Usually in a first course on differential geometry we learn some classical results on the geometry of curves and surfaces in the ordinary euclidean space, and just later in more advanced courses we learn systematically the concepts and the tools of the analysis on manifolds, one of whose pillars is the Frobenius' Theorem.

In order to remark the continuity between the two stages, it would be nice, for example, to present the Frobenius' Theorem together with some of its application in the realm of classical differential geometry.

Adressing to someone who has had already an introductory course on the differential geometry, and now is taking a course on smooth manifolds, what are results from classical differential geometry of curves and surfaces that I could present as good illustrations of Frobenius' Theorem?

Any suggestion is welcome.