bibkey: ganatra2018frobenius authors: Sheel Ganatra year: 2018 title: Math 535a Differential Geometry doi: null url: https://sheelganatra.com/spring2018_math535a/ claim: The course states the classical Frobenius criterion that a smooth constant-rank distribution is integrable exactly when it is involutive. strata_touched: [] license: citation-only triage: anchor

Frobenius and the actual transport distribution

The course’s lecture schedule entry on vector bundles, sub-bundles and distributions explicitly states “a distribution is integrable if and only if it is involutive”; subsequent entries discuss the straightening of a nonzero vector field and the Frobenius proof. This is a course-level locator for the classical theorem, without an original-publication claim, DOI or numbered proof in the page.

The FIB boundary geometry volume §22 assumes a smooth rank-three subbundle of the actual state tangent bundle and uses this criterion inside its interface-to-leaf deduction. It proves bilinearity of the transverse bracket, gives a direct commuting-flow construction, and computes a nonintegrable rank-three example on a four-dimensional state space. The theorem’s local integral-leaf dimension does not determine the ambient state dimension, identify physical position or equate a pointwise vector product with a Lie bracket.