cianfa72 said:
Ok, so is your ##\{(p, f(p))\}## in post#7 an element of the tangent bundle ##TM## ?
No, this was just a point on the manifold with its function value. E.g. a point on Earth and its air pressure. The tangent bundle points in the direction along isobars, since as you correctly observed, my function was a level set, an isobar. My ##f## in the picture is a walk on Earth along isobars. The tangent vectors tell me where I have to go to from point to point, parameterized by time, the second coordinate of my flow (walk).
cianfa72 said:
Ok, so your function ##f:\,M\longrightarrow M## is parametrized by the parameter ##t##
Yes, but: Forget the manifold! Forget the function! It might well be that I made some technical errors since I only wanted to describe the situation, and that was a tangent bundle, neither a function nor a manifold. If you want to have it rigorous then read all the links above, yours and mine, or even better a textbook or lecture note.
You have to make a decision:
Either you want to know how it works in general, then you must not care about ##M## or ##f## because this entire thread is about ##X.## No ##p\in M##, no ##f\in C^\infty (M).## Then you can use the image as an example of a vector field and try to understand the two theorems.
Or you want technical precision. In that case, you have to provide the entire framework including particularly the notations. Every single author writes his derivatives differently. And all those names: frames, vector bundles, subbundles, etc. are finally coordinate systems and derivatives. Best would be a publically available lecture note with the two theorems you mentioned - no Wikipedia, no images, no insight articles. This way, we could seriously talk and use the same language.
The straightening theorem says that a vector field can locally be described by a differential operator of the first coordinate by a suitable coordinate system.
The Wikipedia link is insufficient for a qualified discussion. It is poorly written. "Let ##f=(f_1,\ldots.f_n).##" What does that mean? ##n## components of what function? A map from ##M## or from ##\mathbb{R}^m## with ##m=n ## or ##m\neq n## to what? Obviously coordinates. But that has to be guessed from context. You cannot read it there. I seriously recommend avoiding Wikipedia if you want to learn something. Choose a lecture note. There are hundreds available around the world.
The Frobenius theorem says that a vector field is completely integrable if it forms a subalgebra of the Lie algebra, i.e. of the tangent space.
The same is true here:
do not consult Wikipedia, search for a lecture note. The search key "Frobenius Theorem + differential geometry + pdf" would probably do. If not, try differential topology instead.
You have asked why the straightening theorem is called the one-dimensional case of the Frobenius theorem. It actually says that the Frobenius theorem is a generalization for higher dimensions, i.e. the other way around. The straightening theorem says we can write ##X=\sum_{k=1}^{n} c_k \dfrac{\partial }{\partial x_k}## as ##X=\dfrac{\partial }{\partial y_1}## by a suitable choice of local coordinates. This is clearly completely integrable since ##\left[\alpha\dfrac{\partial }{\partial y_1},\beta\dfrac{\partial }{\partial y_1}\right]=0.##
This is no longer automatically true in higher dimensions. If we have ##X=\sum_{k=1}^{n} c_k \dfrac{\partial }{\partial x_k}## and ##Y=\sum_{k=1}^{n} d_k \dfrac{\partial }{\partial x_k}## then we cannot assume that the two vector fields commute (which answers your very first question). We have a mess of mixed partial derivatives in ##[X,Y]##, so the order of integrations matters. Complete integrability means that we only want to integrate using ##X## and ##Y##. The mixed partial derivatives must therefore be expressable as linear combinations of ##X## and ##Y## again, i.e. ##[X,Y] \in \operatorname{span}\{X,Y\}.## That's the Frobenius theorem for two vector fields. For even more (linear independent) vector fields the theorem says that we have complete integrability (involutivity) if those vector fields build a Lie subalgebra of the tangent space. If you look at the pictures on Wikipedia then you will notice how hard it is to draw an image for the Frobenius theorem where ##[X,Y]\neq 0.## I assume that you won't find one in lecture notes either. We run out of dimensions too quickly to get an impression of, say four linear independent vectors.
If we consider both theorems in the light of differential equation systems, then the straightening theorem allows us a solution with only one integration along ##y_1.## If we have more independent variables, then we will have likely to integrate over more variables. But we aren't allowed to switch orders: ##[X,Y]=Z\neq 0.## That gives us the next variable ##Z## and so on. However, if ##[X,Y]=\alpha X+\beta Z## then we do not need new variables when we switch the order. This condition is equivalent to the statement that ##\operatorname{span}\{X,Y\}## is a two-dimensional Lie algebra, a Lie subalgebra of all possible vector fields. It allows us integrations without having to add new variables.
The equations are more complicated if we have to solve for more than two variables. However, as long as the corresponding vector fields build a Lie subalgebra, we at least don't have to add even more variables if we switch the order of integration.