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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Poincaré duality and quantum (co)homology for $S^2 \times S^2$
Consider the symplectic manifold $(M,\omega):=(S^2 \times S^2, \omega_{FS}\oplus \omega_{FS})$. The homology $H_*(M;\mathbb{C})$ has as a basis the 4 non-trivial classes: $[pt],[M],A$ and $B$, where
$...
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Hamiltonian and Lagrangian correspondence
I'm trying to clarify how we get a Hamiltonian directly from a Lagrangian using the Legendre transform. Let me give some preliminaries for my question to make sense.
A Hamiltonian system is a triple ...
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Computing Hamiltonian vector field
Let $R=(R_1,R_2) \in \mathbb{R}^2$ and consider the product of $2$-spheres $S^2 \times S^2 \subset \mathbb{R}^3 \times \mathbb{R}^3$ with cartesian coordinates $(u_1,u_2)=((x_1,y_1,z_1),(x_2,y_2,z_2))$...
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The defining matrix of a symplectic matrix
Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me.
A $2n\times 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition:
$$M^T\...
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Path components of space of almost complex structures
Let $M$ be a closed oriented smooth 4-manifold which admits an almost complex structure. The Ehresmann-Wu theorem states that a class $c\in H^2(M;\mathbb{Z})$ is realizable as the first Chern class of ...
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Showing that a 2-form on an odd dimensional space is not degenerate
On an odd-dimensional space $\mathbb R^{2n+1}$ with coordinates $x_1...x_n;y_1...y_n;t$ consider the following 2-form:
$$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$
where $\omega^1$ is any 1-...
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When is the unit ball in $\text{Ham}(M,\omega)$ with respect to the Hofer metric contractible?
Associated to a symplectic manifold $(M^{2n},\omega)$ is the group of (compactly supported) Hamiltonian diffeomorphisms $\text{Ham}(M,\omega)$. This group is equipped with the famous Hofer metric $d_H$...
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Symplectic forms on $\mathbb{C}P^3$
Combining results of Gromov, McDuff, Taubes shows that any two symplectic forms on $\mathbb{C}P^2$ with the same total volume are symplectomorphic (e.g. to the Fubini-Study form $\omega_{FS}$). See ...
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Casimir Functions of Poisson Structure
I have just started to read Poisson Geometry and was working my way through some problems. This may be basic for many, but I am just trying to understand the way to work through it. The problem says:
...
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Symplectic manifolds (a problem in Arnold's classical mechanics book)
I'm working my way through V.I. Arnold's Mathematical Methods of Classical Mechanics. In the process I'm trying to wrap my head around one of the problems in the chapter on Symplectic Manifolds. In ...
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open submanifold of symplectic manifold by considering tangent spaces
This may be a slightly vague/ill-posed question. I apologize in advance.
Suppose we have a symplectic manifold $(M,\omega)$ and a smooth submanifold $U$.
Suppose at any point $m \in U$ we have that ...
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Does group of symplectomorphisms preserve Hopf fibration
Consider the closed ball of radius c := $B^4(c)$, with the usual symplectic form coming from $\mathbb{R}^4$.
In the proof of Lemma 2.1 of the following paper(https://arxiv.org/pdf/math/0207096.pdf) ...
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Infinitesimal generator of a smooth $S^1$-action over $\mathbb C$
Question: Consider the symplectic manifold $(\mathbb C, \omega_0 = \frac{i}{2} {\rm d}z \wedge {\rm d}\bar z)$ and a smooth $\Bbb S^1$-action over $\mathbb C$ given by $$(t,z) \mapsto t^kz$$ for some ...
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Existence of regular values of a toric action.
From a book of M. Audin (p.57)
Consider a torus $T$ acting effectively on a compact symplectic
manifold $(W,\omega)$ with moment map $\mu\colon W\to\mathfrak t^*$.
We know that in this case ...
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What is the connection between these two notions of a “vector field”?
I'm reading a physics text which dives into some mathematical concepts that are new to me. Consider a function $F(q,\dot q)$ (dynamical variable) where here $q$ and $\dot q$ represent sets of local ...
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Is the normal bundle invariant under homotopy?
Let $(M^{2n},J)$ be a complex $n$-manifold and $u:\mathbb{CP}^1\to M$ an immersed $J$-holomorphic curve. Then there exists a short exact sequence of vector bundles $$0\to T\mathbb{CP}^1\to u^{\ast}TM\...
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Equivalent definitions of Hyperkahler manifolds.
I am reading the paper HYPERKAHLER METRICS ON COTANGENT BUNDLES OF ¨
HERMITIAN SYMMETRIC SPACE by OLIVIER BIQUARD AND PAUL GAUDUCHON.
Suppose $M$ is a manifold with a triple $(g,I,J)$ where g is a ...
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Definition of bivector field on a manifold.
