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Poisson manifold: Examples, Further topics & Formal definition

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.

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Poisson manifold topic overview

The analysis highlights Examples, Further topics and Formal definition as prominent areas in the source structure around Poisson manifold.

Related topics
127
Source areas
9
Connected nodes
136
Extracted relationships
170
Concept neighborhoods
58
Bridge connections
136

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Introduction · 28 topics
Examples · 21 topics
Further topics · 19 topics
Overview · 18 topics
Formal definition · 14 topics
Poisson cohomology · 14 topics
Integration of Poisson manifolds · 6 topics
Symplectic leaves · 4 topics
Poisson submanifolds · 3 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Introduction

Formal definition

Symplectic leaves

Examples

Poisson cohomology

Integration of Poisson manifolds

Poisson submanifolds

Further topics

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Poisson manifold connects Entity context

The extracted context around Poisson manifold shows recurring relationship patterns in the source. For example, Poisson manifold → Alan Weinstein, Although, Carl Gustav Jacob Jacobi, For, Hamiltonian, In, Indeed, Jacobi, Jacobi's, Lie, More, Moreover, Poisson, Poisson's, Poisson-commutes, Rightarrow, Siméon Denis Poisson, Sophus Lie, What Another extracted example is Poisson manifold → CC-BY-SA-3, Francesco Cattafi, ISSN, July, Lock-gray-alt-2, Lock-green, Lock-red-alt-2, PDF, Poisson, Science, The, This, Wikidata Q117054291, WikiJournal, Wikipedia, Wikisource-logo. Use these groups to spot repeated connection types before inspecting the individual relationships.

Poisson manifold

Top relations

related to history · 19
Poisson manifold → Alan Weinstein, Although, Carl Gustav Jacob Jacobi, For, Hamiltonian, In, Indeed, Jacobi, Jacobi's, Lie, More, Moreover, Poisson, Poisson's, Poisson-commutes, Rightarrow, Siméon Denis Poisson, Sophus Lie, What
related to References · 16
Poisson manifold → CC-BY-SA-3, Francesco Cattafi, ISSN, July, Lock-gray-alt-2, Lock-green, Lock-red-alt-2, PDF, Poisson, Science, The, This, Wikidata Q117054291, WikiJournal, Wikipedia, Wikisource-logo
related to Examples · 13
Poisson manifold → Casimir, Consider, For, Given, If, Killing, Lie, Lie-Poisson, More, One, Poisson, Then, This
related to Other types of submanifolds in Poisson geometry · 13
Poisson manifold → Another, For, However, In, It, Lie, Lie-Poisson, Phi, Poisson, Similarly, TC, The, Weinstein
related to Other examples and constructions · 11
Poisson manifold → Any, Given, If, Jacobiator, Let, Lie, M/G, Omega, Poisson, The Cartesian, This
related to Linearisation problem · 9
Poisson manifold → Cartan, Due, For, If, It, Lie, Other, Poisson, The
related to Poisson cohomology · 9
Poisson manifold → In, Notice, Omega, Poisson, Rham, Schouten-Nijenhuis, The Poisson, TM, Using
related to Symplectic groupoids · 9
Poisson manifold → Among, Conversely, Equivalently, Lagrangian, Lie, Moreover, Omega, Poisson, The
related to Integration of Poisson manifolds · 8
Poisson manifold → Any Poisson, Gamma, Lie, Omega, Poisson, Several, The, TM
related to Symplectic realisations · 8
Poisson manifold → Crainic-Fernandes, Hamiltonian, Indeed, Lie, Poisson, Roughly, This, While

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

displaystyle poisson pi symplectic manifold structure lie bracket mathfrak vector field bivector manifolds structures mathcal given form smooth algebra space

