Theorems of Frobenius

The Frobenius theorems are fundamental results in differential geometry and Lie theory that address when certain geometric or algebraic structures can be smoothly decomposed into simpler pieces. Specifically, they provide conditions under which a set of directions or vector fields defines a foliation—a partition of a space into smooth, evenly arranged subspaces. In essence, the theorems tell us when it's possible to break down complex spaces into well-behaved layers, and they guarantee such decompositions when certain compatibility conditions, like closure under specific operations, are met.