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# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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### How to check conditions for Liouville-Arnold theorem?

Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem: Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...
2answers
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### Symplectisation as a functor between appropriate categories

Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its ...
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### Example of overtwisted contact manifold with finitely many periodic Reeb orbits

Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits? An example is given by the irrational ellipsoid in $(\mathbb{R}^4,\omega_\text{st})$, which is not ...
1answer
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### Moment map interpretation of Einstein equation

Einstein's famous equation relates the geometry of a (4-dimensional) manifold to the matter content in that manifold. Is there a way to obtain Einstein's equation as a moment map? More precisely, ...
0answers
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Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ... 1answer 166 views ### One-Forms in Functional Space? I am presently reading this paper on covariant phase space and I have difficulty understanding the following formalism developed: In the paper (section$2.2$, pg.$12$), the authors have introduced ... 0answers 94 views ### A non-Kaehler manifold complex and symplectic in exactly one way Does there exist a closed connected smooth manifold that admits exactly one (up to biholomorphism) integrable complex structure and exactly one (up to symplectomorphism and rescaling) symplectic ... 1answer 112 views ### Lagrangian torus fibrations and Arnol'd-Liouville theorem Let$(X, \omega)$be a closed symplectic manifold of dimension$2n$and$\pi: X \rightarrow Q$a Lagrangian torus fibration. Let$F_q$denote the fiber at$q \in Q$. It is claimed in a paper of ... 1answer 135 views ### Log Calabi-Yau surfaces without maximal boundaries Let$X$be a smooth projective surface over$\mathbb{C}$,$D\subset X$is an effective divisor.$(X,D)$is a log Calabi-Yau pair if$K_X+D$is a principal divisor. The complement$M=X\setminus D$is a ... 1answer 104 views ### Orientable surface bundle Is it true that every orientable surface bundle can be made into a symplectic fibration?If yes, why? What about the particular case that$M$is a connected compact 4-manifold? 1answer 115 views ### Reference for the nearby Lagrangian conjecture for$T^*S^1$I am looking for a reference for the proof of the nearby Lagrangian conjecture of$T^*S^1$, that is, that every exact and compact Lagrangian submanifold of the cylinder is Hamiltonianly isotopic to ... 1answer 115 views ### Compactly supported symplectomorphisms of$D^2$I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of$D^2 \times D^2$being contractible. Consider the dimensional disk$D^2 \...

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