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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An Hamiltonian diffeomorphism is also a Poisson diffeomorphism
Let $(M,\{-,-\})$ be a Poisson manifold.
An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth ...
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Hamiltonian diffeomorphisms on a Poisson manifold
Let $(M,\{-,-\})$ be a Poisson manifold.
An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth ...
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Roadmap for learning symplectic geometry, starting from differential geometry
I am a Physics Major. I have done an undergraduate level course of differential geometry. I want to get into symplectic and Riemannian geometry, but I would like to start over from differential one ...
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A vector field is Poisson iff its flow is a Poisson diffeomorphism
I'm studying Poisson geometry from "Lectures on Poisson Geometry" (Crainic, Fernandes, Marcut). Let $M$ be a Poisson manifold and let $X$ be a vector field on $M$. We define $X$ to be ...
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Reconciling two characterizations of symplectic potential
The reference text is N.M.J. Woodhouse's Geometric Quantization.
Let $Q$ be a smooth manifold with coordinates $\{q_i\}$, and $M=T^*Q$ be its cotangent bundle with coordinates $(p_i, q_i)$ (so that $\{...
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Understanding the Correspondence of Sections to Right and Left Invariant Vector Fields in Ping Xu's 'Momentum Maps and Morita Equivalence
I want to understand this passage on the articule: Ping Xu. "Momentum Maps and Morita Equivalence".
By $A → P$ we denote the Lie algebroid of $Γ\rightrightarrows P$, where the anchor map is ...
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The map not induced by hamiltonian flow.
Some time ago I needed to solve the following problem:
We have a torus with $\mathbb{T}^2$ with coordinates $(x \ mod \ 1, y \ mod \ 1)$ and symplectic form $\omega = dx \wedge dy$. Given function $F :...
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A way to realize the $H^1(B;\mathbb{Z}/2)$ action on Pin- and Spin- structure
A Pin-structure on a real vector bundle $F^n\to B$ is a $Pin_n$ principal bundle $P^\#\to B$ with an isomorphism $P^\#\times_{Pin_n}\mathbb{R}^n\cong F$. A familiar property shows that $H^1(B;\mathbb{...
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What properties does the flow of a Hamiltonian vector field have compared to the flow of a symplectic vector field?
Let $(P,\omega)$ be some real $2n$-dim symplectic manifold. A symplectic vector field, $Z\in\mathfrak{X}_{sp}(P)$, is one for which the Lie derivative satisfies $\mathcal{L}_Z \omega =0$ or, ...
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Preserving the symplectic 2-form vs phase space volume
Say I have a Hamiltonian system of $N$ particles in 3D-3V phase space. I'm using some sort of update scheme taking the system from $t^{n-1}$ to $t^{n}$ to $t^{n+1}$. I want to know if the update ...
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Poisson bracket with the symplectic form
Consider the following extract from a text:
I can't seem to varify the second equality in the definition. Maybe the author meant to write $(\frac{\partial f}{\partial \mathbf{x}})^T J \frac{\partial ...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.3-4. (fubini study form is closed)
In exercise 5.3-4. in Marsden's book I'm asked to
prove that $\mathbf d \Omega^{fs} = 0$ on $\mathbb P \mathcal H$ directly,
where $\mathbb P \mathcal H$ is an arbitrary projective Hilbert space (...
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Proof of Square Zero Property for the Total Differential in the de Rham Double Complex of a Lie Groupoid
I am trying to understand the de Rham double complex associated to a Lie groupoid, and I am having trouble proving a fundamental property of the total differential, i.e., that it squares to zero.
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Proving isotropy of a groupoid multiplication graph in a quasi-presymplectic groupoid
I'm studying quasi-presymplectic groupoids and I've come across the following proposition which I'm finding challenging to prove. Any help or guidance would be greatly appreciated.
Given a Lie ...
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Approximating the symplectic flow of Hamiltonian systems with $H = \frac{1}{2}p^TM^{-1}p + V(q)$
Consider a symplectic map $\phi_t(p,q)\in \mathbb{R}^{2n}$, that solves or approximates the Hamiltonian system $$\dot{p} = - \nabla V(q),\quad\dot{q} = M^{-1}p,$$ for Hamiltonian $H = \frac{1}{2}p^TM^{...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.2-3. (symplectic map is immersion)
I'm either confused with the definition of symplectic forms / immersions or the way exercise 5.2-3 in Marsden's 'Introduction to Mechanics and Symmetry' was stated.
It reads as follows
Exercise 5.2-3....
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conic Lagrangian submanifold with boundary
I'm reading through the paper "Lagrangian Intersection and the Cauchy Problem" by Melrose and Uhlmann, and I'm having trouble with the definition of intersecting pair of Lagrangian manifolds....
