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Given a smooth manifold $\mathcal{M}$ that is a fibre bundle over two different base spaces, i.e., there are $\Pi_1:\mathcal{M}\rightarrow B_1$ and $\Pi_2:\mathcal{M}\rightarrow B_2$, if I can prove t …

Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n …

I already asked this in the physics forum but without much attention, so I thought it might attract more attention here. Is there an integral expression for the Poisson bracket that can be derived fro …

Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image …

Given a $2n$-dimensional symplectic manifold $\mathcal{M}$ and two different lagrangian fibrations $\pi_1:\mathcal{M}\rightarrow \Gamma_1$ and $\pi_2:\mathcal{M}\rightarrow \Gamma_2$, with $\Gamma_1, …

Assume $\mathcal{M}$ is a compact symplectic $2n$-dimensional manifold with a Hamiltonian action of the torus $\mathbb{T}^n$. Given a family of functions $F=(f_1,\ldots,f_n)$ defining an integrable sy …