Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
1answer
29 views

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 $...
1
vote
0answers
21 views

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 ...
0
votes
1answer
29 views

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 $(u_1,u_2)=((x_1,y_1,z_1),(x_2,y_2,z_2))$...
0
votes
1answer
15 views

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\...
2
votes
1answer
53 views

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 ...
0
votes
1answer
40 views

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-...
0
votes
0answers
14 views

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 $d_H$...
1
vote
0answers
34 views

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 ...
1
vote
0answers
38 views

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: ...
1
vote
0answers
57 views

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 ...
0
votes
1answer
36 views

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 ...
2
votes
1answer
41 views

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) ...
3
votes
1answer
38 views

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 ...
0
votes
0answers
15 views

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 ...
1
vote
0answers
42 views

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 ...
8
votes
1answer
83 views

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\...
1
vote
0answers
28 views

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 ...
0
votes
0answers
18 views

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 ...
1
vote
0answers
35 views

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 $...
2
votes
2answers
97 views

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 ...
3
votes
1answer
37 views

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 \...
1
vote
1answer
37 views

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 ...
3
votes
1answer
30 views

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}...
4
votes
1answer
55 views

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 ...
3
votes
0answers
48 views

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 ...
0
votes
0answers
34 views

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? ...
1
vote
1answer
50 views

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 ...
3
votes
0answers
31 views

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 ...
5
votes
0answers
57 views

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 ...
0
votes
1answer
171 views

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 ...
4
votes
0answers
82 views

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 ...
1
vote
0answers
41 views

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. ...
2
votes
1answer
59 views

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\...
3
votes
0answers
56 views

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 ...
1
vote
1answer
38 views

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 ...
1
vote
0answers
63 views

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 ...
4
votes
1answer
121 views

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)=...
4
votes
0answers
49 views

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?
1
vote
0answers
27 views

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 ...
3
votes
1answer
87 views

$df_1,…,df_k$ 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 $F:=(f_1,...,f_k):M\to \mathbb{R}^k$, ...
3
votes
0answers
94 views

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 $$\...
2
votes
3answers
170 views

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 ...
3
votes
1answer
54 views

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 ...
1
vote
1answer
51 views

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 ...
1
vote
0answers
33 views

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 ...
1
vote
1answer
55 views

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|...
3
votes
2answers
62 views

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 $Y$....
2
votes
0answers
45 views

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 ...
2
votes
0answers
30 views

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},\...
0
votes
1answer
50 views

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{\...