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Dmitri Panov
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ThisLet me conisder the case when the distribution of planes is just a remak, noof codimension 1 and explain why in this case it is enough to have $C^1$ smoothness in order to ensure the answerexistence of the folitation. 

In the case when the distribution is of codimension 1, you can reformulateformulate Frobenius Theorem, so that it uses just one derivative 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 $C^1$. Though it is not clear forLet me if this garatiniesgive a sketch of the integrablilityproof 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

This is just a remak, no the answer. In the case the distribution is of codimension 1, you can reformulate Frobenius Theorem, so that it uses just one derivative. 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 $C^1$. Though it is not clear for me if this garatinies the integrablility.

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

Source Link
Dmitri Panov
  • 28.9k
  • 4
  • 92
  • 161

This is just a remak, no the answer. In the case the distribution is of codimension 1, you can reformulate Frobenius Theorem, so that it uses just one derivative. 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 $C^1$. Though it is not clear for me if this garatinies the integrablility.