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.
Best Answer
First, anything that is proved using the Frobenius theorem can also be proved using the existence and uniqueness theorem for ODE's and the fact that partials commute. The theorem is used in differential geometry to show that local geometric assumptions imply global ones. Here are a few examples that come to mind:
1) If a submanifold of $R^n$ has zero second fundamental form, it is an affine subspace of $R^n$. A similar statement holds if the second fundamental form is a constant nonzero multiple of the metric.
2) More generally, if you have an abstract Riemannian $n$-manifold with a symmetric $2$-tensor that satisfies both the Gauss and Codazzi-Mainardi equations, then it determines an isometric immersion of the manifold into $(n+1)$-dimensional Euclidean space unique up to rigid motion. In an elementary course, you might present the $2$-dimensional case.
3) A Riemannian manifold with vanishing sectional curvature is isometric to the standard flat metric on $R^n$. Again, the $2$-dimensional case can be presented in an elementary course.
4) A Riemannian manifold with constant sectional curvature $1$ ($-1$) is locally isometric to $S^n$ ($H^n$).
Whether you prefer to use ODE's or Frobenius when proving these statements depends on how you set everything up. In particular, if you set everything up using differential forms ("moving frames"), then it is natural to use Frobenius. If you set things up in co-ordinates, it is perhaps more direct to use ODE's instead.