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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Symplectic Form Preserved by Orthogonal Transformation
I'm trying to prove that the symplectic form
$$\omega = d(\cos\theta) \wedge d\phi$$
is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
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Trivialization of a path of tamed almost complex structures
I am wondering if the following result is true:
Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
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Prove that the $2$ form defines a symplectic structure
Prove that the $2$ form
$$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$
defines a symplectic structure on $\mathbb{R}_x^4$.
My definition of as ...
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Question about symplectic tranformations
Suppose I know that two vectors $\vec{a}$ and $\vec{b}$ are perpendicular in a given basis spanned by basis vectors $\vec{x}$. Now suppose I transform to another basis $\vec{x'}$ using a symplectic ...
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Non-degenerate solutions to constant Hamiltonian flow
As I'm trying to work my way through Dietmar Salamon's "Notes on Floer Homology", I'm having trouble with the very first exercise.
Let $(M, \omega)$ be a compact symplectic manifold. Let $H$ be a ...
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Lagrangian subspaces
Let $\Lambda_{n}$ be the set of all Lagrangian subspaces of $C^{n}$, and $P\in \Lambda_{n}$. Put $U_{P} = \{Q\in \Lambda_{n} : Q\cap (iP)=0\}$. There is an assertion that the set $U_{P}$ is ...
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Computation of a pullback of a two form
If we have a Lagrangian immersion from $C^{2}$ to $C^{4}$ defined like this
\begin{align}
\notag
\phi : (x,y,u,v) \to (x, y, u, v, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, ...
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Hamiltonian Isotopy in Symplectic geometry
In any standard symplectic geometry/topology textbook, the concept of Hamiltonian isotopy was introduced:
$(M, \omega)$ is a sympplectic manifold. Given a symplectic isotopy
$\phi_t : M \rightarrow ...
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Symplectic Forms
Let $(M, \omega)$ be a symplectic manifold, so that $\omega$ is a non-degenerate 2-form. If $\dim M = 2n$ why does $\omega$ being non-degenerate imply that $\underbrace{\omega \wedge \ldots \wedge ...
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Symplectic geometry as a prequisite for Heegaard Floer homology
I would like to study Heegaard Floer homology in the future in the connection to knot theory.
I read a wikipedia article and it seems that I need to first learn a symplectic geometry (topology?). I ...
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Torsion-free $G$-Structures
I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of ...
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Area of flux homomorphism in symplectic topology
Let $(M,\omega)$ be a symplectic manifold. Let $f:M \to \mathbb R$ be a smooth function. We have vector fields $X_f$ defined by $\omega(X_f,)=df$. Let $\phi_t$ be the flow of $X_f$ and let $\gamma: ...
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symplectic lie algebra is simple
The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\
-I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...
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what are the holomorphic curves in $T^{*}S^3$ with boundary on the zero section?
How would I characterize such things? Is the minimal spanning (real) surface of a (real) curve in $S^3$ contained entirely in that $S^3$?
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Why does symplectic geometry have many applications in mathematics
It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you!
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Darboux's theorem in the symplectic geometry
From the Darboux's theorem in the symplectic geometry, we know that symplectic manifolds with the same dimension is locally "equivalence". I have a little puzzle with the meaning of "equivalence".
...
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Normal Bundle of Twistor lines
I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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Index of a Symmetric Matrix
In Hansjorg Geiges' introductory textbook on Symplectic Geometry, is defined a projective conic given by $q^tAq=0$ where $A$ is a symmetric matrix of rank 3 and index 2. What does "index 2" mean? I ...
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Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.
I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles.
...
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2answers
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Symplectic 2-Torus
Consider the 2-torus $T=S^1\times S^1$ with symplectic form $\omega=d\theta\wedge d\varphi$ and the vector field $X=\partial_\theta$. I wonder if $X$ is hamiltonian. In other words, is $\iota_X\omega$ ...
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Visualizing diffeomorphisms
This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow ...
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Special Kaehler manifolds
If we have complex vector space $V=T^{*}C^{m}$ with standard complex symplectic form $\Omega =\sum_{i=1}^{m}dz^{i}\wedge dw^{i}$, and if $\tau : V\to V$ is standard real structure of $V$ with set of ...
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Why is the moduli space of flat connections a symplectic orbifold?
In her Lectures on Symplectic Geometry on page 159, Ana Cannas da Silva writes "It turns out that $\mathcal{M}$ is a finite-dimensional symplectic orbifold."
Can somebody give me a reference for ...
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Symplectic 2-Sphere
Consider the sphere $S^2\subset\mathbb{R}^3$ in cylindrical coordinates $(\theta, z)$ (away from poles $z=\pm 1$) with symplectic structure $\omega=d\theta\wedge dz$.
I want to show that the vector ...
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Why should a symplectic form be closed?
Thanks for reading my question. I'm wonder why a symplectic form should be closed. I found many different answers in the internet, but it sounds like a technical requirement (if we omit this ...
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$Sp(V)$ acts transitively on $V^*-\{0\}$ where $\Omega$ here is symplectic 2 form
Let $\dim(V)=6$. Show that $Sp(V,\Omega)$ acts transitively on $V^*-\{0\}$, where $\Omega$ here is a symplectic 2 form on $V$. ($V^*$ here is algebraic dual of $V$)
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transformation of symplectic structure by a matrix
Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have
$$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$
Let ...