Poisson manifold relationships Subject–Predicate–Object triples

TTTA extracted 170 structured relationships around Poisson manifold. Examples in this analysis include Poisson manifold → is a → smooth manifold endowed with a Poisson structure and Poisson manifold → is a → complex manifold M. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Poisson manifoldis asmooth manifold endowed with a Poisson structure0.90text
Poisson manifoldis acomplex manifold M0.90text
Poisson manifoldis aclass in the first Poisson cohomology group0.90text
Poisson manifoldis ahighly non trivial problem0.90text
Poisson manifoldrelated to Deformation quantisationThe0.60section
Poisson manifoldrelated to Deformation quantisationPoisson0.60section
Poisson manifoldrelated to Deformation quantisationThis0.60section
Poisson manifoldrelated to ExamplesGiven0.60section
Poisson manifoldrelated to ExamplesPoisson0.60section
Poisson manifoldrelated to ExamplesLie0.60section
Poisson manifoldrelated to ExamplesOne0.60section
Poisson manifoldrelated to ExamplesFor0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Poisson manifold bring nearby vocabulary together. In this analysis, examples include Displaystyle, Poisson and Pi. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Poisson manifold
    • Displaystyle
    • Poisson
    • Pi
    • Symplectic
    • Structure
    • Lie
    • Bracket
    • Manifolds
    • Structures
    • Bivector
    • Map
    • Linear
  • poisson manifold
    • Displaystyle
    • Poisson
    • Pi
    • Symplectic
    • Structure
    • Lie
    • Bracket
    • Manifolds
    • Structures
    • Bivector
    • Given
    • Map
  • smooth manifold
    • Poisson
    • Functions
    • Pi
    • Displaystyle
    • Symplectic
    • Cdot
    • Partial
    • Bracket
    • Bivector
    • Space
    • Given
    • Frac
  • symplectic manifold
    • Poisson
    • Pi
    • Displaystyle
    • Symplectic
    • Given
    • Bracket
    • Lie
    • Mathcal
    • Called
    • Smooth
    • Infty
    • Structure
  • phase space
    • Vector
    • Functions
    • Bracket
    • Structure
    • Mathbb
    • Linear
    • Called
    • Symplectic
    • Displaystyle
    • Leaves
    • Algebra
    • Mathcal
  • hamiltonian mechanics
    • Vector
    • Fields
    • Every
    • One
    • Infty
    • Cdot
    • Form
    • Functions
    • Space
    • Displaystyle
    • Smooth
    • Map
  • poisson bracket
    • Displaystyle
    • Pi
    • Symplectic
    • Structure
    • Cdot
    • Lie
    • Bracket
    • Poisson
    • Space
    • Mathbb
    • Functions
    • Smooth
  • vector space
    • Hamiltonian
    • Field
    • Fields
    • Space
    • Vector
    • Functions
    • Bracket
    • Linear
    • Infty
    • Structures
    • Cdot
    • Structure

Connections between topic areas Semantic bridges

For Poisson manifold, one of the stronger structural bridges in this analysis connects Poisson manifold with Introduction. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Poisson manifold — Introduction · splits 108 ⟂ 29
Poisson manifold — Examples · splits 115 ⟂ 22
Poisson manifold — Further topics · splits 117 ⟂ 20
Poisson manifold — Overview · splits 118 ⟂ 19
Poisson manifold — Formal definition · splits 122 ⟂ 15
Poisson manifold — Poisson cohomology · splits 122 ⟂ 15
Poisson manifold — Integration of Poisson manifolds · splits 130 ⟂ 7
Poisson manifold — Symplectic leaves · splits 132 ⟂ 5
Poisson manifold — Poisson submanifolds · splits 133 ⟂ 4

Map overview Semantic statistics

Poisson manifold

Nodes137
Edges136
Triples170
Avg. degree1.99
Density0.014599
Components1

Source & methodology

TTTA analyzes the structure around Poisson manifold to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Examples, Further topics & Formal definition, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Poisson manifold · EN edition · Analysis: TopicsToTalkAbout

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