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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Regarding embeddings and homological/cohomological injectivity

Possibly low-level question here: if one has an embedding $M\hookrightarrow X$, call it $\iota$, then this is of course an immersion, so the differential maps $\iota_{\ast}:T_{p}M\rightarrow ...
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Is this computation of the Christoffel coefficients on a Kähler manifold correct?

Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of ...
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Darboux coordinates on projective spaces

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of ...
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Are the symplectic leaves of a Poisson manifold submanifolds?

In "Introduction to Mechanics and Symmetry" by Marsden and Raţiu it is written, in chapter 10, page 347, example b, that "[s]ymplectic leaves need not be submanifolds". In "Lectures on Poisson ...
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Level sets and lagrangian submanifolds

I would like to find all regular values for a map on a symplectic manifold such that the level sets are lagarangians. Precisely, the following example : The $4$-dimensional symplectic manifold ...
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Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,\omega)$ along a compact symplectically embedded submanifold $(M,\sigma)$ results in another compact manifold $(\tilde{X},\tilde{\omega})$ ...
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Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, ...
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What is a sympelctic bundle

What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that. PS: This symplectic ...
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107 views

Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory. So ...
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40 views

What is a pseudo-Kähler manifold?

I am reading a text which says that if a symplectic manifold is pseudo-Kähler, then there exists a unique symplectic connection on it. Since this a side remark without significance to the core of that ...
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Topological intuition about a hamiltonian vector field

Could I ask a conceptual question? If you have a symplectic manifold ($M$, $\omega$) and a real valued function $f : M \to \mathbb{R}$, you can define a hamiltonian vector field $X$ corresponding to ...
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Liouville's theorem and the Wronskian

Liouville's theorem states that under the action of the equations of motion, the phase volume is conserved. The equations of motion are the flow ODE's generated by a Hamiltonian field $X$ and the ...
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What is a Hamiltonian in a Poisson algebra?

Classical physics on the phase space $T^* M$ (with $M$ a smooth manifold) is done mostly in the following way: one endows $T^*M$ with a Riemannian structure $g^*$ (that will give the kinetic term) and ...
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Is there a more formal way to prove Liouville's theorem on the conservation of symplectic volume?

By Liouville's theorem the symplectic volume of a symplectic manifold $(M,\omega)$ is preserved under symplectomorphisms. One usually uses the language of classical mechanics to show that and performs ...
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Is this the pullback of a Hamiltonian flow?

In this reference just in the beginning the author gives the theorem (Theorem 1) of the conservation of a Hamiltonian flow $\phi_t$. According to it this means that $$ \frac{d}{dt}\phi_t^* H = 0$$ I ...
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Kähler metrics on the coadjoint orbits of a compact Lie group

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. It is well-known that each orbit for the coadjoint representation of $G$ on $\mathfrak{g}^*$ carries a canonical symplectic structure, ...
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Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type \begin{equation}(1)\quad \ddot x + f(x)=0 \end{equation} ...
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Pullback of a Hamiltonian

I understand that a Hamiltonian vector field $H$ creates a Hamiltonian flow $\phi_t$. Now, in order to prove that the Hamiltonian is conserved one uses the following \begin{eqnarray*} ...
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55 views

Is the group of symplectomorphisms of a symplectic manifold the symplectic group?

The group of symplectomorphisms of a symplectic manifold $M$ is a subgroup of the group of diffeomorphisms $GL(n)$, actually it is a subgroup of $SL(n)$. My question is whether this group of ...
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Help with the definition of a bilinear form $\omega$

According to this for $V$ a $2n$ (real) dimensional space any bilinear form $\omega: V \times V \to \mathbb{R}$ induces a linear map $\tilde{\omega}: V \to V^*$ via $$ \tilde{\omega}(v) := \omega(v, ...
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Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
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$G$-invariant functions on Manifolds

In a paper I saw the following statement: Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
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Polynomial functions on a smooth manifold

If one views $\Bbb R ^{2n}$ as the cotangent bundle of $\Bbb R ^n$, with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$, then in order to do classical Hamiltonian mechanics on it one considers ...
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Is time-1 map of a Hamiltonian vector field defined on a cylinder always twist?

Suppose I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are ...
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first Chern class of pair (X,D)

Let $(X,D)$ be a pair of projective variety $X$ and $D$ is a simple normal crossing divisor on $X$ then is it correct that $$c_1(X,D)=c_1(X)+[D]$$ where $[D]$ is the current of integration
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Alternative to Arnold's mathematical methods

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics. Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to ...
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Is there a way to compute the Poincaré dual of the following type of degree $(2n-2)$ de Rham class?

