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.

learn more… | top users | synonyms (1)

0
votes
0answers
76 views

a question about pre-symplectic manifold

Let $(M,\omega)$, is pre-symplectic. Then can we say, ker$ \omega$ is subbundle of tangent bundle $TM$?
2
votes
1answer
69 views

Hamiltonian reduction in symplectic geometry

If $V$ is symplectic and $W^\perp \subseteq W\subseteq V$, then why is $W$ a pre-symplectic vector space? Why is $W/W^\perp$ symplectic?
2
votes
2answers
276 views

why symplectic form should be closed when we work on a manifold

For defining the symplectic space $(V, \omega)$ where $V$ is a vector space, it doesn't necessary to add the condition $d\omega=0$. But, when we work on a manifold instead of vector space, then we ...
1
vote
2answers
245 views

Application of the implicit function theorem

Assume that the equation $F(x,y,p)=0$ defines a regular submanifold $M$ of $R^3$. Consider the projection $\pi :M \rightarrow R^2$, given by $\pi (x,y,p)=(x,y)$. By the implicit function theorem, in ...
1
vote
1answer
145 views

canonical one-form

The canonical one-form is defined here: http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&pg=PA128#v=onepage&q&f=false My problem is ...
5
votes
0answers
97 views

Scattering of waves by an obstacle

I wish to study the paper by Melroe and Taylor: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv in Math, Vol 55 (3), 1985, 242–315. ...
4
votes
2answers
173 views

Self-intersection number of a complex curve in complex projective space

I'm currently trying to get a grip on actually calculating some differential-geometric definitions. I'm looking at the following map from $\mathbb{CP}^{1}$ to $\mathbb{CP}^2$ : ...
3
votes
2answers
130 views

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 ...
2
votes
0answers
46 views

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 ...
0
votes
1answer
46 views

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 ...
2
votes
1answer
56 views

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 ...
3
votes
1answer
89 views

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 ...
5
votes
1answer
142 views

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 ...
1
vote
0answers
78 views

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 ...
2
votes
2answers
133 views

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 ...
2
votes
0answers
72 views

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: ...
0
votes
2answers
48 views

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 ...
1
vote
1answer
501 views

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". ...
1
vote
0answers
69 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
2
votes
0answers
126 views

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 ...
10
votes
1answer
322 views

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. ...
2
votes
1answer
42 views

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 $S^3$?
1
vote
1answer
98 views

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 ...
3
votes
1answer
84 views

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 ...
5
votes
2answers
343 views

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 ...
3
votes
2answers
228 views

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 $\iota_X\omega$ ...
3
votes
1answer
208 views

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 ...
5
votes
1answer
145 views

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 ...
0
votes
1answer
84 views

$Sp(V)$ 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 $V$)
1
vote
0answers
75 views

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 ...
2
votes
2answers
194 views

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 $\mathfrak g \times M \to M$ ...
13
votes
1answer
579 views

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 ...
2
votes
0answers
218 views

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 ...
1
vote
1answer
82 views

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 ...
2
votes
1answer
111 views

$1$-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 ...
4
votes
1answer
114 views

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. ...
2
votes
1answer
162 views

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 ...
2
votes
1answer
263 views

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 $C_1,C_2$. ...
7
votes
1answer
203 views

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 ...
1
vote
1answer
75 views

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 $C=0$ ...
1
vote
2answers
373 views

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 ...
1
vote
0answers
96 views

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 ...
2
votes
1answer
62 views

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 ...
2
votes
1answer
262 views

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 ...
3
votes
2answers
107 views

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 ...
1
vote
1answer
53 views

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 ...
1
vote
0answers
130 views

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 $r$) ...
5
votes
1answer
281 views

$4$-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 ...
2
votes
1answer
160 views

Symplectic submanifolds and first integrals

I was working with symplectic submanifolds when I posed the following question: Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard ...
10
votes
2answers
346 views

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 ...