Questions tagged [symplectic-geometry]
Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
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What could have been a good motivation / intuition behind the Dorfman Bracket condition?
I'm a graduate student in Mathematics, currently learning Dirac structures in Differential Geometry. However, I cannot make sense of how the Dorfman bracket condition comes all of a sudden. It ...
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Understanding the symplectic double extension
Let $(B,ω')$ be a symplectic Lie algebra. and the $\delta$ is a symplectic derivation, and $z\in B$.
The double extension of $B$ is: $g=Re ⊕ B ⊕ Rd$ as:
Central Extension : $I = Re ⊕ B,ドルof $B$ with ...
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Families of Almost Complex Structures that Determine a Symplectic Form [closed]
If you have a given symplectic form $\omega \in \Omega^{2}(M)$ on a smooth manifold $M$ it determines a space $\mathcal{J}(M,\omega)$ of compatible almost complex structures. Now, given a manifold $M$ ...
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How to prove the Novikov covering is a covring?
Consider a connected symplectic manifold $(W,\omega),ドル fixed a point $x_0\in W$.In Hamiltonian Floer cohomology theory, we want to construct a Action functional on the contractible loop space $\...
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The pullback of the Fubini-Study form on $\mathbb{CP}^{n}$ is the standard scaling-invariant symplectic form on $\mathbb{C}^{n+1}\setminus\{0\}$
Consider the complex projective space $\mathbb{CP}^{1}$. We work locally in the homogeneous coordinate chart
$$
[z_{0}:z_{1}]\mapsto u:=\frac{z_{1}}{z_{0}}.
$$
In these coordinates, the Fubini-Study ...
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Hodge star of a $(0,q)$-form wedged with the symplectic form on a symplectic manifold with a compatible almost complex structure
Let $(X,\omega)$ be a symplectic manifold of dimension 2ドルn$ and $J$ be an almost complex structure compatible with $\omega$. In particular, $g(v,w):=\omega(v,Jw)$ defines a Riemannian metric, and ...
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Closed monotone 6ドル$-manifold with semisimple $QH^*$ but no nondisplaceable Lagrangian torus?
Let $(M^{6},\omega)$ be a closed, simply-connected, monotone symplectic manifold, i.e.
$$
[\omega]=\lambda,円 c_{1}(TM)\quad\text{in }H^{2}(M;\Bbb R),\qquad\lambda>0.
$$
For many monotone manifolds (...
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Poisson structure on dual of lie algebra
Let $G$ be a Lie group and $\mathfrak g$ it's Lie algebra. I'm trying to check that the Poisson structure on the quotient manifold $\mathfrak g^* \cong T^*G/G$ is given by
$$\{f,g\}_{\mathfrak g^*}(\...