> [!definition] > > Let $X$ be a [[Manifold|manifold]] of class $C^p$ modelled on the [[Banach Space|Banach space]] $E$. Let > $ > TX = \bigsqcup_{p \in X}T_pX > $ > be the disjoint union of [[Tangent Space|tangent spaces]] of $X$, and let $\pi: TX \to X$ by mapping tangent vector in $T_pX$ to $p$. Let $(U, \varphi) \in X$ be a [[Atlas|chart]], then since $T_pX \iso E$, there exists a bijection > $ > \tau_U: \pi^{-1}(U) \to U \times E \quad (p, v) \mapsto (p, \hat v) > $ > If $(V, \psi)$ is another chart, then the translation map > $ > \tau_{U} \circ \tau_{V}^{-1}: \psi(U \cap V) \times E \to \varphi(U \cap V) \times E > $ > where > $ > (\hat p, \hat v) \mapsto (\varphi \circ \psi^{-1}, D(\varphi \circ \psi^{-1})(p)(\hat v)) > $ > induces a map > $ > U \cap V \to \text{Laut}(E) \quad p \mapsto D(\varphi \circ \psi^{-1})(p) > $ > Since $\varphi \circ \psi^{-1}$ is [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions), the above map is of class $C^{p - 1}$, which makes $TX$ a [[Vector Bundle|vector bundle]] of class $C^{p - 1}$, known as the **tangent bundle**. > [!definition] > > Let $f: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]], define > $ > f_* = Tf: TX \to TY > $ > by > $ > (x, v) \mapsto (f(x), df_x(v)) > $ > where $df$ is the [[Differential|differential]] of $f$, then $Tf$ is a [[VB-Morphism|VB-morphism]] of class $C^{p - 1}$, known as the **tangent map**/**global differential**. > > *Proof*. Since $f$ is a $C^{p}$-morphism and $df_x$ is a bounded linear map, $Tf$ is compatible with the projection maps. Now, let $(U, \psi) \in X$ and $(V, \varphi) \in Y$ be chart such that $f(U) \subset V$ and $f_{U, V} \in C^p$, then the local interpretation of the map with respect to the bundle charts > $ > x \mapsto (\tau_V)_{f(x)} \circ df \circ (\tau_{U})^{-1}_{\tau_U(x)} = D(f_{U, V})(x) > $ > is of class $C^{p - 1}$. Hence $f$ is compatible with the bundle charts, making it a VB-morphism of class $C^{p - 1}$. > [!theorem] > > The map from the [[Category|category]] of $C^p$-manifolds to the category of $C^{p - 1}$ vector bundles, defined by $X \mapsto TX$ and $f \mapsto Tf$ is a [[Functor|functor]].