We define the notion of a distribution $\Delta$ on a manifold $M$. We explain what is means for a distribution to be integrable, and prove the Local Frobenius Theorem which gives a necessary and sufficient condition for a distribution to be integrable.

The global version of this theorem—which will be proved next lecture—is the cornerstone of an area of differential geometry called foliation theory. This semester we will use the (global) Frobenius Theorem to prove the Lie Correspondence Theorem and the Quotient Manifold Theorem. Next semester, we will use the Frobenius Theorem to show that flat connections on vector bundles have trivial restricted holonomy groups.