Definition 2.1.1 (Integrable).label Let $X$ be a $C^{p}$ ($p \ge 2$) manifold, $E \subset TX$ be a subbundle, and $x_{0} \in X$, then $E$ is integrable at $x_{0}$ if there exists a submanifold $x_{0} \in Y \subset X$ such that

is an isomorphism. If $E$ is integrable at every $x_{0} \in X$, then $E$ is integrable.