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When does the $\mathfrak g$-invariance of the symplectic form imply $G$-invariance?
Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let $M$ be a smooth manifold. Suppose $G$ acts on $M$, $G \to \text{Diff}(M)$. This naturally induces an action $\mathfrak g \times M \to M$ ...
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A proof of simply connectedness of a symplectic quotient
Let $\rho$ be a unitary representation of a torus $G$ on $\mathbb{C}^n$. The action of $\rho$ is Hamiltonian with a moment map $\mu:\mathbb{C}^n \to \mathfrak{g}^*$. Here $\mathfrak{g}^*$ is the dual ...
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Tautological 1-form on the cotangent bundle
I'm trying to understand a little bit about symplectic geometry, in particular the tautological 1-form on the cotangent bundle. I'm following Ana Canas Da Silva's notes.
On page 10 she describes the ...
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basis free volume form for a symplectic vector space
It's easy to show, using a symplectic basis, that if $\omega$ is a symplectic form on a $2n$-dimensional vector space $V$, then $\omega^n \neq 0$. I'd like to be able to prove it without choosing a ...
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Geodesic Flow is the flow of the Hamiltonian Vector field of $|\xi|$
Let $M$ be a complete Riemannian manifold with metric $g$. The geodesic flow is the one-parameter family of diffeomorphisms $\phi_t$ on $T^*M$ defined as follows. If $\xi \in T^* M$ is a vector based ...
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Will this set of functions form coordinates on a symplectic manifold?
Consider a symplectic manifold $(M, \omega)$. Let us define a concept of a complete set of observables:
A set of functions $f_i : M \to \mathbb R$ form a complete set of observables if any ...
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$1$-form on a symplectic manifold.
If $\omega$ is a $1$-form on a symplectic manifold, will it be closed? It seems to be trivial that if $\sigma$ is symplectic structure on a manifold $M$, then the induced map
$$\sigma^\vee: TM\to ...
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Symplectic Chart
I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following:
If $B$ is a zero neighborhood in Banach space. ...
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1answer
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What is an “invariant form” of a group?
I have often seen this phrase used in at least two frequent contexts,
One uses the notation of $\omega_{AB}$ (the matrix $\{ [0 , I],[-I,0]\}$) to denote the symplectic form for $USp$ group.
One ...
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1answer
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Symplectic reduction: involutive and non-involutive first integrals
Suppose I have a Hamiltonian $H$ with the phase space $\mathcal{M}$, a symplectic manifold with a symplectic 2-form $\omega.$ Now assume that the Hamiltonian system has two first integrals $C_1,C_2$. ...
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Moment map of the action of $\operatorname{SO}(3)$ on the sphere
The moment map of the action of $\operatorname{SO}(3)$ on the sphere can be thought of as inclusion from $S^2$ into $\mathbb R^3$ by identifying $\mathfrak{so}(3)$ (the Lie algebra of ...
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Symplectic submanifolds
Suppose I have the symplectic manifold $(M, \omega)$. Now consider a function $C: M \rightarrow \mathbb{R}$ whose differential is non-zero. Then restricting to the submanifold of $M$ given by $C=0$ ...
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Regarding Legendre transform from tangent bundle to cotangent bundle
(I'm a complete beginner at differential geometry)
I'm studying about constrained systems, in which we "map a Lagrangian system from a tangent to a cotangent bundle. Hamiltonian dynamics then appears ...
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A vector field on a symplectic submanifold intersecting the symplectic complement
Consider a 4-dim symplectic vector field $X$ on the symplectic manifold $(M, \omega)$ in $\mathbb{R}^4$ with $\omega= \sum_{i=1}^2 dy_i \wedge dx_i$. Moreover, the linear terms of $X$ are given by ...
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1answer
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Open sets of symplectic manifolds
Suppose I have a symplectic manifold $(\mathcal{M}, \omega)$. Does it hold that any open subset of $(\mathcal{M}, \omega)$ is a symplectic submanifold?
The statement trivially holds for smooth ...
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is the geodesic flow on Hyperbolic Plane completely integrable?
I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...
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Any intuitive examples of symplectic vector space?
Recently I come into symplectic vector space and its properties in my linear algebra class.
However, this interesting thing is so different from the usual inner-product spaces I've met before, and I ...
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Energy displacement of a cylinder is at most $\pi r^2$.
I want to show that the energy displacement of $Z^{2n}(r)$ is at most $\pi r^2$.
In the textbook of Mcduff and Salamon they write that I should identify the two dimensional ball (with radius $r$) ...
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Proving that a particular submanifold of the cotangent space is Lagrangian
I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
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Given an integral symplectic matrix and a primitive vector, is their product also primitive?
Given a matrix $A \in Sp(k,\mathbb{Z})$, and a column k-vector $g$ that is primitive ( $g \neq kr$ for any integer k and any column k-vector $r$), why does it follow that $Ag$ is also primitive? Can ...
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Symplectic positive definite matrix.
I want to prove that any symmetric positive definite symplectic matrix, $A$, and any real number $\alpha >0$, also $A^{\alpha} \in \operatorname{Sp}(2n)$.
I was given a hint to decompose ...
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Showing that some symplectomorphism isn't Hamiltonian
I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it.
I know ...
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$4$-form $ \omega \wedge \omega$ vanishes on $S^4$
If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
