Thursday, 6 October 2011

dg.differential geometry - Frobenius Theorem for subbundle of low regularity?

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 C1 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 AwedgedA=0. This quantity is well defined is A is C1. Let me give a sketch of the proof that AwedgedA=0 garanties existence of the foliation is A is C1.



The proof is by induction



1) Consider the case n=2. In this case it is a standard fact of ODE, that for a C1 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 R3. 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 C1
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 C1 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 Lv(A)=d(iv(A))+iv(dA)=iv(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

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