# 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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### 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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### 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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### 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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### 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 ...
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 ...