Frobenius Theorem (differential Topology) - Introduction


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. Suppose that fki(x) are a collection of real-valued C1 functions on Rn, for i = 1, 2, ..., n, and k = 1, 2, ..., r, where r < n, such that the matrix (fki) has rank r. Consider the following system of partial differential equations for a real-valued C2 function u on Rn:

begin{matrix} L_1u stackrel{mathrm{def}}{=} sum_i_f_1^i(x)frac{partial u}{partial x^i} &= 0\ L_2u stackrel{mathrm{def}}{=} sum_i_f_2^i(x)frac{partial u}{partial x^i} &= 0\ dots&\ L_ru stackrel{mathrm{def}}{=} sum_i_f_r^i(x)frac{partial u}{partial x^i} &= 0

One seeks conditions on the existence of a collection of solutions u1, ..., unr such that the gradients

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 i, j = 1, 2,..., 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.

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