> [!definition] > > Let $X_1$, $X_2$ be [[Manifold|manifolds]], $\pi_1: W_1 \to X_1$ and $\pi: W_2 \to X_2$ be [[Vector Bundle|vector bundles]]. A **VB-morphism**[^1] $\pi_1 \to \pi_2$ consists a pair of [[Manifold Morphism|morphisms]] > $ > f_M: X_1 \to X_2 \quad f_B: W_1 \to W_2 > $ > such that the following two conditions are satisfied: > > **Projection Compatibility:** The following diagram commutes > $ > \begin{CD} > W_1 @>{f_B}>> W_2 \\ > @V{\pi_1}VV @VV{\pi_2}V \\ > X_1 @>>{f_M}> X_2 > \end{CD} > $ > and the induced map for each $p \in X_1$ > $ > (f_B)_p: W_p \to W_{f(p)} \in L(W_p, W_{p}) > $ > is a [[Bounded Linear Map|bounded linear map]]. > > **Chart Compatibility:** For each $p \in X_1$, there exists trivialising maps > $ > \tau_1: \widetilde{U_1} \to U_1 \times F_1 > $ > at $p$ and > $ > \tau_2: \widetilde{U_2} \to U_2 \times F_2 > $ > at $f_M(p)$, such that $f_M(U_1) \subset U_2$ and the map > $ > U_1 \to L(F_1, F_2) \quad x \mapsto (\tau_2)_{f_M(x)} \circ (f_B)_x \circ (\tau_1^{-1}) > $ > is of class [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions). > > If $F_1$ and $F_2$ are finite-dimensional, then chart compatibility is implied by the projection compatibility. [^1]: Definition taken from Lang's *Differential and Riemannian Manifolds* chapter 3, but I made up the condition names.