> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] modelled on $E$, $\pi$ be a $C^{p - 1}$-[[Vector Bundle|vector bundle]] modelled on $F$, and $L^k$ be the $k$-[[Tensor|tensor]] [[Differentiable Functor|functor]]. Then the induced bundle $L^k\pi$ is the $k$-**tensor bundle** of $\pi$. > [!theorem] > > Let $\xi \in \mathbf{S}(L^k\pi)$ and $\eta \in \mathbf{S}(L^r\pi)$. Define their **tensor product** to be a mapping > $ > \xi \otimes \eta: X \to L^{k + r}\pi \quad p \mapsto \xi_p \otimes \eta_p > $ > then $\xi \otimes \eta \in \mathbf{S}(L^{k + r}\pi)$. > > *Proof*. Let $\tau: \pi^{-1}(U) \to U \times F$ be a trivialising map for $\pi$, and $L^j\tau: \pi^{-1}(U) \to U \times L^jF$ be the corresponding trivialising maps. In local coordinates, > $ > (\wh{\xi \otimes \eta})_p = \wh\xi_p \otimes \wh\eta_p > $ > the tensor product is bilinear and bounded. Hence $\xi \otimes \eta \in \mathbf{S}(L^{k + r}\pi)$. > [!theorem] > > Let $\mathbf{S}(\pi)$ be the space of [[Sections of Vector Bundles|sections of]] $\pi$, and $\lambda \in \mathbf S(L^k\pi)$ be a section of $L^k\pi$, then $\lambda$ induces a multilinear map > $ > \lambda: \mathbf{S}(\pi)^k \to C^{p - 1}(X) \quad \lambda(\xi^1, \cdots, \xi^k)(p) = \lambda_p(\xi^1_p, \cdots, \xi^k_p) > $ > Let $\tau: \pi^{-1}(U) \to U \times F$ be a trivialising map for $\pi$, and > $ > L^k\tau: L^k\pi^{-1}(U) \to U \times L^k(F) \quad (L^k\tau)_p = L^k\tau_p > $ > be the corresponding trivialising map for $L^k\pi$, then the composition can be taken locally as > $ > \lambda(\xi^1, \cdots, \xi^k)(p) = \td \lambda(\wh p) (\td \xi^1(\wh p), \cdots, \td \xi^k(\wh p)) > $ > > *Proof*. For each $p \in U$, $\tau_p \in L(\pi_p, F)$ is an isomorphism, and $L^k(\tau)_p = L^k(\tau_p^{-1})$. Thus the local representations are > $ > \td\xi^j: U \to F \quad p \mapsto \tau_p \circ \xi^j_p > $ > and > $ > \td\lambda: U \to L^k(F) \quad p \mapsto L^k(\tau)_p \circ \lambda_p = \lambda_p \circ \tau_p^{-1} > $ > where the composition is on each argument. From here, > $ > \begin{align*} > \lambda(\xi^1, \cdots, \xi^k)_p &= \lambda_p(\xi^1_p, \cdots, \xi^k_p) \\ > &= \lambda_p(\tau_p^{-1} \circ \tau_p \cdot \xi_p^{1}, \cdots, \tau_p^{-1} \circ \tau_p \cdot \xi_p^{k}) \\ > &= \td\lambda_p \cdot (\td\xi_p^1, \cdots, \td \xi_p^k) > \end{align*} > $ > Now, with respect to these trivialising maps, the composition $L^k(F) \times F^k$ is $k + 1$-linear and bounded, and thus is differentiable. By the differentiability of $\lambda$ and $\bracs{\xi^j}_1^k$, $\lambda(\xi^1, \cdots, \xi^k)$ is of class $C^{p - 1}$. > [!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. > > and let $\lambda: \mathbf{S}(\pi)^k \to C^{p - 1}(X)$ be a $k$-[[Multilinear Map|linear]] map, then for each $p \in X$, there exists $\lambda_p \in L^k(\pi_p)$ such that > $ > \lambda(\xi^1, \cdots, \xi^k) = \lambda_p (\xi^1, \cdots, \xi^k) > $ > and the mapping $p \mapsto \lambda_p$ is a section of $\mathbf{S}(L^k\pi)$. > > *Proof*. With condition $(1)$ and $(2)$, the modules $(C^{p - 1}(X), \mathbf{S}(\pi))$ forms a [[Pointwise Linear Decomposition|pointwise decomposition system]], so $\lambda$ admits a pointwise decomposition. Now, locally, $\lambda$ pointwise can be represented as a tensor product of the value of the frame. By extracting coefficients, we find that $\lambda$ is $C^{p - 1}$.