be a [[Vector Bundle|vector bundle]] of class $C^p$ modelled on $F$. Let $p \in X$ and $(U, \varphi)$ be a [[Atlas|chart]] at $p$, and $\tau: \pi^{-1}(U) \to U \times F$ be a trivialising map. Then the composition > $ > \begin{CD} > \pi^{-1}(U)@>\tau>> U \times F @>{\varphi \times \text{Id}}>> E \times F > \end{CD} > $ > forms a bundle chart, denoted as $(\pi^{-1}(U), \psi)$. Let $\xi: X \to \pi$ be a right inverse of $\pi$, then $\xi$ has **local representation** > $ > \begin{CD} > E @>{\varphi^{-1}}>> U @>{\xi}>> \pi^{-1}(U) @>{\psi}>> E \times F @>{\text{proj}_2}>> F > \end{CD} > $ > denoted as $\td \xi$. Depending on the context, we can take this shorter variant that starts on the manifold instead > $ > \begin{CD} > U @>{\xi}>> \pi^{-1}(U) @>{\psi}>> E \times F @>{\text{proj}_2}>> F > \end{CD} > $ > [!definition] > > Let $X$ be a $C^p$ [[Manifold|manifold]] modelled on $E$, and let $\pi$ be a [[Vector Bundle|vector bundle]] of class $C^p$ modelled on $F$, and define $\mathbf{S}(X, \pi)$ (or simply $\mathbf{S}(X)$) as the space of all [[Section|sections]] of $\pi$. For any $\xi, \eta \in \mathbf{S}(X)$ and $\lambda \in \real$, define > $ > (\lambda \xi + \eta)_p = \lambda \xi_p + \eta_p > $ > then $\mathbf{S}(X)$ forms a [[Vector Space|vector space]] over $\real$. For any $f \in C^p$, define > $ > (f \xi)_p = f(p) \cdot \xi_p > $ > then $\mathbf{S}(X)$ is a [[Module|module]] over $C^p(X)$. > > *Proof*. Let $(U, \varphi)$ be a [[Atlas|chart]] at $p$ and $(\td U, \psi)$ be a bundle chart containing $\pi^{-1}(p)$, then $\xi$ and $\eta$ have local representations > $ > \td\xi: E \to F \quad \td\eta: E \to F > $ > In which case, since the bundle chart is linear on each fibre, > $ > \td{(\lambda \xi + \eta)} = \lambda \td\xi + \td \eta \quad \td{f\xi} = \td f \cdot \td\xi > $ > the resulting maps have $C^p$ local representation around $p$. Hence they are also in $\mathbf{S}(X)$. # Component Functions > [!definition] > > Let $X$ be a $C^p$-manifold modelled on $E$, and $\pi$ be a vector bundle of class $C^p$ modelled on $\real^n$. Let $\xi: X \to \pi$ be a right inverse of $\pi$. If $(U, \varphi)$ is a chart at $p$ and $\tau: \pi^{-1}(U) \to U \times \real^n$ is a trivialising map, then $\xi$ can be written in local coordinates > $ > \td\xi: E \to \real^n \quad v \mapsto (\xi_1(v), \cdots, \xi_n(v)) = \xi_i(p)e^i \quad \td\xi = \xi_ie^i > $ > where $\seqf{\xi_i}$ are the **component functions** of $\xi$, and $\xi|_U$ is a [[Manifold Morphism|morphism]] if and only if each of its component functions are [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions). > > *Proof*. $\xi$ is $C^p$ if and only if $\tilde \xi$ is $C^p$, if and only if each component is $C^p$. > [!theorem] > > Let $X$ be a $C^p$-manifold modelled on $E$, and $\pi$ be a vector bundle over $X$ modelled on $\real^n$. If $U \subset X$ is open, $\seqf{\xi_i}$ is a local [[Frame|frame]], and $\seqf{\xi_i^*}$ is its [[Coframe|dual]], then the mapping > $ > \mu: \pi^{-1}(U) \to U \times \real^n \quad (p, v) \mapsto (\xi_1^*v, \cdots, \xi_n^*v) > $ > is a trivialising map for $\pi$. > > If $\eta: X \to \pi$ is a right inverse of $\pi$, then $\eta$ can be written in local coordinates > $ > \eta: U \to \real^n \quad p \mapsto (\angles{\xi_1^*(p), \eta_p}, \cdots, \angles{\xi_n^*(p), \eta_p}) \quad \eta = \eta^i\xi_i > $ > where $\eta^i = \angles{\xi_i^*, \eta}$ are the **component functions of** $\eta$ **with respect to** $\seqf{\xi_i}$. Hence any smoothness criterion also applies to these components. > > *Proof*. Let $\tau: \pi^{-1}(V) \to V \times \real^n$ be another trivialising map and assume without loss of generality that $V \subset U$. For any $p \in U$, the map $v \mapsto \sum_i \xi_i \angles{\xi_i^*, v}$ is the identity, so the map $v \mapsto \xi_i(p)v^i$ is the inverse of $\mu_p$. > > Let $\{\td\xi_i\}_1^n$ with $\td\xi_i(p) = \tau_p \circ \xi_i$, and $T_p$ be the matrix with $\{\td\xi_i(p)\}_1^n$ as its columns. Then for any $v \in \real^n$, > $ > Tv = \td\xi_i v^i = \tau_p \circ \xi_i v^i = \tau_p \circ \mu_p^{-1} = (\tau \circ \mu^{-1})_p > $ > is the transition map. As the map $p \mapsto T_p$ is $C^p$, $\tau$ and $\mu$ are VB-equivalent, so $\mu$ is a trivialising map. # Smoothness Criterion > [!theorem] > > Let $X$ be a $C^p$-manifold modelled on $E$, $\pi$ be a vector bundle of class $C^p$ modelled on $F$, and $\xi: X \to \pi$ be a right inverse of $\pi$. Then the following are equivalent: > 1. $\xi$ is $C^p$. > 2. For every point $p \in X$, there exists $U \in \cn^o(p)$, where $\xi|_U: U \to \pi$ is $C^p$. > 3. For every point $p \in X$, there exists $U \in \cn^o(p)$, a chart $(U, \varphi)$, and a trivialising map $\tau: \pi^{-1}(U) \to U \times F$, with respect to which $\xi$ has $C^p$ local representation. > 4. There exists an open cover $\seqi{U}$ of $X$, such that $\xi|_{U_i}: U_i \to \pi$ is $C^p$ for all $i \in I$. > 5. There exists an open cover $\seqi{U}$ of $X$, where for each $U_i$ there exists a chart $\varphi_i$ and a trivialising map $\tau_i$, with respect to which $\xi$ has $C^p$ local representation. > > If $F = \real^n$, then the following are equivalent to the above: > > 6. For every point $p \in X$, there exists $U \in \cn^o(p)$, a chart $(U, \varphi)$, and a trivialising map $\tau_i: \pi^{-1}(U) \to U \times \real^n$, and $\xi$ has $C^p$ component functions with respect to them. > 7. There exists an open cover $\seqi{U}$ of $X$, where for each $U_i$ there exists a chart $\varphi_i$ and a trivialising map $\tau_i$, and $\xi$ has $C^p$ component functions with respect to them. > > *Proof*. > > $(1) \Rightarrow (2)$: Let $p \in X$, then by definition there exists a bundle chart $(\pi^{-1}(U), \td\varphi)$ and a chart $(U, \varphi)$ at $p$, with respect to which $\xi|_{U}$ has $C^p$ local representation. Therefore $\xi|_U$ is $C^p$. > > $(2) \Rightarrow (3)$: Since the translation maps are $C^p$-isomorphisms, we can assume without loss of generality that $\td\varphi = (\varphi \times {Id}) \circ \tau$ for some trivialising map $\tau$. > > By taking a neighbourhood/covering set for each $p$, $(2) \Leftrightarrow (4)$ and $(3) \Leftrightarrow (5)$. > > $(4) \Rightarrow (1), (5) \Rightarrow (1)$: In both situations, there exists an open cover $\seqi{U}$ of $X$, where each $\xi|_{U_i}$ is $C^p$, by the [[Gluing Lemma]], $\xi$ is $C^p$. > > Since $C^p$-smoothness of components is equivalent to $C^p$-smoothness of local