Given a closed, connected, symplectic manifold $(X^{2n},\omega)$, is there a systematic method to computing the Poincaré dual surface to degree $(2n-2)$ classes of the form $$[\omega]^{n-2}\cup B + ...
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Root spaces for symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$

Consider the symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$ over a field $K$. I know that the root system is given by $C_n=\{\pm 2e_j, \pm e_j \pm e_k:j,k=1 \cdots n, j \neq k\} $ where ...
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sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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A very general question about blow-up for experienced symplectic topologists and algebraic geometers

I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant ...
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Exact sequences of sympletic vector spaces

Is it true that if one has an exact sequence of vector spaces: $0 \to V \to W \to S \to 0$ Such that W is a sympletic vector space of even complex dimension and S has even real dimension and V has ...
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Does the Poisson bivector give rise to an integrable distribution?

I am reading the book Lectures on the geometry of Poisson manifolds, by Izu Vaisman. To a Poisson structure $\{\cdot,\cdot\}$ on a manifold $M$ we associate the Poisson bivector field ...
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Hamiltonian vector field - confusion

I am trying to understand the definition of a Hamiltonian vector field. I take as my symplectic manifold the sphere $$p_1^2+p_2^2+p_3^2=1$$ with symplectic form $$\omega=p_1 dp_2 \wedge dp_3+ p_2 dp_3 ...
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What's the best place to learn Quantum Homology

I am a PhD student and I am trying to learn quantum homology. I already know some of the analysis, but I am struggling to really find a good readable reference which covers enough of the analysis to ...
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Suitable reference for learning symplectic geometry

I am interested in studying symplectic geometry by myself and I'm looking for a good text to use as a reference in the way. I am a bit lost because I've found a lot of notes and books on the subject ...
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KAM theorem for symplectic maps using Generalized Implicit Function Theorems

I've been studying KAM theory for a while and as many of you surely know, there exist many methods in proving "KAM theorems" for different settings. Most of the literature deal with the persistence of ...
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Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension.

See the answer here. Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension:$$ U(n) = O(2n, ...
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Delzant theorem for polyhedra

Delzant theorem says that there is a 1-1 correspondence between compact toric symplectic manifolds (modulo equivariant symplectomorphism) and the Delzant polytopes (modulo lattice isomorphism). The ...
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Computing real de Rham cohomology of Hironaka's 3-manifold example

I have read the construction of Hironaka's famous 3-manifold example: in short, it is a union of two smooth curves $C$ and $D$ in a smooth projective 3-manifold $P$ which intersect each other at two ...
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What does the notation $G\times_P\mathfrak{p}^\perp$ mean, for $P\subset G$ Lie groups?

Suppose $G$ is a Lie group, $P$ a Lie subgroup with $\mathfrak{p}$ the associated Lie algebra. What object is $G\times_P\mathfrak{p}^\perp$? I don't understand what the $\times_P$ means, ...
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Showing Hofer's metric is bi-invariant

Im trying to show that the Hofer metric on a symplectic manifold is bi-invariant but im struggling. Firstly, given the flow $\rho_t$ of a hamiltonian $H_t$, the Hofer metric is $$ ...
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Darboux's Theorem Alternate Proof

I've been given the task of proving Darboux's theorem through non-standard means. Definitions Let $(M,\phi)$ be a symplectic manifold. $\mathcal{F}_{\text{SP}(V)}(M)$ is the bundle of frames ...
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How the canonical symplectic form acts

I've read that the canonical symplectic form $\omega$ on $\mathbb R^{2n}$ is given by $$\omega=\sum_{i=1}^n dp_i\wedge dq_i,$$ where $(p_1,\dots,p_n,q_1,\dots,q_n)$ are the coordinates on $\mathbb ...
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Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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Is this statement on symplectic maps completely general or does it need some extra hypotheses (as the ones with which I proved it)?

A lemma from McDuff-Salamon says that $\psi:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ is symplectic iff $\{F,G\}\circ\psi=\{F\circ\psi,G\circ\psi\}$. I proved that. Then there is an exercise showing that ...
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Showing that the intersection of two particular vector spaces has codimension 1 in the smaller of the two spaces

Here is the setting: I have a compact symplectic manifold $(X^{2n},\omega)$ and a compact symplectically embedded submanifold $(M^{2d},\sigma)$; that is, $\iota^{\ast}\omega=\sigma$. The dimension 2d ...
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Corresponding toric variety for n-simplex

Let $P $ be a Delzant polytope and $X_P $ be a corresponding Toric variety. I want to see if $P=\sum $ be a n-simplex then $X_P=\mathbb P^n$
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Computing the signature of the intersection form on the middle cohomology of compact, symplectic, non-Kaehler manifolds…

For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index ...
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construct a connection such that a given tensor is parallel wrt it

Let $\omega$ be a symplectic form on a smooth manifold $M$. How does one construct a connection on $TM$ such that $\omega$ is parallel to it? It's easy to construct a connection on a dual bundle ...
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Does $\{f,g\}$ mean anything when neither $f,g$ are the hamiltonian of a system?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...