> [!definition] > > Let $X$ and $Y$ be $C^p$-[[Manifold|manifolds]], and $\pi: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]]. A $C^p$-morphism $\sigma: Y \to X$ is a **section of $\pi$** if it is a right inverse. > > If $\sigma$ is only defined on an open set of $Y$, then $\sigma$ is a **local section** of $\pi$. > [!theorem] > > Let $X$ and $Y$ be $E$ and $F$ manifolds, respectively. Let $\pi: X \to Y$ be a $C^p$-morphism, then $\pi$ is a [[Submersion|submersion]] at $p \in X$ if and only if there exists a local section whose image includes $p$. > > *Proof*. Let $p \in X$, $(U, \psi) \in X$ be a chart at $p$, and $(V, \phi) \in Y$ be a chart at $f(p)$. Suppose that $\pi$ is a submersion at $p$, then $D(\pi_{U, V})(p)$ is surjective and its kernel splits $E$ as $\ker\braks{{D(\pi_{U, V})}(p)} \oplus E_2$. Then $\pi_{U, V}|_{E_2}$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism, and its inverse is a local section. > > Now suppose that $\sigma$ is a local section, then > $ > D(\pi_{U, V} \circ \sigma_{V, U})(\pi(p)) = D(\pi_{U, V})(p) \circ D(\sigma_{V, U})(\pi(p)) > $ > so $D(\pi_{U, V})(p)$ is surjective. Moreover, $\im{D(\sigma_{V, U})(\pi(p))}$ is a closed subspace of $E$ with $D(\pi_{U, V})(p)$ restricted to it being an isomorphism. Therefore the kernel splits and $\pi$ is a submersion.