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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Subspace of a quotient space is Lagrangian

Let $(V, \omega)$ be a symplectic vector space. Let $W \subset V$ be coisotropic and let $U \subset V$ be Lagrangian. Show that the quotient space $ ((U \cap W)+W^{\perp})/W^{\perp} \subset W/W^{\perp}...
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Gromov width of weighted projective planes

I am interested in knowing the Gromov width of (the complement of the three orbifold points of) weighted projective planes $\Bbb{CP}(a,b,c)$. Let me emphasize that I am mainly interested in upper ...
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Symplectic Reduction of 3-D Chern Simons Theory

So, I'm new to gauge theories and symplectic reduction and was trying to analyze the Chern Simons theory in three dimensions. I have a few questions regarding the steps towards reduction. First off, ...
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Calculation of the Lie derivative of the fundamental one-form in three different ways

I am a physicist who is trying to understand more formal differential geometry in the context of classical mechanics. I came across three ways of computing the Lie derivative of differential one-forms....
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Question about tangent space of a level set of function.

In McDuff and Salamon's book p343, they claimed: The hypersurface $\Sigma$ is of dimension $2n−2$ and the tangent space at $p$ is $T_p\Sigma= \{ v ∈ T_pM | dG(p)v = dH(p)v = 0\}.$ Here, M is a ...
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Question about image point Hamiltonian function [closed]

Are all points in the image of a Hamiltonian function regular value, Why? If not, is 0 a regular value?
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Intersection number of two simple closed curves on a torus

The isotopy classes of oriented simple closed curves on the torus, are classified by primitive vectors in $\mathbb{Z}^2$, namely, $\{(p, q)| gcd(p,q) = 1 \}$. Prove that for two simple closed curves, $...
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Why should we introduce symplectic width?

I'm a beginner in Symplectic topology and I'm learning the book Introduction to Symplectic Topology. In the second chapter of this book, they introduce the notation symplectic width $w_L(A)$ for ...
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Calculating Lie algebra representation in coherent state basis from Kaehler potential

In this paper by Onofri, they mention an explicit formula (eq. 13) for calculating a Lie algebra representation on the space of coherent states. $$ [\hat{X}\psi](z) = (H(z,\bar{z}) + iXf(z,\bar{z}))\...
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Legendrian in the cosphere bundle

Let $M$ and $N$ be dual lattices of rank $n$, i.e. $N\cong \text{Hom}(M, \Bbb{Z})$. Let $M_\Bbb{R}=M\otimes \Bbb{R}$ and similarly $N_\Bbb{R}$. Then $T^n\cong M_\Bbb{R}/M$ is the $n$-torus and we can ...
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Why $SO(3)$ action is exact?

In McDuff&Salamon's book(P207 Example 5.3.1), they claim that Consider the diagonal action of $SO(3)$ on the phase space $\mathbb{R}^6 $ (with the standard symplectic structure) by $\psi_\Phi(x,y)...
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Triviality of Sp(TM)

Let M be a symplectic manifold of dimension $2n$ and $TM$ denote its tangent bundle. Let Sp(TM) denote the bundle over M whose fibers are linear maps preserving symplectic structure on M. Is Sp(TM) ...
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Question about calculation the equality of Hamiltonian vector field

In McDuff&Salamon's Book, they claimed that since $X_{g^{-1}\xi g}=\psi^*_g X_\xi$, hence the functions $H_{g^{-1}\xi g}$ and $\psi^*_g H_\xi$ generate the same Hamiltonian vector field. Here, $...
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Question about orbit of $S^1$ action on symplectic manifold

In Salamon&McDuff's book, they claim Denote by $\alpha_X \in H_1(M,Z)$ the homology class which is represented by the orbits of the $S^1$ action. All these orbits are homologous because M is ...
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Question about Hamiltonian action

In Salamon&Mcduff's book p195, they claim The question obviously arises as to which circle actions are Hamiltonian. An obvious necessary condition is that a Hamiltonian action on a compact ...
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Rescaling the contact form affects the Reeb vector field

Let $(M,\xi)$ be a contact manifold with contact form $\alpha$ and Reeb vector field $R$. The contact form can be rescaled by any positive smooth function $f$ to obtain a new contact form $\alpha_f:=f\...
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Question about Nijenhuis tensor

In Mcduff&Salamon's Book, they assume $\partial_X Y=\sum_j \xi_j \frac{\partial Y}{\partial x_j}$, when $X=\sum_j \xi_j \frac{\partial }{\partial x_j}$. There is a conclusion I can not understand ...
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Symmetric Symplectic matrix preseve Lagrangian space