I am reading this article:
http://arxiv.org/abs/1112.5037v1 . In this, it defines a symplectic manifold as a manifold equiped with a nondegenerate bivector field $\pi$ that is Poisson. l want to ...
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Method of generating functions; Do we always get a symplectomorphism?
I am reading this very popular introduction on symplectic geometry by Cannas. The story goes as follows: Having manifolds $X_1, X_2$ , we want to construct symplectomorphisms between $M_1,M_2$ (where $...
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2answers
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Existence of harmonic symplectic structure on a symplectic manifold (which is cohomologue to the initial symplectic structure)
Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric.
Is there a symplectic structure $\omega '$ which is a harmonic $2$-form? Can one choose such ...
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1answer
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Critical points of action functional are periodic Reeb orbits
Let $(M,\alpha)$ be a contact manifold. It is well-known that the critical points of the action functional, given by
\begin{align}
A: C^\infty(S^1,M) \to \mathbb{R} \\
\gamma \mapsto \int_\gamma \...
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1answer
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How do we know the solutions to the Hamilton Jacobi equation are seperable?
So I usually start my questions here with "I'm sure this is a stupid question", but I am even more convinced than usual that it's true this time. We are trying to find a solution $S$ of the Hamilton ...
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1answer
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Two definitions of the space of pointed Riemann surfaces
I was reading Wendl's notes on closed holomorphic curves, and could not seem to figure out the following assertion. I think I am missing some fact or the other about Riemann surfaces.
Let $\mathcal{M}...
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1answer
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Bubbling off in Deligne-Mumford compactification
I'm trying to understand an argument in Casim Abbas' 'An Introduction to Compactness Results in Symplectic Field Theory'. Here's where I stuck: (It's on chapter 3.3.2, Adding additional marked points ...
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Model for symplectic geometry
An almost symplectic structure on a smooth even dimension manifold $M$ can be viewed as a reduction of structure group $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$ for the principal frame ...
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Prerequisites for symplectic reduction and geometric quantization
I am setting out to read Guillemin's "Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-spaces" and am not sure my background is enough. What are the prerequisites of reading this book? ...
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1answer
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Metaplectic group is the unique double covering of symplectic group
The proof that $Mp(2n,\mathbb{R})$ is the unique connected double cover of $Sp(2n,\mathbb{R})$ uses the fact that the fundamental group of the latter is infinite cyclic (the integers). I have not ...
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Cotangent bundle of the flag variety $T^\ast \mathcal{B}$ as complex and real symplectic manifold?
Let $G_\mathbb{C}$ be a semisimple Lie group and $G$ an compact real form of it.
For example, take $G_\mathbb{C} = SL(n,\mathbb{C})$ and $G = SU(n)$. Pick a Borel subgroup of $G_\mathbb{C}$.
The ...
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Where does the symplectic structure on coadjoint orbits of Lie groups on their Lie algebras come from?
I have read in several places that if $\Omega$ is the coadjoint orbit of $\zeta \in \mathfrak{g}^*$, the map from $G \to \Omega$ that sends $g \mapsto Ad^*(g)(\zeta)$ gives a surjection, and taking ...
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the manifold of $ D=\{x,y,z,w \mid xw-yz \le 0\}$
What geometric object $({{x,y,z,w}})$ is defined by the equation $xw-yz\le 0$ in $\mathbb R^4$? Using parametrics, can we describe this manifold? Is it possible to prove a diffeomorphism between this ...
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Lie algebra-valued differential forms, exactness, closedness, and the Moyal product
This is my attempt to prove something (I'm not even sure if it's true to begin with) using somewhat loose arguments. I present here all the steps and ideas and I would be extremely grateful if someone ...
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Geometrical Quantization and Connections
Really I think this question boils down to what the physical significance of a connection is.
Physically, we can think of a symplectic manifold $(\mathcal{M},\omega)$ as essentially a phase space.
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1answer
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Non-linear path between symplectic forms in $\mathbb{R}^4$
Give an example of a pair of symplectic forms $\omega_0,\omega_1$ in $\mathbb{R}^4$, which:
$1)$ induce the same orientation (i.e., the volume forms $\omega_0\wedge\omega_0$ and $\omega_1\wedge\...
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Vertical $J$-holomorphic spheres
Let $\pi:M\to N$ be a smooth fiber bundle with fiber $F$ and let $g$ be a Riemannian metric on $M$. Using $g$ we may split the tangent bundle of $M$ as $$TM\cong TF\oplus \pi^{\ast}TN,$$ where $TF$ is ...
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1answer
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Cotangent lift of the commutator of 2 vector fields
Given a smooth manifold $X$ and a vector field $u$ on $X$, we can perform a cotangent lift to get a vector field $\tilde u$ on $T^*X$. Explicitly, this can be done by thinking of $u$ as defining an ...
