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May
4
comment About the geometry of completely integrable systems
Dear DamienC: Thank you very much, "perhaps" I have realized what I was missing. This dialogue with you was very useful to me.
May
4
comment About the geometry of completely integrable systems
Dear DamienC, By the theorem of Carathedory-Jacobi-Lie I am able to extend $f_1,\ldots, f_n$ to a local symplectic chart $f_1,\ldots,f_n,g_1,\ldots,g_n$ around an arbitrary point $x$. So taking level sets of $g_1,\ldots,g_n$, I get a Lagrangian submanifold $\Sigma$ passing through $x$ and transversal to $\mathcal{F}$. But I am unable to exclude that the intersection of $Sigma$ with some leaf of $mathcal{F}$ has $x$ as accumulation point.
May
4
revised About the geometry of completely integrable systems
corrected spelling
May
4
comment About the geometry of completely integrable systems
Dear DamienC, thanks a lot for your attention. About the question: my doubt arose just because I was unable to conclude the reported statement, or to construct a counterexample, within the hypothesis in the initial setting. Thanks also for the linked first chapter, surely it will be useful.
May
3
asked About the geometry of completely integrable systems
Apr
29
comment Does an abelian group acting on a riemaniann manifold define an othogonal foliation?
I have added a fifth tag differential-geometry as more comprehensive, Are you contrary?
Apr
29
revised Does an abelian group acting on a riemaniann manifold define an othogonal foliation?
edited tags
Apr
29
revised To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
another attempt to fix an error in TeX; deleted 4 characters in body
Apr
29
revised To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
added 30 characters in body
Apr
29
revised To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
deleted 13 characters in body; deleted 12 characters in body
Apr
29
revised To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
Trying to correct LaTeX
Apr
29
revised To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
trying to correct LaTeX
Apr
29
answered To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
Apr
28
comment Foliating R^3 with straight lines
@Brendan Foreman: Nice comment. Wouldn't you develop it as an answer?
Apr
28
revised The Jacobi Identity for the Poisson Bracket
corrected wrong notation
Apr
27
comment The Jacobi Identity for the Poisson Bracket
@Josè Figueroa-O'Farrill: Thanks for the attention. In my answer, I tried to highlight this point that was already in your answer. But my approach is lowbrow with respect to the highbrow answer of Jonathan.
Apr
25
comment Lie group operation and tangent vectors
About some of following answers: Excuse me, but is not the content of the question exactly to prove that $T_{e,e}\mu(\xi,\eta)=\xi+\eta$ for any $\xi,\eta\in T_eG$? So we should not appeal to it in a proof. But we should point out that this expression is just a consequence of the canonical identification of $T(G\times G)$ with the direct product $TG\times TG$.
Apr
24
comment The Jacobi Identity for the Poisson Bracket
@Josè Figueroa-O'Farrill: When, for an arbitrary almost-symplectic manifold, we again construct the bracket, is correct that $d\omega(X_f,X_g,X_h)$ is equal to the Jacobiator $J(f,g,h)$? or I am making same mistake?
Apr
24
revised The Jacobi Identity for the Poisson Bracket
added 581 characters in body
Apr
23
comment The Jacobi Identity for the Poisson Bracket
I find your answer to be the right complement to Jose's answer. Thanks.