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Geometric Intuition for Hamiltonian Actions
Let $G$ be a connected Lie group acting on the symplectic manifold $(M,\omega)$. In the definition of a Hamiltonian action one requires that the moment map $\mu\colon M\xrightarrow{} \mathfrak{g}^\ast$...
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When does a smooth projective toric variety admit a symplectic structure?
I'm trying to understand the relationship between toric varieties and their associated polytopes. There is a very crisp result in the case of symplectic toric varieties, namely that there is a 1:1 ...
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general form of group elements of the symplectic group
For the unitary group $U(n)$, its group elements are of the form $e^{iH}$, with $H$ being a $n\times n $ hermitian matrix.
Do we have a similar expression for the group element of the symplectic group ...
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Symplectomorphism taking a Lagrangian to its isotopic copy
Suppose $\iota_0:L\xrightarrow{}M$ is a compact Lagrangian submanifold in a symplectic manifold $(M,\omega)$. Let $\iota_t:L\times I\xrightarrow{} M$ be an isotopy and denote $\iota_1(L)$ by $L’$. I ...
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Hamiltonian vector field and fibers
Let's assume that $(M,\omega)$ is a symplectic manifold, $N$ an arbitrary manifold and we have some differentiable map $\phi:M\to N$, with conneted fibers.
If we denote by $\phi^{-1}(p)$ the fiber ...
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Explicit solution for Harmonic Oscillators "solution in quadratures"
So I was reading about integrable systems and of course the harmonic oscillator in one dimension was mentioned as an example. After some simple calculations the author ended up with the following ...
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Are all equal area map projections local symplectomorphisms or not all of them?
I'm not a mathematician (I am a cartographer) and I want to know wether all equal area map projections are local symplectomorphisms or not. It is no doubt that the surfaces of a 2-sphere and Earth ...
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Can we recover dynamics from coisotropic reduction?
I am trying to understand the significance of coisotropic reduction in a mechanical way.
Let $(M,\omega)$ be a symplectic manifold and $i: N \hookrightarrow M$ a coisotropic submanifold i.e. $T_qN^{\...
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When are Infinitesimal $\mathfrak{g}-$actions Lie homomorphisms?
A short background from Smooth Manifolds, by Lee.
Given $G$ a Lie group and a right $G$-action on a smooth manifold $M$, we have an infinitesimal $\mathfrak{g}:=Lie(G)$-action over $M$. That is, a Lie ...
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Reflexivity of infinite dimensional vector spaces and existence of non degenerate bilinear forms
How can an infinite dimensional vector space like (most of) the Lebesgue or Sobolev spaces even be reflexive? If an infinite dimensional vector space cannot be isomorphic to its dual space, the how ...
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How big can a wedge of 2-forms be?
The comass of a 2-form $\alpha$ is the maximal value of $\alpha(u,v)$
for a pair of unit vectors $u,v$. The symplectic form $\alpha$ on
$\mathbb R^{2n}$ has the property that $|\alpha^{\wedge n}| = n!...
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The natural action of $Sp(V)$ on $V$ is Hamiltonian with the co-moment map given by: $\tilde{\mu} : sp(V) \to C^\infty(V); A \mapsto \tilde{\mu}_A$,
Given a symplectic vector space $(V, \Omega)$, consider the Lie group $G := Sp(V)$, consisting of all symplectomorphisms $\phi: V → V$. Show that if $(·, ·)$ is an invariant inner product, then the ...
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Examples of degenerate (Floer) Hamiltonian orbits?
Say I have a periodic hamiltonian $H: S^1 \times M \longrightarrow M$ defined on a symplectic manifold $M$.
Then, a $1$-periodic Hamiltonian orbit of $H$ is the same thing as a fixed point of the ...
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Prove that the natural action of $G$ on $\mathbb{C}^2$ is Hamiltonian with the moment map $µ$
We are given the complex special linear group $SL(2, \mathbb{C})$ which is a Lie group $G$, with its corresponding Lie algebra $sl(2, \mathbb{C})$ denoted by $g$. We are to prove that the natural ...
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Prove that if $M$ is simply connected and $H^2(g) = 0$, then $M$ is symplectomorphic to an adjoint orbit.
A manifold $M$ is said to be homogeneous if there exists a transitive action $G ↷ M$. Let $(M,ω)$ be a homogeneous symplectic manifold, i.e., there exists a transitive and symplectic action $G ↷ M$. ...
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What is a naive reason for "wrapped" Floer homology?
Many first courses on Floer homology, which I have seen, use the question of nondisplaceability as a motivation. People usually begin with saying that they want to study the question whether or not, ...
