> [!definition] > > Let $X, Y$ be two $C^p$ [[Manifold|manifolds]] and $f: X \to Y$ be a map. If for any $x \in X$, there exists a [[Atlas|chart]] $(U, \psi)$ at $x$ and a chart $(V, \varphi)$ at $f(x)$ such that $f(U) \subset V$ and the map > $ > f_{U, V} = \varphi \circ f \circ \psi^{-1}: \psi(U) \to \varphi(V) > $ > is a [$C^p$ morphism](Space%20of%20Continuously%20Differentiable%20Functions), then $f$ is a $C^p$-**morphism**, and $f_{U, V}$ is the *coordinate representation* of $f$ with respect to the charts $(U, \psi)$ and $(V, \varphi)$. > [!theorem] Morphism Fact Rundown > > - $C^p$-morphisms are [[Continuity|continuous]]. > - Composition of $C^p$ morphisms is a $C^p$ morphism. > [!theorem] > > Let $X, Y$ be $C^p$ manifolds, $f: X \to Y$ be a $C^p$ morphism, then for *any* pair of charts $(U, \psi)$ at $x$ and $(V, \varphi)$ at $f(x)$, $f_{U, V} \in C^p$. > > *Proof*. Let $(U, \psi)$ and $(V, \varphi)$ be a set of charts such that $f(U) \subset V$ and $f_{U, V} \in C^p$. Let $(U', \psi')$ and $(V', \varphi')$ be charts at $x$ and $f(x)$, respectively, then > $ > \begin{align*} > f_{U', V'} &= \varphi' \circ f \circ \psi'^{-1} \\ > &= \underbrace{(\varphi' \circ \varphi^{-1})}_{C^p} \circ \underbrace{(\varphi \circ f \circ \psi^{-1})}_{f_{U, V} \in C^p} \circ \underbrace{(\psi \circ \psi'^{-1})}_{C^p} > \end{align*} > $ > [!definition] > > Let $X$ and $Y$ be $\real^m$ and $\real^n$-[[Manifold|manifolds]] of class $C^p$ ($p \ge 1$), respectively, and $f: X \to Y$ be a [$C^p$ morphism](Manifold%20Morphism.md). Let $p \in X$, then the **rank** of $f$ at $p$ is the rank of [$df_p$](Differential.md). If the rank of $f$ at every point on $X$ is the same, then $f$ is of **constant rank**.