Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. In its most elementary form, the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneous partial differential equations. Let be a collection of C1 functions, with r < n, and such that the matrix ( f ik ) has rank r. Consider the following system of partial differential equations for a C2 function u : Rn → R: One seeks conditions on the existence of a collection of solutions u1, ..., un−r such that the gradients ∇u1, ..., ∇un−r are linearly independent. The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators Lk satisfy a certain integrability condition known as involutivity. Specifically, they must satisfy relations of the form for 1 ≤ i, j ≤ r, and all C2 functions u, and for some coefficients ckij(x) that are allowed to depend on x. In other words, the commutators must lie in the linear span of the Lk at every point. The involutivity condition is a generalization of the commutativity of partial derivatives. In fact, the strategy of proof of the Frobenius theorem is to form linear combinations among the operators Li so that the resulting operators do commute, and then to show that there is a coordinate system yi for which these are precisely the partial derivatives with respect to y1, ..., yr.