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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When is a symplectic connection a symplectomorphism?

Let $(M, \omega)$ be a symplectic manifold and $\nabla$ a symplectic connection, i.e. an element of $\Omega^1 (M, \operatorname{End} (TM)).$ Denote by $\operatorname{End}_{\omega} (TM)$ a bundle of ...
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Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
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local to global definition of symplectic form on cotangent bundle

Show that the form $\omega$ defined locally as $$\omega = \sum dx_i \wedge d\xi_i$$ is globally well-defined on $T^*M$ and restricted to the zero section of $T^*M$ vanishes. Here we consider $M$ to be ...
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Tangent space to inverse of a regular value in Hamiltonian system

I don't quite understand what is meant by restriction of vector field $X_H$ to a fiber of $J$, where $(M,ω,H)$ is Hamiltonian system and $ω$ is a symplectic form and $J$ is the moment map.
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Proof that Morse complex is a complex using coherent orientation

I'm reading the book Morse Homology by M. Schwarz, which aims to develop Morse homology in strict analogy with Floer homology. For orientation matters, the book follows the paper A. Floer and H. ...
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A function constant on the orbits of $G$?

Let $G$ be an $n$-dimensional compact Lie group. Let $G \times M \rightarrow M$ be a group action on a smooth manifold $M$, let $f \in C^{\infty} (M; \mathbb{R})$. Construct a function $f_{avg}: M \...
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Proving Hamiltonian symplectomorphism commutes with given symplectomorphism

I'm trying to prove this identity which is mentioned at the very beginning of this paper by Dostoglou and Salamon "Self-Dual instants and holomorphic curves". Let $(M,\omega)$ be a closed symplectic ...
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Inverse images of coadjoint orbits are coisotropic submanifolds?

Problem Let $(M, \omega)$ be a symplectic manifold. Let $G$ be a connected, compact Lie group acting on $M$. Let $J: M \rightarrow \mathfrak{g}^{*}$ be the moment map. Let $\eta$ be a regular value of ...
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Elements in the coadjoint orbit are regular values?

Let $(M, \omega)$ be a symplectic manifold and let $G$ be a compact, connected Lie group acting on it. Let $J: M \rightarrow \mathfrak{g}^{*}$ be the moment map. Assume that $\eta$ is a regular value ...
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1-Form and 2-Form in cotangent bundle with time dependecy

I have these question but first of all a bit of context. I have a Lagrangian $L(q,\dot q,t)\in C^\infty(\textbf{T}M \times \mathbb{R})$. We know that the taulogical 1-form in $\textbf{T}M$ is $\theta=...
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Relation between Hochschild homology and cohomology

Let $A$ be an associative algebra, then we have the Hochschild chain complex, namely: $.. \to A^{\otimes 3} \xrightarrow{d_2} A^{\otimes 2} \xrightarrow{d_1} A$, where, for example, $d_1 (a \otimes b)...
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Understanding implications of the equivariant Darboux-Weinstein theorem

I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here: The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example ...
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Topology on space of contact structures

hope everyone is staying safe during this period of the year. In contact geometry, one endows a manifold $M$ with a contact form, which is a one-form $\alpha$ such that $(d\alpha)^n\rvert_{\ker \...
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Confusion in calculating Lie derivative of lifted symplectical vector fields

I am a bit confused about a calculation I did and I do not see my mistake. First of I took any vector field $X \in \mathfrak{X}(M)$ and lifted it as some $\hat{X}\in \mathfrak{X}(T^*M)$ (s.t. $T\pi\...
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Why is it that a bilinear skew-symmetric form is symplectic iff its' flat correspondent is an isomorphism?

I have come across the following claim: Assume a skew-symmetric bilinear form $\omega: V \times V \rightarrow \mathbb{R}$ is symplectic if, by definition, $\forall v \in V\setminus \left\lbrace 0_V \...
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Motivation for Kahler Geometry

I have been studying Symplectic Geometry. Previously I studied Riemannian Geometry. In Symplectic Geometry I learned the existence of an almost complex structure and how some special almost complex ...
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Example of a Hamiltonian Lie group action

I was wondering why the following Lie group action is Hamiltonian. Equip $\mathbb{C}^{k\times n}\cong\mathbb{R}^{2kn}$ with the canonical symplectic form $\omega_0$ on $\mathbb{R}^{2kn}$. We have an ...
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What is symplectic geometry? [closed]

EDIT: Much thanks for answers. As was pointed out, the question as it stands is a little too broad. Nevertheless, I don't want to delete it, because I think that such introduction-style questions can ...
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The Arnol'd conjecture for symplectomorphisms sufficiently close to the identity

