Integrability Conditions For Differential Systems - Examples of Applications

Examples of Applications

In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi, i.e., a collection of 1-forms forming a basis of the cotangent space at every point with which are closed (dθi = 0, i=1,2, ..., n). By the Poincaré lemma, the θi locally will have the form dxi for some functions xi on the manifold, and thus provide an isometry of an open subset of M with an open subset of Rn. Such a manifold is called locally flat.

This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe

.

If we had another coframe, then the two coframes would be related by an orthogonal transformation

If the connection 1-form is ω, then we have

On the other hand,

But is the Maurer–Cartan form for the orthogonal group. Therefore it obeys the structural equation and this is just the curvature of M: After an application of the Frobenius theorem, one concludes that a manifold M is locally flat if and only if its curvature vanishes.