Suppose $\psi$ a symmetric symplectic matrix(identify to symplectic map in $\mathbb{R}^{2n}$), and $V$ a Lagrangian subspace in $\mathbb{R}^{2n}$. Do we have $\psi(V)=V$? I'm trying to prove the ...
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Any symplectic vector bundle is a complex vector bundle

I want to understand the statement in the title : Any $Sp(2n,\mathbb R)$ principal bundle aka symplectic real vector bundle is a complex vector bundle. I understand any real vector bundle can be given ...
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Integrability of Euler top

I'm taking an integrability course, it's a bit too fresh and my physics a bit rusty. I'm stuggling with an exercise about the Euler top. I'd appreciate a hand with some questions, or a least some tips....
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Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?

Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form? If $\omega$ is a symplectic form, then the real cohomology class $[\omega]$...
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Basic questions about Contact metric structure

Let $(M,\xi)$ be a coorientable contact $(2n+1)$-manifold with $\alpha$ and $R$ as contact form and Reeb vector field, respectively. A contact metric structure on $(M,\xi)$ is a pair $(J,g)$ where $J$ ...
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Time derivative of Pushforward equality

In Audin and Damian's "Morse Theory and Floer Homology", In Prop. 5.4.5, there is a statement about the time derivative of a pushforward that I am having trouble understanding. In the last ...
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Definition of fiberwise action

I'm studying Margaret Symington's paper on torus fibrations, and I came across this action: We have a regular Lagrangian Fibration $\pi : (M^{2n}, w^2) \longrightarrow B$ ($(M^{2n}, w^2)$ is a ...
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Equivalence of $\mathcal{L}_{\Pi^{\sharp}(df)} = 0$ to $[\Pi, \Pi]_{SN} = 0$ for Poisson bivectors?

A bivector field $\Pi$ induces a Poisson structure if $[\Pi, \Pi]_{SN} = 0$, where the bracket is the Schouten-Nijenhuis bracket. An $n$-vector field $\Pi$ induces a Nambu-Poisson structure of order $...
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Symplectic structure on tensor products and duals

Suppose $(V,\omega_V)$ and $(W,\omega_W)$ are two finite-dimensional real symplectic vector spaces. I want to know how I can define new symplectic structures. I know $V\oplus W$ comes with a natural ...
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can $g(x)$ be associated to $\sum_n g(n^{-s})$?

While thinking about re-constituting real symplectic manifolds into complex ones via tori fibrations and mirror symmetry I thought about a possible association of objects: $g(x) \to \sum_n g(n^{-s})\...
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Space of Contact Forms in $\mathbb{R}^3$ is contractible. Some fact about Volume Forms

To show that a space of contact forms in $\mathbb{R}^3$ is contractible, it's enough to show that $$\lambda_t=(1-t)\lambda_1+t\lambda_2$$ is a contact form for all $t\in[0,1]$ if $\lambda_0,\lambda_1$ ...
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A symplectic ball embedding into $\mathbb{C}P^2$

I want to show that the following map is a symplectic ball-embedding inside $\mathbb{C}P^2$. The mapping is defined for every $s<1$, $i:B^4(s)\longrightarrow \mathbb{C}P^2$ as follows, $$i(z_1,z_2):...
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Proposition 22.3 Brian Hall - Quantum Theory for Mathematicians

I've been reading Brian Hall Quantum Theory for Mathematicians and I'm struggling with the proof of proposition 22.3. Specifically, how equation (22.5) comes about: i.e. want to prove $[\nabla_{X},\...
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The Lie Bracket of a Poisson Algebroid

Given a Poisson manifold $(M,\pi)$, define a bracket on $\Omega^1(M)$ by $$ [\alpha,\beta]=\mathcal{L}_{\pi^\sharp(\beta)}\alpha-\mathcal{L}_{\pi^\sharp(\alpha)}\beta-d\pi(\alpha,\beta) $$ Anti-...
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What is the definition of the pullback of this map

Let $M$ be a smooth manifold on which acts a compact Lie group $G$. Let's suppose we have a diffeomorphism on $M$ $$f : M \rightarrow M $$ If $\Phi : M \rightarrow \mathfrak{g}^* $ is a map between $M$...
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Cotangent bundle $T^*S^1$ is an exact symplectic manifold.