representations, $(3) \Leftrightarrow (6)$ and $(4) \Leftrightarrow (7)$. # Extensions > [!theorem] > > Let $X$ be a $C^p$ [[n-Manifold|n-manifold]] and $\pi$ be a vector bundle of class $C^{p-1}$ modelled on $F$. Let $U \subset X$ be open, and define $\pi_U = \pi^{-1}(U)$. Then for any $\xi \in \mathbf{S}(U, \pi_U)$ and $p \in U$, there exists precompact neighbourhoods $V, S \in \cn^o(p)$ such that $\ol{V} \subset S \subset \ol{S} \subset U$, and section $\mu \in \mathbf{S}(X)$ where $\mu|_V = \xi|_V$. > > *Proof*. > > ![[bump_setup.png]] > > As $X$ is [[Locally Compact Hausdorff Space|LCH]] and $U \in \cn^o(p)$, there exists $S \in \cn^o(p)$ precompact with $\ol{S} \subset U$, and $V \in \cn^o(p)$ precompact with $\ol{V} \subset S$. Since $\ol{V}$ is closed with $\ol{V} \subset S$, there exists a $C^p$-[[Smooth Bump Function on Manifold|bump function]] $\psi \in C^p(X)$ such that $\psi|_{V} = 1$ and $\psi|_{\ol{S}^c} = 0$. > > ![[glue_setup.png|300]] > > Let $\mu = \psi \cdot \xi$, > $ > \mu: X \to \pi \quad \mu(p) = \begin{cases} > \psi(p) \cdot \xi(p) & p \in U \\ > 0 &p \in S^c > \end{cases} > $ > Since $\psi \cdot \xi|_{S^c \cap U} = 0$, the two definitions agree. With respect to any trivialising map, $\mu|_{S^c}$ always has constant local representation ($0$ everywhere), so $\mu|_{S^c}$ is $C^p$. As $\mu|_U$ and $\mu|_{S^c}$ are both $C^p$, by the [[Gluing Lemma]], $\mu \in \mathbf{S}(X)$. # Pointwise Decomposition > [!theorem] > > Let $X$ be a $C^p$-manifold and $\pi$ be a $C^{p - 1}$ vector bundle. Suppose that > 1. For any $p \in X$, there exists a neighbourhood $U \in \cn^o(p)$ such that $\pi$ admits a [[Frame|frame]]. > 2. For any [[Open Cover|open cover]], $X$ admits a $C^{p - 1}$ [[Partition of Unity|partition of unity]] subordinate to it. > > In particular, if $X$ is modelled on $\real^n$ and $\pi$ is the [[Tangent Bundle|tangent bundle]] or most of its images under [[Differentiable Functor|differentiable functors]], then $X$ and $\pi$ satisfy the above conditions. > > Let $\xi \in \mathbf{S}(X, \pi)$ and $p \in X$ such that $\xi(p) = 0$, then there exists a decomposition > $ > \xi = \lambda_i\xi^i \quad \lambda_i \in C^{p - 1}(X) \quad \xi^i \in \mathbf{S}(\pi) \quad \lambda_i(p) = 0 > $ > Therefore $(C^{p - 1}(X), \mathbf{S}(\pi))$ forms a [[Pointwise Linear Decomposition|pointwise decomposition system]]. > > *Proof*. Using $(1)$, let $U \in \cn^o(p)$ be a neighbourhood of $p$ such that $\pi$ admits a frame. > > With charts and coordinate balls, let $V, W \in \cn^o(p)$ such that $\ol{W} \subset V \subset \ol{V} \subset U$. Apply $(2)$, and let $\bracs{\varphi, \psi}$ be a $C^{p - 1}$-partition of unity subordinate to $V$ and $\ol{W}^c$, such that $\varphi|_W = 1$ and is supported in $U$, and $\psi|_{V^c} = 1$ with $\psi|_{W} = 0$. This allows the first decomposition $\xi = \varphi \xi + \psi \xi$, and localises the problem to $U$. > > Let $\{\xi^i\}_1^d$ be a frame on $U$, then they can be extended to $X$ via $\varphi$ by filling in $0$ outside of $V$. From here, $\xi|_{U} = \lambda_i \xi^i$, so $\xi = \varphi \lambda_i \xi^i + \psi \xi$. Since $\xi(p) = 0$, $\lambda_i(p) = 0$ for each $i$. As $\psi(p) = 0$, we have found the desired decomposition.