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Exterior Derivative and Lie Derivative on infinite dimensional manifolds
Lately I have been trying to understand the chapter in Abraham and Marsden's Foundations of Mechanics on infinite-dimensional Hamiltonian systems. Now that I've finally got a feeling for the canonical ...
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How does a totally complex polarization induce a Kähler structure of the symplectic manifold
A totally complex polarization of a symplectic manifold $(M,\omega)$ is a subbundle $F$ of the complexified tangent bundle $TM_{\mathbb C}:=TM\oplus iTM$ such that
$F$ is integrable in the sense that ...
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Equivalence of Hyper Kahler structure and three non isomorphic symplectic structures
Suppose that $(M,g, I, J,K)$ is a Hyper Kahler manifold, then the two forms:
\begin{alignat*}{3}
\omega_I(v,w)=&g(Iv,w),\qquad &&\omega_J(v,w)=g(Jv,w),\qquad &&\omega_K(v,w)=...
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Lagrangian field theories define a canonical $\mathbb R$-torsor/$\mathbb R$-bundle
Why is it possible to define an $\mathbb R$-torsor/$\mathbb R$-bundle out of the Lagrangian and to construct a projective limit? (I'm guessing this has to do with time evolution acting on the space of ...
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What can we say about the divergence of Hamiltonian vector fields?
Let $M$ be a smooth $n$-dimensioanl manifold. To set some notations
$C^\infty(M)$ denote smooth functions $M \to \mathbb{R}$
$\Omega^k(M)$ denote $k$-forms on $M$
$\tau(M)$ denote vector fields on $M$...
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Lefschetz fibration on product of surfaces
If I understand the literature correctly, then every symplectic 4 manifold (potentially up to connected sum with complex projective space) admits a lefschetz fibration with codomain a two sphere.
In ...
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isotropy of the cotangent lift of a group action
Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the phase space (i.e. coordinates and momentum), which is the appropriate cotangent bundle of the ...
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What is the geometry of $L=\lbrace W \subset V:\text{$W$ is Lagrange subspace of $V$} \rbrace$?
I know the following statement:
Theorem. Let $(V,\omega)$ be a finite dimensional symplectic linear space. Then the symplectic group $Sp(V)$ of $V$ transitively acts on the set $L=\lbrace W \subset V:...
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Show that exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$
I need to prove that for $\pi: E → M$ a vector bundle then exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$. I star with definitions: Let be $\pi : E → M$ a vector bundle, $\pi$ is a ...
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criteria for when the total space of a vector bundle over a toric variety is a toric variety itself
notice that this is not the same as being a toric vector bundle.
also, i'm not interested in the projectivization.
rather, i'm perfectly fine working with smooth quasi-projective toric varieties.
...
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Symplectic vector fields form a Lie Algebra
I am trying to prove that symplectic vector fields with the usual Lie bracket of vector fields form a Lie Algebra. Recall that a vector field $X$ on a symplectic manifold $(M,\omega)$ is symplectic ...
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Smooth points in fibers of moment map
Let $G$ be a connected Lie group acting on a smooth manifold $X$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the induced action on the cotangent space $T^*X$ which is always Hamiltonian ...
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Symplectic vs contact diffeomorphisms and Lagrangian vs Legendrian submanifolds
Let $X,Y$ be symplectic manifolds of dimensions at least 4 (in dimension 2 this is clearly false as pointed out by @Arctic Char in the comments) with symplectic forms $\omega_X,\omega_Y$ and let $f\...
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Computing the virtual dimension of the moduli space $M(L, \beta)$
In Auroux's paper "MIRROR SYMMETRY AND T-DUALITY IN THE COMPLEMENT
OF AN ANTICANONICAL DIVISOR", in section §3 p.7 the author presents the moduli space $M(L, β)$ of J-holomorphic discs with ...
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Type A quivers and flag varieties from Kirilllov
I'm trying to understand section 10.7 of Kirillov's "Quiver representations and quiver varieties", which shares its title with this question. We let Q be a type A quiver of length $\ell$. We ...
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Does a co-exact 1-form generate a Hamiltonian vector field?
Let $M$ be an $n$-dimensional Riemannian orientable manifold. Denote the space of vector fields on $M$ by $V(M)$ and of $k$-forms on $M$ by $\Omega(M)$.
Let $\xi = \delta \beta$ be a co-exact 1-form ...
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Searching for a bound for the integral of a 1-form along a loop.
Consider the submanifold $M$ of $\mathbb R^8$, with coordinates $(x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4)$, defined by the following equations
$$x_1^2+y_1^2+x_2^2+y_2^2=1,$$
$$x_3^2+y_3^2+x_4^2+y_4^2=1,$$
$$...
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