> [!definition] > > Let $X, Y$ be $C^p$-[[Manifold|manifolds]] ($p \ge 1$), $f: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]], and $p \in X$. $f$ is an **submersion at** $p$ if there exists [[Atlas|charts]] $(U, \psi) \in X$ and $(V, \varphi) \in Y$ such that the [[Derivative|derivative]] $Df_{U, V}(\hat p)$ is surjective and its [[Kernel|kernel]] [[Split Subspace|splits]]. > > If $f$ is a submersion at every point, then $f$ is a **submersion**. >