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Symplectic structure on a covering manifold
How to show that a covering manifold of a symplectic manifold admits a symplectic structure? More precisely, let M be a $2n$-manifold and $(N, \omega)$ be a symplectic 2n-manifold. If there exists a ...
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1answer
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Conormal bundle and lagrangian submanifold
Let $Q_1^{n_1},Q_2^{n_2}$ be smooth manifolds, $\phi:Q_1\to Q_2$ a smooth map and:
$$R_\phi:=\{(x, \xi, y,\eta)\mid y=\phi(x), \xi=(d\phi)^*\eta\}\subset T^*Q_1\times T^*Q_2$$
$$\text{graph}(\phi)=...
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Hyperbolic lagrangian in a symplectic $6$-manifold
Is there an example of a closed symplectic $6$-manifold and a closed lagrangian sub-manifold, which is diffeomorphic to a hyperbolic $3$-manifold?
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Is it possible to define an integration measure on a presymplectic manifold induced by the presymplectic structure?
Let us consider a symplectic manifold $(M,\omega)$. By means of this structure, I can easily define a symplectic measure $\mu(M)$ on $M$ by considering the "right number" of wedge product of the ...
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1answer
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$df_1,…,df_k$ linearly independent $\Rightarrow \frac{\omega^n}{n!}=df_1\wedge…\wedge df_k\wedge\sigma$
Let $(M,\omega)$ be a symplectic manifold, $H,f_1,...,f_k\in C^\infty(M)$ non-zero functions with $\{H,f_i\}=0$. If $c\in\mathbb{R}^k$ is a regular value of $F:=(f_1,...,f_k):M\to \mathbb{R}^k$, ...
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Pulling back symplectic structure to $TQ$ - why is a certain term (not) zero?
Let $(Q,g)$ be a pseudo-Riemannian manifold, $(q^1,\ldots, q^n)$ be local coordinates in $Q$ and $(q^1,\ldots, q^n, v^1,\ldots, v^n)$ the induced tangent coordinates in $TQ$. I wanted to check that $$\...
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3answers
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Darboux theorem for $2$-dimensional manifolds
Let $M$ be a $2$-dimensional manifold. Using the fact that every non-vanishing $\alpha\in\Omega^1(M)$ can be written as $\alpha=fdg$ locally for convenient smooth functions $f,g$, prove Darboux's ...
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1answer
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If $F^*\alpha=\alpha$, then $F$ is the cotangent lift of some $\phi$
Let $Q$ be a smooth manifold, $\alpha\in\Omega^1(T^*Q)$ the tautological $1$-form and $\omega:=-d\alpha$. Suppose there is a diffeomorphism $F:T^*Q\to T^*Q$ such that $F^*\alpha=\alpha$. Prove there ...
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1answer
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Euler field is invariant under the cotangent lift
Let $Q$ be a smooth manifold, $\alpha\in\Omega^1(T^*Q)$ the tautological $1$-form and $F:T^*Q\to T^*Q$ a symplectomorphism such that $F^*\alpha=\alpha$ ($F$ could be the cotangent lift of some ...
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The dual Lie algebra in the context of Hamiltonian action
Let $t$ be a Lie algebra. what is the precise structure of the dual lie algebra $t^*$ when we are considering the momentum map associated to a Lie group action on a symplectic ...
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Tautological $1$-form does not depend on coordinates.
In Ana Cannas' Lectures on Symplectic Geometry (page $9$), the tautological $1$-form on the cotangent bundle $T^*Q$ is defined on local coordinates $(U,x_1,...,x_n,\xi_1,...,\xi_n)$ by:
$$\alpha\big|...
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2answers
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Consider symplectic vector fields $X,Y$ and a symplectic connection $\nabla$. Is $\nabla_{X}Y$ symplectic?
Consider a symplectic manifold $(M,\omega)$, together with a symplectic connection $\nabla$, i.e. a torsion-free connection such that $\nabla{\omega} = 0$. Fix two symplectic vector fields $X$ and $Y$....
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contact geometry good reference
I am working on which seems to be a classical result : the isomorphism between the cotangent bundle of projective space (with zero section removed) and the cotangent bundle of its dual projective ...
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Symplectic reduction of a linear symplectic space.
I want to know what is the symplectic reduction of a symplectic linear space.
Suppose $(V,\omega)$ is vector space with a nondegenerate bilinear form $\omega$. We can assume it is $(\mathbb R^{2n},\...
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1answer
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Kähler form and the condition of positive definiteness
I know that a Kähler form $\omega$ looks like:
$$\omega=\frac{i}{2}\sum_{j,k}h_{jk}dz_j\wedge d\overline{z}_k$$
such that $\overline{h_{jk}}=h_{jk}$ ($\omega$ is real), $\partial\omega=0$, $\overline{\...
