Non-integrable tangent distributions

[jostpuur]

What kind of tangent distributions are not integrable? Is there concrete examples with two dimensional non-integrable distributions in three dimensions? When I draw a picture of two smooth vector fields in three dimensions, they always seem to generate some submanifold, indicating integrability.

 

[Chris Hillman]

Examples of the Frobenius Integrability Theorem

For other readers: jostpuur is asking about a fundamental result in the theory of manifolds, the Frobenius integrability theorem, which gives a very simple answer to questions like this: given a one-form $\alpha$, when can we find functions f,g such that $\alpha = f \, dg$? In this case, in terms of DEs we have found an integrating factor for the first order equation $\alpha(h) = 0$. For one-forms on R^3 the problem of finding "integral two-surfaces" through the two-analog of a vector field, which unlike the one-case is nontrivial, reduces to this question.

A special case of the Frobenius integrability theorem says that (assuming the given one-form $\alpha$ is nonzero on the neighborhood of interest) there exist f,g such that $\alpha = f \, dg$ iff $d\alpha = \mu \wedge \alpha$ for some one-form $\mu$.

jostpuur, what book are you reading? This is so important that I am surprised it offers no examples. (I seem to constantly recommend Flanders, Differential Forms with Applications to the Physical Sciences, Dover reprint, which does offer examples.) Does what I've said before help you find an example? If not, ask again, I can give you explicit examples.

 

[jostpuur]

I'm reading lecture notes of the course, which are in pdf form, but not in English. The notes mention Frobenius theorem, but it seems to be in a different form. It says that a tangent distribution is integrable precisely when it is involutive.

Distribution $\Delta$ is defined to be integrable, when for each $p\in M$, there exists a submanifold $N$ so that $p\in N$ and that $T_q N = \Delta_q$ for all $q\in N$.

Distribution is defined to be involutive, if always when $X_p, Y_p\in \Delta_p$, where X and Y are tangent vector fields, also $[X,Y]_p\in \Delta_p$.

Is this theorem the same thing what you were talking about, but only in different form?

 

[Chris Hillman]

Yes, exactly the same. Your course is more sophisticated than I thought, so you should see Flanders (for examples), Spivak (for alternative gender), and also Boothby (for alternative forms of the Frobenius theorem).

 

[jostpuur]

By drawing pictures, it looked like that I would have had a counter example to the Frobenius theorem, but now when got into equations, the flaw in the attempt became apparent. So, it seems everything is fine now.

There seems to be a huge list of books I should read. Too many books, too little time.

 

[jostpuur]

Your course is more sophisticated than I thought

This theorem was only mentioned, and not proved, in the lecture notes. So it could be that the course is not as sophisticated as you were thinking, but I cannot know for sure of course.