In studying symplectic geometry, a relatively easy corollary of Weinstein's Lagrangian Neighbourhood theorem is the following. On a closed symplectic manifold $(M,\omega)$ with $H^1_{\text{dR}}(M)=...
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37 views

Complex vector space with a complex structure is a real vector space

Suppose $V$ is a finite-dimensional vector space with a (linear) complex structure $J$. Further assume that the vectors $u_1,...,u_n$ form a basis of $V$ over $\mathbb{C}$. Then the vectors $u_1,Ju_1,....
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When is it possible to add different powers of a dimensioned quantity? [duplicate]

When is it possible to add different powers of a dimensioned (i.e. non-dimensionless) quantity, e.g. consider power series for dimensioned quantities. For example, when can you exponentiate a ...
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Showing there exists a unique $1$-form $\alpha$ with these properties

Problem: Let $M$ be a smooth manifold. Let $\omega$ be the canonical symplectic form on $T^{*} M$. Prove that there exists a unique $\alpha \in \Omega^{1} (T^{*} M)$ with the following properties: (i)...
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Constant vector field on the torus $\mathbb{T}^{2n}$ is symplectic

Let $\mathbb{T}^{2n}=\mathbb{R}^{2n}/\mathbb{Z^{2n}}$ be the $2n$-torus, which we equip with the unique symplectic form $\omega$ that pulls back to the standard symplectic form on $\mathbb{R}^{2n}$ ...
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To show that the space of all transversal lagrangian subspaces is contractible

Let V be a symplectic vector space. Let $L_0$ be a Lagrangian subspace. Show that the space of all Lagrangian subspaces of V with transverse intersection with $L_0$ is contractible. This is ...
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The submanifold $S^2$ in the symplectic product $S^2\times S^2$

Let $S^2$ come equipped with the usual symplectic form and $S^2\times S^2$ come equipped with the product symplectic form and coordinates $(x,y)$ with $x\in S^2$. Consider the "diagonal sphere" $(x,x)$...
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Introduction to holomorphic symplectic manifolds

What is a good resource to learn basics about holomorphic symplectic manifolds? All references in the Wikipedia article are concerned with real symplectic manifolds, and I'm not sure which basic ...
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Example of a symplectic but non-Hamiltonian vector field on $\mathbb{T}^{2n}$

I want to show that there exists a symplectic vector field on the $2n$ torus $\mathbb{T}^{2n}$, endowed with the unique symplectic form $\omega$ that pullsback to the canonical symplectic form $\...
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Construction of a non-autonomous Hamiltonian diffeomorphism

Let $(M,\omega)$ be a symplectic manifold. I have read that the autonomous Hamiltonian diffeomorphisms (i.e. a Hamiltonian diffeomorphism generated by a time-independant Hamiltonian) form a proper ...
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Associating the Hamiltonian vector field $X_f$ to a function $f$ is a Lie algebra homomorphism

Let $(M, \omega)$ be a complex analytic symplectic variety, i.e. $\omega \in \Gamma(\Lambda^2 \Omega_M)$ is a closed holomorphic 2-form, which is everywhere non-degenerate. We can view $\omega$ as an ...
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Classification of symplectic surfaces and confusion about “symplectomorphism”

So I read that the unique invariant of symplectic surfaces is the total area, i.e. two surfaces are symplectomorphic iff their area is the same. Consider $S^2$ with polar coordinates $(h,\theta)$ ...
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Restriction on the number of component functions if Poisson brackets vanish?

Problem: Let $\omega$ be a symplectic form on a manifold $M$, where $\dim(M) = 2n$, and let $F: (f_1, \ldots, f_k) : M \rightarrow \mathbb{R}^k$ be a submersion. If $H$ is a function such that $\left\...
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symplectic form, tautological 1-form and DeRham cohomology

There's a question in Lee's Introduction to Smooth Manifold that asks to prove that the Grassmannian product of a symplectic form is not exact. However, isn't this incorrect if there exists a ...
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Showing there is a symplectomorphism of $(T^{*} M, \omega_{T^{*}M})$ satisfying this property

Problem: Let $M$ be a manifold, and let $\alpha$ be a closed $1$-form on $M$. Show that there exists a symplectomorphism of $(T^{*} M, \omega_{T^{*} M})$ that maps the zero section to the submanifold $...
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What is the connection map?

In my symplectic geometry homework I have this exercise: Let $Q$ be a manifold, $\pi:T^*Q\to Q$ be the projection map, and $\kappa:T(T^*Q)\to T^*Q$ be the connection map. Prove that the standard ...
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Are coadjoint orbits symplectic manifolds?