We say that a symplectic manifold $(M,\omega)$ is exact if $\omega=d\lambda$ for some $1$-form $\lambda$. Consider a smooth manifold $S^1$. By standard construction, we know that a cotangent bundle $T^...
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Conditions on space of smooth almost complex structures so that it's a banach manifold

In the following paper by A.Abbondandolo and M. Schwarz https://arxiv.org/pdf/math/0408280.pdf in section $1.6$ we consider the set of smooth almost complex structures $\mathcal{J}$ on $T^*M$ ...
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What does it mean by "A continuous family of symplectic forms"

I have difficulty in understanding the meaning of "A continuous family of symplectic forms". I have seen this in many papers on symplectic geometry. Does it mean, we have a one parameter ...
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A variety is the moduli space of structure sheaves of points

In the last paragraph of the first page of this paper, it is mentioned that an $n$-dimensional Calabi-Yau manifold $X$ is the moduli space of structure sheaves of its points and I am not really sure ...
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$\omega$ is a symplectic form then $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$

Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$. Let $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ be the map given by $L^k_{\omega}(\alpha)=\alpha\wedge\omega^k$. Then is it true ...
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Explicit evaluation of symplectic form on cotangent bundle

Let $M$ be a manifold and denote by $T^*M$ its cotangent bundle. Let $(x,U)$ be a coordinate chart so that $x: U\to \mathbb{R}^{n}$. Let $p\in U$ and $v\in T^*_pM$, then we can write $v = v_i dx^i$ ...
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Condition for a Lagrangian submanifold to be totally geodesic

Consider $(M,\omega)$ a symplectic manifold and $L\subset M$ a Lagrangian submanifold. Let $J$ be an almost complex structure and $g:=\omega(.,J.)$ a compatible riemannian metric . Are there ...
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How to learn math after your PhD is finished [closed]

Question: How does someone go about learning advanced topics in Math after they're done with their PhD? Specific example: You've done your undergrad and masters degrees in math and learned from ...
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Symplectic trivialization along path

Let $(M,\omega)$ be a (symplectic) manifold. I want to compute the Maslov index of a loop $\gamma:\mathbb{R}\to M$ directly. In order to do that I have to find a (symplectic) trivialization of $\gamma^...
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volume preserving version of Moser's theorem

There exists an well-known theorem of Moser : Thereom(Moser) Let $M$ be a compact oriented smooth manifold and $\alpha,\beta$ be volume forms whose total volumes are the same. Then, there exists a ...
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"Appropriate" Hamiltonian function of simple pendulum

Consider a simple pendulum of length $\ell$ and mass $m$, where the only force is gravity. If $\theta$ is the angle between the rod and the vertical direction, and $\xi$ is the coordinate along fibers ...
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Bundle isomorphisms for $J$-holomorphic tangent bundle

In Chapter 14 of Lectures on Symplectic Geometry by da Silva, she claims that, if $(M,J)$ an almost complex manifold, then there are real bundle isomorphisms \begin{align*}\pi_{1,0}:TM\otimes\mathbb C&...
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Proving that implicit midpoint method for Hamiltonian systems is symplectic using a criterion

The implicit midpoint rule is defined as $$y_{n+1}=y_n+hJ^{-1}\nabla H\left(\frac{y_{n+1}+y_n}{2}\right).$$ where $y=(p,q)$. I know how to prove that this method is symplectic by hand, using the ...
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Clarification on the definition of a symplectic integrator

According to the notes that I am reading, a numerical one-stop method $y_{n+1}=\Phi_h(y_n)$ is said to be symplectic if, when applied to a Hamiltonian system, the discrete flow $y\mapsto \Phi_h(y)$ is ...
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Understanding a condition for a transformation to be symplectic

I am reading these notes to learn about symplectic mapping, and there is something that I don't understand. Theorem 5 on page 11 says: Let $(p,q)\rightarrow (P,Q)$ be a smooth mapping, close to the ...
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The Fubini-Study form on the complex projective space can be obtained from the Symplectic reduction

For a complex complex plane $\mathbb{CP}^{n}$, there is a natural Kähler form $$\omega=\partial\overline{\partial}\log\sum_{i=0}^{n}z_i\overline{z_i}$$ Which is so called the Fubini-Study form, where $...
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7 votes
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modified Hamiltonians for symplectic methods

I'm interested in methods for numerically integrating Hamiltonian systems $$\begin{align} \dot q & = +\frac{\partial H}{\partial p} \\ \dot p & = -\frac{\partial H}{\partial q} \end{align}$$ ...
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what is the conjugate momentum for this variable?

I am trying to find a canonical transformation for this Hamiltonian: \begin{equation} H(q,p) = p + e^p\cos(q) \end{equation} and I'm trying to find a conjugate momentum $P$ to the variable $Q$ defined ...
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