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 f_ki(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 (f_ki) has rank r. Consider the following system of partial differential equations for a real-valued C2 function u on Rn:

$\left. \begin{matrix} L_1u\ \stackrel{\mathrm{def}}{=}\ \sum_i f_1^i(x)\frac{\partial u}{\partial x^i} &= 0\\ L_2u\ \stackrel{\mathrm{def}}{=}\ \sum_i f_2^i(x)\frac{\partial u}{\partial x^i} &= 0\\ \dots&\\ L_ru\ \stackrel{\mathrm{def}}{=}\ \sum_i f_r^i(x)\frac{\partial u}{\partial x^i} &= 0 \end{matrix}\right\}.$ (1)

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

are linearly independent.

The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators L_k 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 ck_ij(x) that are allowed to depend on x. In other words, the commutators must lie in the linear span of the L_k 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 L_i so that the resulting operators do commute, and then to show that there is a coordinate system y_i for which these are precisely the partial derivatives with respect to y₁, ..., y_r.

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Frobenius Theorem (differential Topology) - Introduction

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