Let $G$ be a Lie group and $\mathfrak{g}$ be it's Lie algerba. Consider the coadjoint orbit for an element $x \in \mathfrak{g}: O_x$. It is well known that $$O_x \simeq G/G_x$$ where $G_x$ is the ...
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Lagrangian mechanics: geodesic paths are critical points of the action

Let $(Q,g)$ be a Riemannian manifold, let $\mathcal{L}(q,v)=\frac12 g_q(v,v)$ be a Lagrangian and consider the action $S=\int_a^b\mathcal{L}(q(t),\dot{q}(t))dt$. I want to show that any geodesic path ...
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Symplectomorphisms of $S^2$

It is well-known that $S^2$ is a Kähler manifold with complex structure $J$ (as the Riemann sphere), Kähler metric $g$ (induced from $S^2\subset\mathbb{R}^3$), and Kähler form $\omega$ (volume form of ...
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Does a Hamiltonian-preserving, symplectic vector field provide an integral of motion?

Consider the hamiltonian system $(M, \omega, H)$ with hamiltonian vector field $X$ defined by $$ \tag{1}\label{1} \iota_{X}\omega = -dH $$ It is a simple matter of applying definitions to show that a ...
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Curvature of canonical connection on 4-manifold with self-dual harmonic 2 form

Let $X$ be an compact oriented Riemannian 4-manifold with $b_2^+\geq1$. Let $\omega$ be a self-dual harmonic two form vanishing transversely. On $X\setminus \omega^{-1}(0)$, the spinor bundle $S_+$ ...
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Does the complex structure of a Kähler manifold preserves the Lie algebra of symplectic vector fields

Let $(M, \omega, g, J)$ be a Kähler manifold with symplectic form $\omega$, Riemannian metric $g$ and complex structure $J$. Question: If $X$ is a symplectic vector field, is $JX$ also symplectic?
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Expression for $\nabla_{JX} Y$ on a Kähler manifold

Let $(M, \omega, g, J)$ be a Kähler manifold with symplectic form $\omega$, Riemannian metric $g$ and complex structure $J$. I'm looking for a formula that gives an expression for $\nabla_{J X} Y$, ...
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Spheres have contact structure but no symplectic structure

So I just learnt that $S^{2n}$ for n>1 doesn't have a symplectic structure. I saw the proof uses that the 2nd deRahm cohomology group of higher even dim spheres is 0. But I have been asked to show ...
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Two non isotopic symplectic forms on the Torus

Part 1 - How can I find two non isotopic symplectic forms on $\mathbb T^2$ ? All I know is that the usual symplectic form on $\mathbb R^2$ can be "projected" onto $\mathbb R^2/\mathbb Z^2$ thereby ...
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On the proof of lemma from symplectic topology

Let $X$ be riemannian manifold, $\Phi:X\to \mathbb{R}$ smooth and $grad \Phi:X\to TX$ complete vectorfield, that is there is global flow $\phi_t:X\to X$. Following is from Introduction to symplectic ...
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Conventions on symplectic embeddings vs. symplectomorphisms

As I understand it, a symplectomorphism of vector spaces (resp. smooth manifolds) is required to be an isomorphism (resp. a diffeomorphism), while there is the broader category of symplectic ...
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Structure theorem of symplectic modules

My question comes from the content presented on slide 31 of the following presentation given by Jean-Pierre Tignol (unfortunately, I do not have access to the main reference on the subject, that is ...
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Eigenvectors, generalized eigenvectors and stability

I'm currently studying the stability conditions for symplectic matrices, but I've some doubts that probably stem directly from basic linear algebra. Let $M$ be a 4x4 real symplectic matrix. Let $V$ ...
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Given a contact manifold give a contact form on $M × \mathbb R^2$

$M$ is contact manifold with contact form $w$. I need to give a contact form on $M×\mathbb R^{2}$ The hints provided are: $M×\mathbb R$ has an exact symplectic form $d(e^{t}w) = dm$ say $M×\mathbb ...
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Let $W$ be the maximal symplectic subspace of a presymplectic vector space $(V,\omega)$. Then $W^\omega=\text{Rad}(\omega)$.

Let $(V,\omega)$ be a presymplectic vector space and let $$\text{Rad}(\omega)=\{v\in V\colon\omega(v,v')=0\,\,\forall v'\in V\}.$$ Let $(W,\omega|_W)$ be a maximal symplectic subspace, i.e. $W$ is ...
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Hypersurface of contact type

I have the following definitions for a contact structures and hypersurface of contact type in my lecture: 1)A contact structure on a manifold $W^{2n+1}$ is a hyperplane field $\xi \subset TW$ which ...

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