Frobenius theorem

Frobenius theorem provides a way to determine when a collection of vector fields (directions) on a space can be combined to form a smooth surface or subspace through every point. Essentially, it states that if the directions are consistent in how they interact—meaning their "commutator" (a measure of their combined change) can be written as a combination of the original directions—then these directions define a well-behaved, smooth subspace called an integrable distribution. This helps identify when complex geometric structures can be neatly "foliated" into smooth layers, like the pages of a book, based on their directional properties.