Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$. Bracket is a second order operator, hence subbundle $E$ needs to be $C^2$.
Are there any generalizations for subbundle which is $C^1$, $C^{1+smth}$?
Thank you, Z.
Let me conisder the case when the distribution of planes is of codimension 1 and explain why in this case it is enough to have $C^1$ smoothness in order to ensure the existence of the folitation.
In the case when the distribution is of codimension 1, you can formulate Frobenius Theorem in terms of 1-forms. Namely you can define a non-zero 1-form $A$, whose kernel is the distribution. The smoothness of this 1-form will be the same as the smoothness of the distribution. Now, you can say that the distribution is integrable if $A\wedge dA=0$. This quantity is well defined is A is $C^1$. Let me give a sketch of the proof that $A\wedge dA=0$ garanties existence of the foliation is A is $C^1$.
The proof is by induction
1) Consider the case $n=2$. In this case it is a standard fact of ODE, that for a $C^1$ smooth distribution of directions on the plane the integral lines are uniquelly defined.
2) Conisder the case $n=3$. We will show that the foliation exists locally near any point, say the origin $O$ of $R^3$. The 1-form A, that defines the distribution is non vanishing on one of the coordinate planes, say $(x,y)$ plane in the neighborhood of $O$. Take a $C^1$ smooth vector field in the neigborhood of $O$ that is transversal to planes $z=const$ and satisfies $A(v)=0$. Take the flow correponding to this vector field. The flow is $C^1$ smooth and moreover it preserves the distribution of planes $A=0$. Indeed, dA vanishes on the planes A=0 (by the condition of integrability), and we can apply the formula for Lie derivative $L_v(A)=d(i_v(A))+i_v(dA)=i_v(dA)$. Finally, we take the integral curve of the restriction of $A=0$ to the plane $(x,y)$ and for evey curve conisder the surface it covers unders the flow of $v$. This gives the foliation.
This reasoning can be repeated by induction.
A good refference is Arnold, Geometric methods of ordinary differential equations. I don't know if this book was transalted to English
You can even get a Frobenius theorem for Lipshitz vector fields, which need not span everywhere (the rank can drop along closed subsets). The state of the art in this domain is done by the control theorists, particularly Sussmann and Agrachev. Here is an abstract for a recent article which will have other earlier references:
Journal of Differential Equations Volume 243, Issue 2, 15 December 2007, Pages 270-300. F. Rampazzo and H.J. Sussmann
Abstract: We generalize the classical Frobenius Theorem to distributions that are spanned by locally Lipschitz vector fields. The various versions of the involutivity conditions are extended by means of set-valued Lie derivatives—in particular, set-valued Lie brackets—and set-valued exterior derivatives. A PDEs counterpart of these Frobenius-type results is investigated as well.
